quantum computer essay

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National Academy of Engineering. Frontiers of Engineering: Reports on Leading-Edge Engineering from the 2018 Symposium. Washington (DC): National Academies Press (US); 2019 Jan 28.

Frontiers of Engineering: Reports on Leading-Edge Engineering from the 2018 Symposium.

• Hardcopy Version at National Academies Press

Quantum Computing: What It Is, Why We Want It, and How We're Trying to Get It

SARA GAMBLE .

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Quantum mechanics emerged as a branch of physics in the early 1900s to explain nature on the scale of atoms and led to advances such as transistors, lasers, and magnetic resonance imaging. The idea to merge quantum mechanics and information theory arose in the 1970s but garnered little attention until 1982, when physicist Richard Feynman gave a talk in which he reasoned that computing based on classical logic could not tractably process calculations describing quantum phenomena. Computing based on quantum phenomena configured to simulate other quantum phenomena, however, would not be subject to the same bottlenecks. Although this application eventually became the field of quantum simulation, it didn't spark much research activity at the time.

In 1994, however, interest in quantum computing rose dramatically when mathematician Peter Shor developed a quantum algorithm, which could find the prime factors of large numbers efficiently. Here, “efficiently” means in a time of practical relevance, which is beyond the capability of state-of-the-art classical algorithms. Although this may seem simply like an oddity, it is impossible to overstate the importance of Shor's insight. The security of nearly every online transaction today relies on an RSA cryptosystem that hinges on the intractability of the factoring problem to classical algorithms.

• WHAT IS QUANTUM COMPUTING?

Quantum and classical computers both try to solve problems, but the way they manipulate data to get answers is fundamentally different. This section provides an explanation of what makes quantum computers unique by introducing two principles of quantum mechanics crucial for their operation, superposition and entanglement.

Superposition is the counterintuitive ability of a quantum object, like an electron, to simultaneously exist in multiple “states.” With an electron, one of these states may be the lowest energy level in an atom while another may be the first excited level. If an electron is prepared in a superposition of these two states it has some probability of being in the lower state and some probability of being in the upper. A measurement will destroy this superposition, and only then can it be said that it is in the lower or upper state.

Understanding superposition makes it possible to understand the basic component of information in quantum computing, the qubit. In classical computing, bits are transistors that can be off or on, corresponding to the states 0 and 1. In qubits such as electrons, 0 and 1 simply correspond to states like the lower and upper energy levels discussed above. Qubits are distinguished from classical bits, which must always be in the 0 or 1 state, by their ability to be in superpositions with varying probabilities that can be manipulated by quantum operations during computations.

Entanglement is a phenomenon in which quantum entities are created and/or manipulated such that none of them can be described without referencing the others. Individual identities are lost. This concept is exceedingly difficult to conceptualize when one considers how entanglement can persist over long distances. A measurement on one member of an entangled pair will immediately determine measurements on its partner, making it appear as if information can travel faster than the speed of light. This apparent action at a distance was so disturbing that even Einstein dubbed it “spooky” ( Born 1971 , p. 158).

The popular press often writes that quantum computers obtain their speedup by trying every possible answer to a problem in parallel. In reality a quantum computer leverages entanglement between qubits and the probabilities associated with superpositions to carry out a series of operations (a quantum algorithm) such that certain probabilities are enhanced (i.e., those of the right answers) and others depressed, even to zero (i.e., those of the wrong answers). When a measurement is made at the end of a computation, the probability of measuring the correct answer should be maximized. The way quantum computers leverage probabilities and entanglement is what makes them so different from classical computers.

• WHY DO WE WANT IT?

The promise of developing a quantum computer sophisticated enough to execute Shor's algorithm for large numbers has been a primary motivator for advancing the field of quantum computation. To develop a broader view of quantum computers, however, it is important to understand that they will likely deliver tremendous speed-ups for only specific types of problems. Researchers are working to both understand which problems are suited for quantum speed-ups and develop algorithms to demonstrate them. In general, it is believed that quantum computers will help immensely with problems related to optimization, which play key roles in everything from defense to financial trading.

Multiple additional applications for qubit systems that are not related to computing or simulation also exist and are active areas of research, but they are beyond the scope of this overview. Two of the most prominent areas are (1) quantum sensing and metrology, which leverage the extreme sensitivity of qubits to the environment to realize sensing beyond the classical shot noise limit, and (2) quantum networks and communications, which may lead to revolutionary ways to share information.

• HOW ARE WE TRYING TO GET IT?

Building quantum computers is incredibly difficult. Many candidate qubit systems exist on the scale of single atoms, and the physicists, engineers, and materials scientists who are trying to execute quantum operations on these systems constantly deal with two competing requirements. First, qubits need to be protected from the environment because it can destroy the delicate quantum states needed for computation. The longer a qubit survives in its desired state the longer its “coherence time.” From this perspective, isolation is prized. Second, however, for algorithm execution qubits need to be entangled, shuffled around physical architectures, and controllable on demand. The better these operations can be carried out the higher their “fidelity.” Balancing the required isolation and interaction is difficult, but after decades of research a few systems are emerging as top candidates for large-scale quantum information processing.

Superconducting systems, trapped atomic ions, and semiconductors are some of the leading platforms for building a quantum computer. Each has advantages and disadvantages related to coherence, fidelity, and ultimate scalability to large systems. It is clear, however, that all of these platforms will need some type of error correction protocols to be robust enough to carry out meaningful calculations, and how to design and implement these protocols is itself a large area of research. For an overview of quantum computing, with more detail regarding experimental implementations, see Ladd et al. (2010) .

In this article, “quantum computing” has so far been used as a blanket term describing all computations that utilize quantum phenomena. There are actually multiple types of operational frameworks. Logical, gate-based quantum computing is probably the best recognized. In it, qubits are prepared in initial states and then subject to a series of “gate operations,” like current or laser pulses depending on qubit type. Through these gates the qubits are put in superpositions, entangled, and subjected to logic operations like the AND, OR, and NOT gates of traditional computation. The qubits are then measured and a result obtained.

Another framework is measurement-based computation, in which highly entangled qubits serve as the starting point. Then, instead of performing manipulation operations on qubits, single qubit measurements are performed, leaving the targeted single qubit in a definitive state. Based on the result, further measurements are carried out on other qubits and eventually an answer is reached.

A third framework is topological computation, in which qubits and operations are based on quasiparticles and their braiding operations. While nascent implementations of the components of topological quantum computers have yet to be demonstrated, the approach is attractive because these systems are theoretically protected against noise, which destroys the coherence of other qubits.

Finally, there are the analog quantum computers or quantum simulators envisioned by Feynman. Quantum simulators can be thought of as special purpose quantum computers that can be programmed to model quantum systems. With this ability they can target questions such as how high-temperature superconductors work, or how certain chemicals react, or how to design materials with certain properties.

• CONCLUSIONS AND OUTLOOK

Quantum computers have the potential to revolutionize computation by making certain types of classically intractable problems solvable. While no quantum computer is yet sophisticated enough to carry out calculations that a classical computer can't, great progress is under way. A few large companies and small start-ups now have functioning non-error-corrected quantum computers composed of several tens of qubits, and some of these are even accessible to the public through the cloud. Additionally, quantum simulators are making strides in fields varying from molecular energetics to many-body physics.

As small systems come online a field focused on near-term applications of quantum computers is starting to burgeon. This progress may make it possible to actualize some of the benefits and insights of quantum computation long before the quest for a large-scale, error-corrected quantum computer is complete.

• Born M. The Born-Einstein Letters. London: Walker; 1971.
• Feynman RP. Simulating physics with computers. International Journal of Theoretical Physics. 1982; 21 (6-7):467–488.
• Ladd TD, Jelezko F, Laflamme R, Nakamura Y, Monroe C, O'Brien JL. Quantum computers. Nature. 2010; 464 (7285):45–53. [ PubMed : 20203602 ]
• Shor PW. Algorithms for quantum computation: Discrete logarithms and factoring. Proceedings of the 35th Annual Symposium on Foundations of Computer Science; 1994. pp. 124–134. https://arxiv ​.org/abs/quant-ph/9508027 .
• Cite this Page GAMBLE S. Quantum Computing: What It Is, Why We Want It, and How We're Trying to Get It. In: National Academy of Engineering. Frontiers of Engineering: Reports on Leading-Edge Engineering from the 2018 Symposium. Washington (DC): National Academies Press (US); 2019 Jan 28.
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• Published: 10 January 2022

40 years of quantum computing

Nature Reviews Physics volume  4 ,  page 1 ( 2022 ) Cite this article

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This year we celebrate four decades of quantum computing by looking back at the milestones of the field and forward to the challenges and opportunities that lie ahead.

In science there are few true eureka moments experienced by lone geniuses, but rather a continuous exchange and development of ideas that drive the collective human curiosity in new directions. Before a new field of research is born, there is usually a time when many similar ideas are in the air and scientists start to see something new forming, but cannot quite put their finger on it. Then someone manages to articulate a new concept, opening a new direction. From there, it may take years or even decades until the full implications are grasped. Such is the case of quantum computing.

In the early 1980s a deep connection between physics and computation was becoming evident. Twenty years earlier, Rolf Landauer had linked thermodynamics and information . In 1980, mathematician Yuri Manin mentioned in the introduction of his book Computable and Uncomputable (in Russian) the idea of a quantum automaton that used superposition and entanglement (see the English translation in ref. 1 ) and Paul Benioff discussed 2 a microscopic quantum mechanical Hamiltonian as a model of Turing machines. Then, in May 1981, a conference on the ‘ Physics of Computation ’ organized by MIT and IBM brought together physicists and computer scientists. Among the participants were some well-known scientists like Freeman Dyson, John Archibald Wheeler or Richard Feynman, Landauer and Benioff, and others whose names resonate with anyone having worked in quantum computing: Charles Bennett, Tommaso Toffoli, Edward Fredkin. The talks were published the next year in the International Journal of Theoretical Physics .

It is difficult to tell to what extent these papers were influenced by the discussions at the meeting or whether the ideas presented had been articulated by individual scientists beforehand. Most contributors referenced the other papers, except Feynman who did not cite anyone (although he did credit Fredkin for inspiration) and just transcribed his keynote speech with its colloquialisms (“Nature isn’t classical, dammit.”). His paper 3 has become a landmark in quantum computation and simulation, and has been credited for the birth of these fields. Feynman took the ideas that were in the air — computation is a physical process, perhaps even a quantum mechanical one — then turned them around by asking how to compute (simulate) physics. He showed that “quantum mechanics can’t seem to be imitable by a local classical computer”, but could be tacked by “quantum computers — universal quantum simulators”. Manin had had a similar intuition 1 (“the quantum behaviour of the system might be much more complex than its classical simulation”), but he did not develop it further.

Although it is hard to assign a single moment in time as the starting point of quantum computing, as a journal, we like to take the 1982 issue of the International Journal of Theoretical Physics as the crystallization of the idea of a quantum computer. We would also like to credit all the pioneers whose ideas connected quantum mechanics with computing.

From 1982 to today quantum computing has been on a journey with many ups and downs and unexpected encounters. It saw great excitement after Shor’s quantum algorithm for factorization in 1994, followed by the first proposals for building a quantum computer. Hopes were high, but then came the realization of how difficult it would be in practice. No other algorithms to rival the potential of Shor’s were found. Despite disappointment, momentum was not lost and the field branched into different directions. Unexpected connections to fundamental physics and insight into the foundations of quantum mechanics were uncovered and numerous advances were made both in theory and experiment. Things started to pick up again for quantum computing and the past five years have witnessed a renewed commercial interest and the first demonstrations of quantum computers performing tasks that are hard for classical computers, a quantum advantage.

To celebrate four decades of quantum computing we put together a Collection of relevant content from our pages. As we have done in the past, we will revisit milestone papers and their legacy in ‘then and now’-type retrospective pieces. We will also look ahead with a Roadmap article and other upcoming content. Watch this space.

Mathematics as Metaphor. Selected Essays of Yuri I. Manin 77–78 (American Mathematical Society, 2007).

Benioff, P. The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines. J. Stat. Phys. 22 , 563–591 (1980).

Article   ADS   MathSciNet   Google Scholar  

Feynman, R. P. Simulating physics with computers. Int. J. Theor. Phys. 21 , 467–488 (1982).

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40 years of quantum computing. Nat Rev Phys 4 , 1 (2022). https://doi.org/10.1038/s42254-021-00410-6

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What Is Quantum Computing?

This article was reviewed by a member of Caltech's Faculty .

Many researchers believe that quantum computers will complement rather than replace our conventional technologies.

Why do we want quantum computers?

Scientists and engineers anticipate that certain problems that are effectively impossible for conventional, classical computers to solve will be easy for quantum computers. Quantum computers are also expected to challenge current cryptography methods and to introduce new possibilities for completely private communication.

Quantum computers will help us learn about, model, and manipulate other quantum systems. That ability will improve our understanding of physics and will influence designs for things that are engineered at scales where quantum mechanics plays a role, such as computer chips, communication devices, energy technologies, scientific instruments, sensors, clocks, and materials .

Just as people could envision few of today's uses of classical computers and related technologies back in the 1950s, we may be surprised by the applications that emerge for quantum computers.

How does a quantum computer work?

Quantum computers share some properties with classical ones. For example, both types of computers usually have chips, circuits, and logic gates. Their operations are directed by algorithms (essentially sequential instructions), and they use a binary code of ones and zeros to represent information.

Both types of computers use physical objects to encode those ones and zeros. In classical computers, these objects encode bits (binary digits) in two states—e.g., a current is on or off, a magnet points up or down.

Quantum computers use quantum bits, or qubits, which process information very differently. While classical bits always represent either one or zero, a qubit can be in a superposition of one and zero simultaneously until its state is measured.

In addition, the states of multiple qubits can be entangled , meaning that they are linked quantum mechanically to each other. Superposition and entanglement give quantum computers capabilities unknown to classical computing.

Qubits can be made by manipulating atoms, electrically charged atoms called ions, or electrons, or by nanoengineering so-called artificial atoms, such as circuits of superconducting qubits, using a printing method called lithography.

Learn more: Quantum Computers Animated, from Caltech's Institute for Quantum Information and Matter [VIDEO]

Do quantum computers exist?

Nascent quantum computers have existed in various forms for more than a decade . Several technology companies already have working quantum computers and make them available together with related programming languages and software development resources.

The technology with the broadest potential uses, in which quantum gates control qubits through logical operations, is in fast-moving, early development. Today, computers of this type generally have fewer than 100 qubits. The qubits are kept in a quantum state inside nested chambers that chill them to near absolute zero temperature and shield them from magnetic and electric interference.

This technology reached a milestone in 2019 , when a quantum computer completed a specific calculation in a sliver of the time a classical supercomputer would have needed to solve the same problem. The feat is considered a proof of principle; the use of this type of quantum computer to solve practical problems is expected to be years away.

A different approach to quantum computing, called quantum annealing , is further along in development but limited to a specific kind of calculation. In this approach, a quantum computer housed in a cryogenic refrigerator uses thousands of qubits to quickly approximate the best solutions to complex problems. The approach is limited to mathematical problems called binary optimization problems, which have many variables and possible solutions. Some companies and agencies have purchased this type of computer or rent time on new models to address problems related to scheduling, design, logistics, and materials discovery.

When will broadly useful quantum computers be available?

It may be years before general-purpose quantum computers can be applied to a variety of practical problems. To do useful work, they probably will require thousands of qubits. Scaling up brings challenges.

Large numbers of qubits are harder to isolate, and if they interact with molecules or magnetic fields in their environment, they collapse or decohere, losing the essential but fragile properties of superposition and entanglement. The more qubits there are, the more likely the machine is to make errors as individual qubits are disturbed by the environment.

Theorists and experimentalists develop strategies to reduce errors, lengthen the time that qubits can stay in quantum states, and increase the system's fault tolerance , preserving its accuracy even in the presence of errors.

Researchers are inventing new designs for qubits and quantum computers and enhancing existing technology. Established and newer strategies will take time to scale up, increase in reliability, and demonstrate their potential.

How has Caltech influenced quantum computing?

From its beginnings, the field of quantum computing has been shaped by Caltech. Breakthroughs have come from alumni and current Caltech scientists and engineers, some of whom are affiliated with Caltech centers such as the Institute for Quantum Information and Matter and its precursors; the Kavli Nanoscience Institute ; the new AWS Center for Quantum Computing ; and JPL , a NASA laboratory managed by Caltech. Working together across engineering and science and with colleagues worldwide, these researchers have

• forecast quantum-mechanical devices in 1959 and quantum computers in 1981;
• performed the first experiment realizing quantum teleportation , which can transmit information over great distances;
• created Shor's algorithm , which showed that quantum computers have potential to solve problems that classical computers cannot;
• stored entangled quantum states in a memory device for the first time;
• conceptualized a method for correcting errors by drawing on entanglement to protect information from disturbances in the local environment;
• theorized materials that can physically encode and protect information ; and
• developed methods to verify that quantum computers are calculating correctly .

Dive Deeper

Caltech and Amazon Partner to Create New Hub of Quantum Computing

Physicists Create Elusive Particles That Remember Their Pasts

Conventional Computers Can Learn to Solve Tricky Quantum Problems

• Quantum computing for the very curious

Andy Matuschak and Michael Nielsen

Part of a series of essays in a mnemonic medium which makes it almost effortless to remember what you read.

• How the quantum search algorithm works
• How quantum teleportation works
• Quantum mechanics distilled

by Andy Matuschak and Michael Nielsen

Presented in a new mnemonic medium which makes it almost effortless to remember what you read.

Part I: The state of a qubit

Part ii: introducing quantum logic gates, part iii: universal quantum computing.

Our future projects are funded in part by readers like you.

Special thanks to our sponsor-level patrons, Adam Wiggins , Andrew Sutherland , Bert Muthalaly , Calvin French-Owen , Dwight Crow , fnnch , James Hill-Khurana , Lambda AI Hardware , Ludwig Petersson , Mickey McManus , Mintter , Patrick Collison , Paul Sutter, Peter Hartree , Sana Labs , Shripriya Mahesh , Tim O'Reilly .

If humanity ever makes contact with alien intelligences, will those aliens possess computers? In science fiction, alien computers are commonplace. If that's correct, it means there is some way aliens can discover computers independently of humans. After all, we’d be very surprised if aliens had independently invented Coca-Cola or Pokémon or the Harry Potter books. If aliens have computers, it’s because computers are the answer to a question that naturally occurs to both human and alien civilizations.

Here on Earth, the principal originator of computers was the English mathematician Alan Turing. In his paper, published in 1936 Alan M. Turing, On Computable Numbers, with an Application to the Entscheidungsproblem (1936). , Turing wasn’t trying to invent a clever gadget or to create an industry. Rather, he was attacking a problem about the nature of mathematics posed by the German mathematician David Hilbert in 1928. That sounds abstruse, but it’s worth understanding the gist of Hilbert and Turing’s thinking, since it illuminates where computers come from, and what computers will become in the future.

Through his career, Hilbert was interested in the ultimate limits of mathematical knowledge: what can humans know about mathematics, in principle, and what (if any) parts of mathematics are forever unknowable by humans? Roughly speaking, Hilbert’s 1928 problem asked whether there exists a general algorithm a mathematician can follow which would let them figure out whether any given mathematical statement is provable. Hilbert’s hoped-for algorithm would be a little like the paper-and-pencil algorithm for multiplying two numbers. Except instead of starting with two numbers, you’d start with a mathematical conjecture, and after going through the steps of the algorithm you’d know whether that conjecture was provable. The algorithm might be too time-consuming to use in practice, but if such an algorithm existed, then there would be a sense in which mathematics was knowable, at least in principle.

In 1928, the notion of an algorithm was pretty vague. Up to that point, algorithms were often carried out by human beings using paper and pencil, as in the multiplication algorithm just mentioned, or the long-division algorithm. Attacking Hilbert’s problem forced Turing to make precise exactly what was meant by an algorithm. To do this, Turing described what we now call a Turing machine : a single, universal programmable computing device that Turing argued could perform any algorithm whatsoever.

Today we’re used to the idea that computers can be programmed to do many different things. In Turing’s day, however, the idea of a universal programmable computer was remarkable. Turing was arguing that a single, fixed device could imitate any algorithmic process whatsoever, provided the right program was supplied. It was an amazing leap of imagination, and the foundation of modern computing.

In order to argue that his machine could imitate any algorithmic process, Turing considered what operations a human mathematician could perform when carrying out an algorithm. For each such operation, he had to argue that his machine could always do the same thing. His argument is too long to reproduce in full here, but it’s fun and instructive to see the style of Turing’s reasoning:

Computing is normally done by writing certain symbols on paper. We may suppose this paper is divided into squares like a child’s arithmetic book. In elementary arithmetic the two-dimensional character of the paper is sometimes used. But such a use is always avoidable, and I think that it will be agreed that the two-dimensional character of paper is no essential of computation. I assume then that the computation is carried out on one-dimensional paper, i.e. on a tape divided into squares. … The behavior of the computer [Turing is referring to the person performing an algorithm, not the machine!] at any moment is determined by the symbols which he is observing, and his “state of mind” at that moment. We may suppose that there is a bound B B B to the number of symbols or squares which the computer can observe at one moment. If he wishes to observe more, he must use successive observations. We will also suppose that the number of states of mind which need be taken into account is finite.

Obviously, this was an informal and heuristic argument! Invoking a child’s arithmetic book, or someone’s mental state is not the stuff of a rigorous, bulletproof argument. But Turing’s argument was convincing enough that later mathematicians and scientists have for the most part been willing to accept it. Turing’s machine became the gold standard: an algorithm was what we could perform on a Turing machine. And since that time, computing has blossomed into an industry, and billions of computers based on Turing’s model have been sold.

Still, there’s something discomforting about Turing’s analysis. Might he have missed something in his informal reasoning about what an algorithm is? In 1985, the English physicist David Deutsch suggested a deeper approach to the problem of defining what is meant by an algorithm David Deutsch, “Quantum theory, the Church-Turing principle and the universal quantum computer” (1985). . Deutsch pointed out that every algorithm is carried out by a physical system, whether it be a mathematician with paper-and-pencil, a mechanical system such as an abacus, or a modern computer. Deutsch then considered the following question (I've slightly rephrased to make it easier to read):

Is there a (single) universal computing device which can efficiently simulate any other physical system?

If there was such a device, you could use it to perform any algorithm whatsoever, because algorithms have to be performed on some kind of physical system. And so the device would be a truly universal computer. What’s more, Deutsch pointed out, you wouldn’t need to rely on informal, heuristic arguments to justify your notion of algorithm, as Turing had done. You could use the laws of physics to prove your device was universal.

So let’s come back to our opening question: will aliens have computers? Deutsch’s question above is a simple, fundamental question about the nature of the universe. It’s the kind of question which alien counterparts to Deutsch could plausibly come to ponder. And the alien civilizations of which they are a part would then be led inexorably to invent computers.

In this sense, computers aren’t just human inventions. They are a fundamental feature of the universe, the answer to a simple and profound question about how the universe works. And they have likely been discovered over and over again by many alien intelligences.

There’s a wrinkle in this story. Deutsch is a physicist with a background in quantum mechanics. And in trying to answer his question, Deutsch observed that ordinary, everyday computers based on Turing’s model have a lot of trouble simulating quantum mechanical systems Researchers such as Yu Manin and Richard Feynman had previously observed this, and as a result had speculated about computers based on quantum mechanics. . In particular, they seem to be extraordinarily slow and inefficient at doing such simulations. To answer his question affirmatively, Deutsch was forced to invent a new type of computing system, a quantum computer . Those quantum computers can do everything conventional computers can do, but are also capable of efficiently simulating quantum-mechanical processes. And so they are arguably a more natural computing model than conventional computers. If we ever meet aliens, my bet is that they’ll use quantum computers (or, perhaps, will have quantum computing brains). After all, it’s likely that aliens will be far more technologically advanced than current human civilization. And so they’ll use the computers natural for any technologically advanced society.

This essay explains how quantum computers work. It’s not a survey essay, or a popularization based on hand-wavy analogies. We’re going to dig down deep so you understand the details of quantum computing. Along the way, we’ll also learn the basic principles of quantum mechanics, since those are required to understand quantum computation.

Learning this material is challenging. Quantum computing and quantum mechanics are famously “hard” subjects, often presented as mysterious and forbidding. If this were a conventional essay, chances are that you’d rapidly forget the material. But the essay is also an experiment in the essay form. As I’ll explain in detail below the essay incorporates new user interface ideas to help you remember what you read . That may sound surprising, but uses a well-validated idea from cognitive science known as spaced-repetition testing. More detail on how it works below. The upshot is that anyone who is curious and determined can understand quantum computing deeply and for the long term.

That said, you need some mathematical background to understand the essay. I’ll assume you’re comfortable with complex numbers and with linear algebra – vectors, matrices, and so on. I’ll also assume you’re comfortable with the logic gates used in conventional computers – gates such as AND, OR, NOT, and so on.

If you don’t have that mathematical background, you’ll need to acquire it. How you do that depends on your prior experience and learning preferences – there’s no one-size-fits-all approach, you’ll need to figure it out for yourself. But two resources you may find helpful are: (1) 3Blue1Brown’s series of YouTube videos on linear algebra; and (2) the more in-depth linear algebra lectures by Gil Strang . Try them out, and if you find them helpful, keep going. If not, explore other resources.

It may seem tempting to try to avoid this mathematics. If you look around the web, there are many flashy introductions to quantum computing that avoid mathematics. There are, for instance, many rather slick videos on YouTube. They can be fun to watch, and the better ones give you some analogies to help make sense of quantum computing. But there’s a hollowness to them. Bluntly, if they don’t explain the actual underlying mathematical model, then you could spend years watching and rewatching such videos, and you’d never really get it. It’s like hanging out with a group of basketball players and listening to them talk about basketball. You might enjoy it, and feel as though you’re learning about basketball. But unless you actually spend a lot of time playing, you’re never going to learn to play basketball. To understand quantum computing, you absolutely must become fluent in the mathematical model.

As you know, in ordinary, everyday computers the fundamental unit of information is the bit. It’s a familiar but astonishing fact that all the things those computers do can be broken down into patterns of 0 0 0 s and 1 1 1 s , and simple manipulations of 0 0 0 s and 1 1 1 s . For me, I feel this most strongly when playing video games. I’ll be enjoying playing a game, when I’ll suddenly be hit by a realization of the astounding complexity behind the imaginary world visible on my screen:

Source Copyright Wildfire Games, used under a Creative Commons Attribution-Share Alike 3.0 license . .

Underlying every such image is millions of pixels, described by tens of millions of bits. When I move the game controller, I am effectively conducting an orchestra, tens of millions strong, organized through many layers of intermediary ideas, in such a way as to create enjoyment and occasionally sheer delight.

I’ve described a bit as an abstract entity, whose state is 0 0 0 or 1 1 1 . But in the real world, not the world of mathematics, we must find some way of storing our bits in a physical system. That can be done in many different ways. In your computer’s memory chips, bits are most likely stored as tiny electric charges on nanometer-scale capacitors (i.e., little reservoirs of charge), just above the surface of the chip. Old-fashioned hard disks take a different approach, using tiny magnets to store bits. Furthermore, different types of memory use different types of capacitor; different types of hard disk use different approaches to magnetization.

For the most part you don’t notice these differences when you use your computer. Computer designers work very, very hard to make the details of the physical instantiation of the bits invisible not just to the user, but also (often) invisible even to programmers. Many programmers never think about whether a bit is stored in fast on-microprocessor cache memory, in the dynamic RAM chips, or in some type of virtual memory (say, on a hard disk). There are exceptions – programmers working on high-performance programs sometimes do think about these things, to make their programs as fast as possible. But for many programmers it doesn’t much matter how bits are stored. Rather, they can think of the bit in purely abstract terms, as having a state which is either 0 0 0 or 1 1 1 .

In a manner similar to the way conventional computers are made up of bits, quantum computers are made up of quantum bits , or qubits . Just like a bit, a qubit has a state . But whereas the state of a bit is a number ( 0 0 0 or 1 1 1 ) , the state of a qubit is a vector . More specifically, the state of a qubit is a vector in a two-dimensional vector space. This vector space is known as state space. For instance, here’s a possible state for a qubit:

[ 1 0 ] \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] [ 1 0 ​ ]

That perhaps sounds strange! What does it mean that the state of a qubit is a two-dimensional vector? We’re going to unpack the answer slowly and gradually. You won’t have a single epiphany where you think “ahah, that’s what a qubit is!” Rather, you’ll gradually build up many details in your understanding, until you get to the point where you’re comfortable working with qubits, with quantum computations, and more generally with quantum mechanics.

One way qubits are similar to bits: we’ve said absolutely nothing about what the qubit actually is , physically. Maybe the state of the qubit is being stored somehow on an electron, or a photon, or an atom. Or maybe it’s being stored in something stranger, perhaps inside some exotic particle or state of matter, even further removed from our everyday experience.

For our purposes in this essay none of this matters, no more than you should worry about what type of capacitor is storing the bits inside your computer’s RAM. What you should take away is that: (a) qubits have a state; (b) much like a bit, that state is an abstract mathematical object; but (c) whereas a bit’s abstract state is a number, 0 0 0 or 1 1 1 , the state of a qubit is a 2 2 2 - dimensional vector; (d) we call the 2 2 2 - dimensional vector space where states live state space.

Alright, let’s review what we’ve learnt. Please indulge me by answering the questions just below. It’ll only take a few seconds – for each question, think about what you believe the answer to be, click to reveal the actual answer, and then mark whether you remembered or not. If you can recall, that’s great. If not, that’s also fine, just note the correct answer, and continue.

A medium which makes memory a choice

Perhaps you correctly recalled the answers to all three questions just now. Even if so, will you remember the answers in a week? In a year? Human memory is fallible. If your memory is like mine, you might vaguely remember the answers in a week: “what’s the state of a qubit, oh yes, it’s a vector!” But the chances you’ll remember in a month or a year are low. And if you forget such things, you won’t have any durable understanding of quantum computing.

How can we ensure you don’t remember these answers for just a few minutes or a few hours, but well into the future, perhaps even permanently?

One way is for you to be supremely virtuous, to keep coming back and re-reviewing the material until it’s firmly locked in your memory. If you are such a virtuous person, congratulations! But for the other 99 percent of us that’s not likely. What can we do?

For more than a century, cognitive scientists have studied human memory. And they’ve figured out some simple strategies that ensure you’ll remember something permanently. The single most important idea is to re-test you on your knowledge, with expanding time intervals between tests.

As an example, consider the question above: “How many dimensions does the state space of a qubit have?” If you got it right, you’d ideally be tested again in a few weeks. And if you got it right again, you’d be tested again a few months after that. And so on, a gradually expanding schedule. If you get the question wrong on one of those tests, the schedule would contract, so you can relearn the answer.

It turns out that such an expanding schedule is the optimal way to retain information. Each time you’re re-tested your brain consolidates the answer a little better into long-term memory, until eventually it’s permanent.

Spaced-repetition testing is a simple idea, but has profound consequences. First, it doesn’t take much overall time. Because of the expanding test schedule, it typically only takes a few minutes of total review time to memorize the answer to a question for years or decades. I won’t go through the math showing that, but you can see it worked out elsewhere .

Second, spaced-repetition testing gives you a guarantee you will remember the answer to the question. For the most part our memories work in a haphazard manner. We read or hear something interesting, and hope we remember it in future. Spaced-repetition testing makes memory into a choice.

This sounds great, but also like you’ll need to be very disciplined in re-testing yourself. Fortunately, the computer can handle all the scheduling for you. And so this essay isn’t just a conventional essay, it’s also a new medium, a mnemonic medium which integrates spaced-repetition testing. The medium itself makes memory a choice.

This comes at some cost: you’re committing to future review. But consider what that buys you. This essay will likely take you an hour or two to read. In a conventional essay, you’d forget most of what you learned over the next few weeks, perhaps retaining a handful of ideas. But with spaced-repetition testing built into the medium, a small additional commitment of time means you will remember all the core material of the essay. Doing this won’t be difficult, it will be easier than the initial read. Furthermore, you’ll be able to read other material which builds on these ideas; it will open up an entire world.

This spaced-repetition approach is why the questions only require a few seconds to read and answer. They’re not complex exercises, in the style of a textbook. Rather, the questions have a different point: the promise each question makes is that you will remember the answer forever. It’s to permanently change your thinking.

So, I invite you to set up an account by signing in below. If you do so, your review schedule for each question in the essay will be tracked, and you’ll receive periodic reminders containing a link which takes you to an online review session. That review session isn’t this full essay – rather, it looks just like the question set you answered above, but contains instead all the questions which are due, so you can quickly run through them. The time commitment will usually be a few minutes per session – a little more early on, when questions need frequent re-testing, but rapidly dropping off. You can study on your phone while grabbing coffee, or standing in line, or going for a walk, or in transit. The return for that small time commitment is greatly improved fluency in basic quantum computing and quantum mechanics. And that understanding will be internalized, a part of who you are, retained for years instead of days.

To keep this promise, we’re tracking your review schedule for each question, and sending you occasional reminders to check in, and to run through the questions which are due. You can review on your phone while grabbing coffee, or standing in line, or going for a walk, or on transit. The return for that commitment is greatly improved fluency in basic quantum computing and quantum mechanics. And that understanding will be internalized, a part of who you are, retained for years instead of weeks.

Having extolled the virtues of spaced-repetition testing, let’s try another question:

This question is similar to an earlier question: “How many dimensions does the state space of a qubit have?” It may seem inefficient to have such similar questions, but it helps build fluency with the material when you have the “same” information encoded into memory in multiple ways, triggering off different associations. And so many of the questions below have this nature, elaborating ideas in multiple ways.

Connecting qubits to bits: the computational basis states

Let’s get back to understanding qubits. I’ve described what the state of a qubit is, but given no hint about how (or whether) that’s connected to the state of a classical bit. (Henceforth we’ll use the phrase “classical bit” instead of “conventional bit”, after “classical physics”). In fact, there are two special quantum states which correspond to the 0 0 0 and 1 1 1 states of a classical bit. The quantum state corresponding to 0 0 0 is usually denoted ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ . That’s a fancy notation for the following vector:

∣ 0 ⟩ : = [ 1 0 ] . |0\rangle := \left[ \begin{array}{c} 1 \\ 0 \end{array} \right]. ∣ 0 ⟩ : = [ 1 0 ​ ] .

This special state ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ is called a computational basis state .

It’s tempting to see the ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ notation and wonder what all the separate pieces mean – what, for instance, does the ∣ | ∣ mean; what does the ⟩ \rangle ⟩ mean; and so on?

In fact, it’s best to regard ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ as a single symbol, standing for a single mathematical object – that vector we just saw above, [ 1 0 ] \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] [ 1 0 ​ ] . The ∣ | ∣ and ⟩ \rangle ⟩ don’t really have separate meanings except to signify this is a quantum state. In this, ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ is much like the symbol 0 0 0 : both stand for a single mathematical entity. And, as we’ll gradually come to see, a quantum computer can manipulate ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ in ways very similar to how a conventional computer can manipulate 0 0 0 .

This notation with ∣ | ∣ and ⟩ \rangle ⟩ is called the ket notation, and things like ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ are called kets . But don’t be thrown off by the unfamiliar terminology – a ket is just a vector, and when we say something is a ket, all we mean is that it’s a vector.

That said, the term ket is most often used in connection with notations like ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ . That is, we wouldn’t usually refer to [ 1 0 ] \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] [ 1 0 ​ ] as a ket; we’d call it a vector, instead. If you’re a fan of really sharp definitions and strict consistency, this may seem vague and wishy-washy, but in practice doesn’t cause confusion. It’s just a convention to be aware of!

Alright, so ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ is a computational basis state for a qubit, and plays much the same role as 0 0 0 does for a classical bit. It won’t surprise you to learn that there is another computational basis state, denoted ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ , which plays the same role as 1 1 1 does for a bit. Like ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ , ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ is just a notation for a two-dimensional vector, in this case:

∣ 1 ⟩ : = [ 0 1 ] . |1\rangle := \left[ \begin{array}{c} 0 \\ 1 \end{array} \right]. ∣ 1 ⟩ : = [ 0 1 ​ ] .

Again, we’ll gradually come to see that ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ has all the properties we expect of the 1 1 1 state of a classical bit.

Time for a few more questions. A reminder that these have a different purpose to conventional textbook exercises. Textbook exercises are about setting you challenges; the point of the questions below is instead to help you commit the answer to long-term memory.

How to use (or not use!) the questions

Mathematics is a process of staring hard enough with enough perseverance at the fog of muddle and confusion to eventually break through to improved clarity. I’m happy when I can admit, at least to myself, that my thinking is muddled, and I try to overcome the embarrassment that I might reveal ignorance or confusion. Over the years, this has helped me develop clarity in some things, but I remain muddled in many others. — William Thurston, Fields-medal winning mathematician

How should you think about it when you get one of these embedded questions wrong? Often in life testing means being judged, and being judged can be uncomfortable. If you don’t know the answer to a question you may feel like you’ve failed.

But the purpose of these questions has nothing to do with external judgment. Rather, they have two quite different purposes. One purpose is, as explained earlier, to help strengthen your memory, so your new knowledge is consolidated in your long-term memory. The second purpose is to help you diagnose what you do and don’t know, and to help you fill the gaps as rapidly as possible.

In this view, getting a question wrong is useful information. It pinpoints exactly what you need to understand, and puts the onus on you to figure it out. Honestly, the appropriate response to getting a question wrong is probably to throw your hands up in the air and shout “wonderful!”, since it’s at those points of what may appear to be “failure” that your learning rate will be at its highest.

With that said, it’s a common canard in education that “it doesn’t matter if you get something right or wrong, so long as you’re learning”. Unfortunately, that statement is often bullshit. If getting something right or wrong is used to determine your grade, or to influence other people’s opinion of you, then it damn well does matter.

For that reason, I suggest keeping your results private. They are for you alone, and their purpose should be solely to help you learn as rapidly as possible. If they’re not helping you – if they’re making you feel bad, for instance, or being used to judge you – then stop doing the questions.

You’ve probably noticed the questions are self-assessed. If you want to mark yourself “correct” sometimes, even when you’re not, go for it! What impact do you think that will have on your learning? Do you enjoy the slightly transgressive feeling? I must admit that I do. Don’t be embarrassed, if so: this is supposed to be, above all else, fun. Or try marking yourself wrong when you’re correct, or skipping the questions entirely. What impact do these actions have on your learning? The point is to figure out how to engage with the questions to learn as rapidly as possible. And that means experimenting playfully with how you engage, to find what works for you.

How to approach this essay?

This essay is an unusual form. It’s certainly not a product in the conventional startup sense; it’s a research project, an experiment in developing a new and improved form of reading. A conventional product would aim to draw you in, and form a regular, long-term habit. There’d be tens (or hundreds) of millions of words of content for you to read, and lots of people on social media excitedly pointing you toward that content. You’d engage every day, learning more and more and more.

That’s not what’s going on here. “Quantum Computing for the Very Curious” is, instead, like a video game or book or movie, a single one-off project for you to work through. So you commit to it for a while, and then the experience is over. (Also, like a video game, book, or movie, sequels are planned!)

With that said, it’s different from those forms too. Many people consume games, books, and movies as binge activities, hungrily devouring them until complete. “Quantum Computing for the Very Curious” is, by contrast, intentionally an experience spread out over time. Yes, you probably binge at first, working your way quickly through the text over a couple of hours. But then you return occasionally for brief review sessions, prompted by our notifications. That form of spaced testing is the best way for you to remember what you read. Still, while it might be optimal from the point of view of memory formation, it means not using some of the techniques that games, books, and movies use to keep you interested. Nonetheless, I hope you’ll be willing to trust the team at “Quantum Computing for the Very Curious”, and to participate in the experiment.

How can you get the most out of reading in this new mnemonic medium? The ideal is to do an initial read, followed by a few short review sessions over the coming weeks (prompted by our notifications) to help you internalize the ideas. Then (optionally) a followup read, where you can more deeply understand the material. And finally, some sessions of followup review to ensure you remember for the long term:

As you work through this process, we’ll track your overall progress toward completion (meaning: most questions reliably committed to memory), and each time you review we’ll show you that progress.

This is a larger commitment than traditional reading. But for a small factor in effort, you will understand the material far more deeply, and remember it for more than 10x as long. What’s more, while the reading process above looks complex, you’ll be cued at each step by reminders that will help the review process become a habit. Trust the reminders, and this will all happen as a matter of course.

You may find the essay particularly helpful if you’re taking an introductory class on quantum computing. If that’s your situation, I advise you to read the entire essay immediately at the beginning of semester (or even before), answering all the questions as you go. Then continue to follow the procedure described just above, taking a few minutes to complete each review session, prompted by the reminders you’ll be sent. This will make it far easier to understand the rest of the course you’re taking, and help you get much more out of it.

General states of a qubit

The computational basis states ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ and ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ are just two possible states for a qubit. Many more states are possible, and those extra states endow qubits with capabilities not available to ordinary classical bits. In general, remember, a quantum state is a two-dimensional vector. Here’s an example, with a graphical illustration emphasizing the vector nature of the state:

In this example, the state 0.6 ∣ 0 ⟩ + 0.8 ∣ 1 ⟩ 0.6|0\rangle+0.8|1\rangle 0 . 6 ∣ 0 ⟩ + 0 . 8 ∣ 1 ⟩ is just 0.6 0.6 0 . 6 times the ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ vector, plus 0.8 0.8 0 . 8 times the ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ vector. In the usual vector notation that means the state is:

0.6 ∣ 0 ⟩ + 0.8 ∣ 1 ⟩ = 0.6 [ 1 0 ] + 0.8 [ 0 1 ] = [ 0.6 0.8 ] . 0.6|0\rangle + 0.8 |1\rangle = 0.6 \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] + 0.8 \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] = \left[ \begin{array}{c} 0.6 \\ 0.8 \end{array} \right]. 0 . 6 ∣ 0 ⟩ + 0 . 8 ∣ 1 ⟩ = 0 . 6 [ 1 0 ​ ] + 0 . 8 [ 0 1 ​ ] = [ 0 . 6 0 . 8 ​ ] .

I’ve been talking about quantum states as two-dimensional vectors. What I didn’t yet mention is that in general they’re complex vectors , that is, they can have complex numbers as entries. Of course, the example just shown has real entries, as do the computational basis states. But for a general quantum state the entries can be complex numbers. So, for instance, another quantum state is the vector

1 + i 2 ∣ 0 ⟩ + i 2 ∣ 1 ⟩ , \frac{1+i}{2} |0\rangle + \frac{i}{\sqrt{2}} |1\rangle, 2 1 + i ​ ∣ 0 ⟩ + 2 ​ i ​ ∣ 1 ⟩ ,

which can be written in the conventional component vector representation as

[ 1 + i 2 i 2 ] . \left[ \begin{array}{c} \frac{1+i}{2} \\ \frac{i}{\sqrt{2}} \end{array} \right]. [ 2 1 + i ​ 2 ​ i ​ ​ ] .

Because quantum states are in general vectors with complex entries, the illustration above shouldn’t be taken too literally – the plane is a real vector space, not a complex vector space. Still, visualizing it as a plane is sometimes a handy way of thinking.

I’ve said what a quantum state is, as a mathematical object: it’s a two-dimensional vector in a complex vector space. But why is that true? What does it mean, physically, that it’s a vector? Why a complex vector space, and how should we think about the complex numbers? And what’s a quantum state good for, anyway?

These are good questions. But they do take some time to answer. For context consider that the discovery of quantum mechanics wasn’t a single event, but occurred over 25 years of work, from 1900 to 1925. Many Nobel prizes were won for milestones along the way. That includes Albert Einstein’s Nobel Prize, won primarily for work related to quantum mechanics (not relativity, as people sometimes assume).

When some of the brightest people in the world struggle for 25 years to develop a theory, it’s not an obvious theory! In fact, the idea of describing a simple quantum system using a complex vector in two dimensions summarizes much of what was learned over that 25 years. In that sense, it’s quite a simple and beautiful statement. But it’s not an obvious statement, and it’s not unreasonable that it might take a few hours to understand. That’s better than taking 25 years to understand!

As part of that journey toward understanding, let’s get familiar with some more nomenclature commonly used for quantum states.

One of the most common terms is superposition . People will say a state like 0.6 ∣ 0 ⟩ + 0.8 ∣ 1 ⟩ 0.6|0\rangle+0.8|1\rangle 0 . 6 ∣ 0 ⟩ + 0 . 8 ∣ 1 ⟩ is a superposition of ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ and ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ . All they mean is that the state is a linear combination of ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ and ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ . You may wonder why they don’t just say “linear combination” (and sometimes they do), but the reason is pretty much the same reason English-speakers say “hello” while Spanish-speakers say “hola” – the two terms come out of different cultures and different histories.

Another common term is amplitude . An amplitude is the coefficient for a particular state in superposition. For instance, in the state 0.6 ∣ 0 ⟩ + 0.8 ∣ 1 ⟩ 0.6|0\rangle+0.8|1\rangle 0 . 6 ∣ 0 ⟩ + 0 . 8 ∣ 1 ⟩ the amplitude for ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ is 0.6 0.6 0 . 6 , and the amplitude for ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ is 0.8 0.8 0 . 8 .

We’ve learnt that a quantum state is a two-dimensional complex vector. Actually, it can’t be just any old vector, a fact you might have guessed from the very particular amplitudes in some of the examples above. There’s a constraint. The constraint is this: the sums of the squares of the amplitudes must be equal to 1 1 1 .

So, for example, for the state 0.6 ∣ 0 ⟩ + 0.8 ∣ 1 ⟩ 0.6 |0\rangle+0.8 |1\rangle 0 . 6 ∣ 0 ⟩ + 0 . 8 ∣ 1 ⟩ the sum of the squares of the amplitudes is 0. 6 2 + 0. 8 2 0.6^2+0.8^2 0 . 6 2 + 0 . 8 2 , which is 0.36 + 0.64 = 1 0.36+0.64 = 1 0 . 3 6 + 0 . 6 4 = 1 , as we desired.

For a more general quantum state, the amplitudes can be complex numbers, let’s denote them by α \alpha α and β \beta β so the state is α ∣ 0 ⟩ + β ∣ 1 ⟩ \alpha|0\rangle+\beta|1\rangle α ∣ 0 ⟩ + β ∣ 1 ⟩ . The constraint is now that the sum of the squares of the amplitudes is 1 1 1 , i.e., ∣ α ∣ 2 + ∣ β ∣ 2 = 1 |\alpha|^2+|\beta|^2 = 1 ∣ α ∣ 2 + ∣ β ∣ 2 = 1 .

This is called the normalization constraint .

It’s called that because if you think of ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ and ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ as orthonormal vectors, as I drew them earlier,

then the normalization constraint is the requirement that the length of the state is equal to 1 1 1 . So it’s a unit vector, or a normalized vector, and that’s why this is called a normalization constraint.

Summing up all these ideas in one sentence: the quantum state of a qubit is a vector of unit length in a two-dimensional complex vector space known as state space.

We’ve gone through a few refinements of this sentence but that sentence is the final version – there’s no missing parts, or further refinement necessary! That’s what the quantum state of a qubit is. Of course, we will explore the definition further, deepening our understanding, but it will always come back to that basic fact.

One common gotcha in thinking about qubits is to look at the ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ state and think it must be the zero vector in the vector space, often denoted 0 0 0 . But that’s not right at all. The zero vector is at the origin, 0 = [ 0 0 ] 0 = \left[ \begin{array}{c} 0 \\ 0 \end{array} \right] 0 = [ 0 0 ​ ] , while the ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ vector is quite different, ∣ 0 ⟩ = [ 1 0 ] ≠ 0 |0\rangle = \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] \neq 0 ∣ 0 ⟩ = [ 1 0 ​ ]  ​ = 0 . It’s just an unfortunate notational accident that ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ looks as though it should be the 0 0 0 vector. Fortunately, in practice this distinction is easy to get used to, and doesn’t cause confusion. But it’s worth noting.

What does the quantum state mean? Why is it a vector in a complex vector space?

My own conclusion is that today there is no interpretation of quantum mechanics that does not have serious flaws. This view is not universally shared. Indeed, many physicists are satisfied with their own interpretation of quantum mechanics. But different physicists are satisfied with different interpretations. — Steven Weinberg, Nobel Laureate in Physics

Let’s come back to the question of what the quantum state means, and why it’s a vector in a complex vector space.

In the case of classical bits, it’s pretty easy to think about the state. You can think of the 0 0 0 or 1 1 1 states of a bit as corresponding to two very different (but stable) states of a physical system. For instance, a 0 0 0 or 1 1 1 can be indicated by the presence or absence of a hole at some location in a punch card. And so a single punch card can be used to store hundreds or thousands of bits:

Source: Gwern (2006).

Most ways of storing classical bits are variations on this idea. For instance, the dynamic random access memory (RAM) inside your computer is based on the idea of having two tiny metal plates separated by a miniscule gap. Electric charge is stored on those plates, setting up an electric field between them. The 0 0 0 and 1 1 1 states of the bit correspond to two different configurations of charge on the plates. In practice, real dynamic RAM systems use slightly more elaborate ideas, but that’s the heart of it. This is harder to think about than punch cards – most of us don’t have so much experience thinking about moving electric charges around metal plates. But it’s still pretty concrete.

So, how should we think about quantum states? The ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ and ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ states often aren’t difficult to think about, since they often correspond to very concrete physical states of the world, much like classical bits. Indeed, in some proposals they may correspond to different charge configurations, similar to dynamic RAM. Or perhaps a photon being in one of two different locations in space – again, a pretty simple, concrete notion, even if photons aren’t that familiar. There are also many more exotic proposals for the ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ and ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ states. But by and large you can think of the computational basis states as representing a physical system in one of two well-defined configurations.

Of course, qubits have states which aren’t computational basis states, states like 0.6 ∣ 0 ⟩ + 0.8 ∣ 1 ⟩ 0.6|0\rangle+0.8|1\rangle 0 . 6 ∣ 0 ⟩ + 0 . 8 ∣ 1 ⟩ . How should we think about such superposition states? At least in popular media accounts, a very common description is that a state like 0.6 ∣ 0 ⟩ + 0.8 ∣ 1 ⟩ 0.6|0\rangle + 0.8|1\rangle 0 . 6 ∣ 0 ⟩ + 0 . 8 ∣ 1 ⟩ is “simultaneously” in the ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ state and the ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ state.

I must confess, I don’t understand what people mean by this. As far as I can tell, what they’re trying to do is explain the quantum state in terms of classical concepts they’re already familiar with. But I can’t make much sense of it – saying 0.6 ∣ 0 ⟩ + 0.8 ∣ 1 ⟩ 0.6|0\rangle + 0.8|1\rangle 0 . 6 ∣ 0 ⟩ + 0 . 8 ∣ 1 ⟩ is simultaneously in the ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ state and the ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ state seems like word salad, and makes about as much sense to me as Lewis Carroll’s nonsense poem Jabberwocky :

’Twas brillig, and the slithy toves      Did gyre and gimble in the wabe: All mimsy were the borogoves,     And the mome raths outgrabe. …

Of course, lacking any other interpretation it’s tempting to try to impose our classical prejudices on the quantum state. Or, even if you reject that, to get hung up worrying about what a quantum state is. But the trouble is that there is enormous disagreement amongst physicists themselves about how to think about the quantum state. Indeed, many active researchers are trying to understand what the correct way of thinking about the quantum state is, exploring multiple approaches in great depth. So we’re going to defer worrying too much about this until later.

To understand why we’re deferring, suppose someone gives you a Rubik’s cube (or some other challenging puzzle or game) for the first time, all scrambled up. You start to theorize about the best ways of solving it, how to understand it, and so on. But you never actually play around with it, getting familiar with how it behaves.

Often the best way to understand something is to first use it, to get comfortable, to play a lot, and to do lots of informal experiments. As you build familiarity you understand why things are the way they are. At that point, you can go back and better understand the meaning of the basics.

Well, we can think of quantum computing and quantum mechanics as an especially complicated type of puzzle! So the strategy we’re taking is to start with the mathematics of quantum computing – we’ll keep getting familiar with qubits and the quantum state, and developing the consequences. Doing that is how we’ll build up intuition, and will give us the chops needed to come back and think harder about the meaning of the quantum state.

In Part I we learned about the state of a qubit. But in order to quantum compute, it’s not enough just to understand quantum states. We need to be able to do things with them! We do that using quantum logic gates .

A quantum logic gate is simply a way of manipulating quantum information, that is, the quantum state of a qubit or a collection of qubits. They’re analogous to the classical logic gates used in ordinary, everyday computers – gates such as the AND, OR, and NOT gates. And, much like classical gates’ role in conventional computers, quantum gates are the basic building blocks of quantum computation. They’re also a convenient way of describing many other quantum information processing tasks, such as quantum teleportation.

In Part II of this essay we’ll discuss several types of quantum logic gate. As we’ll see, the gates we discuss are sufficient to do any quantum computation. In particular, much as any classical computation can be built up using AND, OR, and NOT gates, the quantum gates we describe over the next few sections suffice to do any quantum computation.

Many of the quantum gates we’ll learn about are based on familiar classical logic gates. But a few are different. Those differences appear innocuous, almost trivial. But in those differences lies the power of quantum computation, and the possibility for quantum computers to be vastly superior to classical computers.

Part II is, frankly, a bit of a slog. Learning quantum gates is like learning basic vocabulary in a new language: there’s no getting round spending a fair bit of time working on it. Still, the spaced-repetition testing should make this basic memory work much easier than is ordinarily the case. And it will prepare you well for some of the more conceptual issues discussed in Part III, where we return to high-level questions about the ultimate nature of computation, and what quantum computers are good for.

The quantum NOT gate

Let’s take a look at our very first quantum logic gate, the quantum NOT gate . As you can no doubt surmise, the quantum NOT gate is a generalization of the classical NOT gate. On the computational basis states the quantum NOT gate does just what you’d expect, mimicking the classical NOT gate. That is, it takes the ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ state to ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ , and vice versa:

N O T ∣ 0 ⟩ = ∣ 1 ⟩ NOT|0\rangle = |1\rangle N O T ∣ 0 ⟩ = ∣ 1 ⟩

N O T ∣ 1 ⟩ = ∣ 0 ⟩ NOT|1\rangle = |0\rangle N O T ∣ 1 ⟩ = ∣ 0 ⟩

But the computational basis states aren’t the only states possible for a qubit. What happens when we apply the NOT gate to a general superposition state, that is, α ∣ 0 ⟩ + β ∣ 1 ⟩ \alpha|0\rangle+\beta|1\rangle α ∣ 0 ⟩ + β ∣ 1 ⟩ ? In fact, it does pretty much the simplest possible thing: it acts linearly on the quantum state, interchanging the role of ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ and ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ :

N O T ( α ∣ 0 ⟩ + β ∣ 1 ⟩ ) = α ∣ 1 ⟩ + β ∣ 0 ⟩ . NOT (\alpha|0\rangle +\beta|1\rangle) = \alpha|1\rangle+\beta |0\rangle. N O T ( α ∣ 0 ⟩ + β ∣ 1 ⟩ ) = α ∣ 1 ⟩ + β ∣ 0 ⟩ .

I’ve been using the notation NOT for the quantum NOT gate. But for historical reasons people working on quantum computing usually use a different notation, the notation X X X . And so the above may be rewritten:

X ( α ∣ 0 ⟩ + β ∣ 1 ⟩ ) = α ∣ 1 ⟩ + β ∣ 0 ⟩ . X (\alpha|0\rangle +\beta|1\rangle) = \alpha|1\rangle+\beta |0\rangle. X ( α ∣ 0 ⟩ + β ∣ 1 ⟩ ) = α ∣ 1 ⟩ + β ∣ 0 ⟩ .

I’ll use the terms X X X gate and NOT gate interchangeably.

Historically, the notation X X X traces its origin to 1927, when the physicist Wolfgang Pauli introduced an operation s x s_x s x ​ (often written σ x \sigma_x σ x ​ in textbooks today) to help describe rotations of certain objects around the x x x spatial axis. This operation later became of interest to people working on quantum computing. But by that point the s s s (and the connection to rotation) was irrelevant, and so s x s_x s x ​ just became X X X .

What we’ve seen so far are very algebraic ways of writing down the way the X X X gate works. There’s an alternate representation, the quantum circuit representation. In a quantum circuit we depict an X X X gate as follows:

The line from left to right is what’s called a quantum wire . A quantum wire represents a single qubit. The term “wire” and the way it’s drawn looks like the qubit is moving through space. But it's often helpful to instead think of left-to-right as representing the passage of time. So the initial segment of wire is just the passage of time, with nothing happening to the qubit. Then the X X X gate is applied. And then the quantum wire continues, leaving the desired output.

Sometimes we’ll put the input and output states explicitly in the quantum circuit, so we have something like:

So that’s the quantum circuit representation of the X X X gate. It is, in fact, our first quantum computation. A simple computation, involving just a single qubit and a single gate, but a genuine quantum computation nonetheless!

There’s a third representation for the X X X gate that’s worth knowing about, a representation as a 2 × 2 2 \times 2 2 × 2 matrix:

X = [ 0 1 1 0 ] . X = \left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right]. X = [ 0 1 ​ 1 0 ​ ] .

To understand in what sense this is a representation of the NOT gate, recall that α ∣ 0 ⟩ + β ∣ 1 ⟩ \alpha|0\rangle + \beta|1\rangle α ∣ 0 ⟩ + β ∣ 1 ⟩ is just the vector [ α β ] \left[ \begin{array}{c} \alpha \\ \beta \end{array} \right] [ α β ​ ] . And so we have:

[ 0 1 1 0 ] ( α ∣ 0 ⟩ + β ∣ 1 ⟩ ) = [ 0 1 1 0 ] [ α β ] = [ β α ] = α ∣ 1 ⟩ + β ∣ 0 ⟩ = X ( α ∣ 0 ⟩ + β ∣ 1 ⟩ ) . \begin{aligned} \left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right] (\alpha|0\rangle + \beta|1\rangle) & = \left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right] \left[ \begin{array}{c} \alpha \\ \beta \end{array} \right] \\ & = \left[ \begin{array}{c} \beta \\ \alpha \end{array} \right] \\ & = \alpha|1\rangle+\beta|0\rangle \\ & = X (\alpha|0\rangle + \beta|1\rangle). \end{aligned} [ 0 1 ​ 1 0 ​ ] ( α ∣ 0 ⟩ + β ∣ 1 ⟩ ) ​ = [ 0 1 ​ 1 0 ​ ] [ α β ​ ] = [ β α ​ ] = α ∣ 1 ⟩ + β ∣ 0 ⟩ = X ( α ∣ 0 ⟩ + β ∣ 1 ⟩ ) . ​

This tells us that X X X and [ 0 1 1 0 ] \left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right] [ 0 1 ​ 1 0 ​ ] act in exactly the same way on all vectors, and thus are the same operation. In fact, as we’ll see later, it turns out that all quantum gates can be thought of as matrices, with the matrix entries specifying the exact details of the gate.

By the way, regarding the X X X gate as a matrix clarifies what may have been a confusing point earlier. I wrote the X X X gate as having the action X ∣ 0 ⟩ = ∣ 1 ⟩ X|0\rangle = |1\rangle X ∣ 0 ⟩ = ∣ 1 ⟩ , X ∣ 1 ⟩ = ∣ 0 ⟩ X|1\rangle = |0\rangle X ∣ 1 ⟩ = ∣ 0 ⟩ . Implicitly – I never quite said this, though it’s true – the X X X gate is a mathematical function, taking input states to output states. But when we write functions we usually use parentheses, so why didn’t I write X ( ∣ 0 ⟩ ) X(|0\rangle) X ( ∣ 0 ⟩ ) and similarly for ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ ? The reason is that for linear functions, i.e., matrices, it’s conventional to omit parentheses and just write X ∣ 0 ⟩ X|0\rangle X ∣ 0 ⟩ – function application is just matrix multiplication.

Quantum wires: why the simplest quantum circuit is often also the hardest to implement

We’ve now seen a simple quantum circuit and quantum logic gate. But it’s not quite the simplest possible quantum circuit. The simplest possible quantum circuit does nothing at all. That is, it’s a single quantum wire:

This circuit is just a single qubit being preserved in time. To be more explicit, if some arbitrary quantum state ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ is input to the circuit, then the exact same state ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ is output (it’s common practice to use the Greek letter ψ \psi ψ to denote an arbitrary quantum state):

Mathematically, this circuit is trivial. But physically it’s far from trivial. In many physical systems, the quantum wire is actually one of the hardest quantum computations to implement!

The reason is that quantum states are often incredibly fragile. If your qubit is being stored in some tiny system – perhaps a single photon or a single atom – then it’s very, very easy to disturb that state. It really doesn’t take much to upset an atom or a photon. And so while quantum wires are mathematically trivial, they can be one of the hardest elements to build in real systems.

That said, there are some systems where quantum wires are easy to implement. If you store your qubit in a neutrino then the state will actually be pretty well preserved. The reason is that neutrinos barely interact with other forms of matter at all – a neutrino can easily pass through a mile of lead without being disturbed. But while it’s intriguing that neutrinos are so stable, it doesn’t mean they make good qubits. The trouble is that since there’s no easy way of using ordinary matter to interact with them, we can’t manipulate their quantum state in a controlled fashion, and so can’t implement a quantum gate.

There’s a tension here that applies to many proposals to do quantum information processing, not just neutrinos. If we want to store the quantum state, then it’s helpful if our qubits only interact very weakly with other systems, so those systems don’t disrupt them. But if the qubits only interact weakly with other systems then that also makes it hard to manipulate the qubits. Thus, systems which make good quantum wires are often hard to build quantum gates for. Much of the art of designing quantum computers is about finding ways to navigate this tension. Often, that means trying to design systems which interact weakly most of the time, but some of the time can be caused to interact strongly, and so serve as part of a quantum gate.

You’ll note that some of the questions above have a different flavor to the questions earlier in the essay. Early questions had cut-and-dried answers. The answer to the question “What’s the standard notation for the quantum NOT gate?” is just: “ X X X ” . But some of the questions above have less well-specified answers. They’re a little fluffy.

This fluffiness may cause you difficulties as you decide how to respond. Your answer to the question “Why is it that systems which make good quantum wires are often hard to build quantum gates for?” may not quite match the answer given. If this is the case, don’t worry. You should mark yourself correct if you’re confident you’ve understood the point, even in terms different from my phrasing. And mark yourself incorrect if the point still needs reinforcement.

A multi-gate quantum circuit

Let’s take a look at a simple multiple-gate quantum circuit. It’s just a circuit with two X X X gates in a row:

It’s worth pausing for a moment, and trying to guess what this circuit does to the input state. It’s worth doing this even if you usually find this kind of guessing frustrating. Even when you get stuck, building up strategies for dealing with stuckness is part of learning any difficult subject. So take a minute or so now.

We’ll try two different ways of figuring out what’s going on. Here’s one approach, based on applying X X X twice to an arbitrary input state, α ∣ 0 ⟩ + β ∣ 1 ⟩ \alpha|0\rangle+\beta|1\rangle α ∣ 0 ⟩ + β ∣ 1 ⟩ . It’s simple algebra, using the fact X X X interchanges the ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ and ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ states:

X ( X ( α ∣ 0 ⟩ + β ∣ 1 ⟩ ) ) = X ( α ∣ 1 ⟩ + β ∣ 0 ⟩ ) = α ∣ 0 ⟩ + β ∣ 1 ⟩ \begin{aligned} X(X(\alpha|0\rangle+\beta|1\rangle)) & = X(\alpha|1\rangle + \beta|0\rangle) \\ & = \alpha|0\rangle + \beta|1\rangle \end{aligned} X ( X ( α ∣ 0 ⟩ + β ∣ 1 ⟩ ) ) ​ = X ( α ∣ 1 ⟩ + β ∣ 0 ⟩ ) = α ∣ 0 ⟩ + β ∣ 1 ⟩ ​

So the net effect is to recover the original quantum state, no matter what that state was. In other words, this circuit is equivalent to a quantum wire:

A second way of seeing this is based on the matrix representation we found earlier for the X X X gate. Observe that if the input is some arbitrary quantum state ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ , then after the first gate the state is X ∣ ψ ⟩ X|\psi\rangle X ∣ ψ ⟩ , and after the second gate the state is X X ∣ ψ ⟩ X X |\psi\rangle X X ∣ ψ ⟩ . Then we observe that the product X X X X X X is

X X = [ 0 1 1 0 ] [ 0 1 1 0 ] = [ 1 0 0 1 ] . \begin{aligned} XX & = \left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right] \left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right] \\ & = \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right]. \end{aligned} X X ​ = [ 0 1 ​ 1 0 ​ ] [ 0 1 ​ 1 0 ​ ] = [ 1 0 ​ 0 1 ​ ] . ​

That is, X X XX X X is the identity operation, and so the output X X ∣ ψ ⟩ XX|\psi\rangle X X ∣ ψ ⟩ is just the original input ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ . In other words, two X X X gates is the same as a quantum wire. And so we’ve arrived at the same conclusion in a different way, using matrix multiplication. Doing such matrix multiplications is a pretty common way of analyzing quantum circuits.

The Hadamard gate

We’ve seen our first quantum gate, the NOT or X X X gate. Of course, the X X X didn’t appear to do all that much beyond what is possible with a classical NOT gate. In this section I introduce a gate that clearly involves quantum effects, the Hadamard gate.

As with the X X X gate, we’ll start by explaining how the Hadamard gate acts on computational basis states. Denoting the gate by H H H , here’s what it does:

H ∣ 0 ⟩ = ∣ 0 ⟩ + ∣ 1 ⟩ 2 H|0\rangle = \frac{|0\rangle +|1\rangle}{\sqrt 2} H ∣ 0 ⟩ = 2 ​ ∣ 0 ⟩ + ∣ 1 ⟩ ​

H ∣ 1 ⟩ = ∣ 0 ⟩ − ∣ 1 ⟩ 2 H|1\rangle = \frac{|0\rangle -|1\rangle}{\sqrt 2} H ∣ 1 ⟩ = 2 ​ ∣ 0 ⟩ − ∣ 1 ⟩ ​

Of course, ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ and ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ aren’t the only quantum states. How does the Hadamard gate act on more general quantum states?

It won’t surprise you to learn that it acts linearly, as did the quantum NOT gate. In particular, the Hadamard gate takes a superposition α ∣ 0 ⟩ + β ∣ 1 ⟩ \alpha|0\rangle + \beta|1\rangle α ∣ 0 ⟩ + β ∣ 1 ⟩ to the corresponding superposition of outputs:

H ( α ∣ 0 ⟩ + β ∣ 1 ⟩ ) = α ( ∣ 0 ⟩ + ∣ 1 ⟩ 2 ) + β ( ∣ 0 ⟩ − ∣ 1 ⟩ 2 ) . H(\alpha|0\rangle + \beta|1\rangle) = \alpha \left( \frac{|0\rangle+|1\rangle}{\sqrt 2} \right) + \beta \left( \frac{|0\rangle-|1\rangle}{\sqrt 2} \right). H ( α ∣ 0 ⟩ + β ∣ 1 ⟩ ) = α ( 2 ​ ∣ 0 ⟩ + ∣ 1 ⟩ ​ ) + β ( 2 ​ ∣ 0 ⟩ − ∣ 1 ⟩ ​ ) .

That’s a mess. We can make it less of a mess by combining the ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ terms together, and also the ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ terms together:

H ( α ∣ 0 ⟩ + β ∣ 1 ⟩ ) = α + β 2 ∣ 0 ⟩ + α − β 2 ∣ 1 ⟩ . H(\alpha|0\rangle + \beta|1\rangle) = \frac{\alpha+\beta}{\sqrt 2} |0\rangle + \frac{\alpha-\beta}{\sqrt 2}|1\rangle. H ( α ∣ 0 ⟩ + β ∣ 1 ⟩ ) = 2 ​ α + β ​ ∣ 0 ⟩ + 2 ​ α − β ​ ∣ 1 ⟩ .

That’s better, but still not pretty! Fortunately, we mostly won’t be dealing with such complex expressions. The only reason I’ve done it here is to be really explicit. Instead of dealing with such explicit expressions, we’ll mostly work with the H H H gate in its circuit and matrix representations (see below). Those let us focus at a more enlightening level of abstraction, rather than messing around with coefficients.

Indeed, much work on quantum computing is about attempting to develop ways of moving from low levels of abstraction to higher, more conceptual levels. Up to now most of our work has been at a very low level, seeming more an exercise in linear algebra than a discussion of a new model of computing. That perhaps seems strange. After all, if you were explaining classical computers to someone, you wouldn’t start in the weeds, with AND and NOT gates and the like. You’d start with a well-designed high-level programming language, and then bounce back and forth between different layers of abstraction. Modern computers aren’t just about logic gates – they’re at least as much about beautiful higher-level ideas: say, lazy evaluation, or higher-order functions, or homoiconicity, and so on.

I wish I could start with high-level abstractions for quantum computers. However, we’re still in the early days of quantum computing, and for the most part humanity hasn’t yet discovered such high-level abstractions. People are still scratching around, trying to find good ideas.

That’s an exciting situation: it means almost all the big breakthroughs are ahead. There’s a sense in which we still understand very little about quantum computing. That might sound surprising: after all, there are great big fat textbooks on the subject . But you could have written a great big fat textbook about the ENIAC computer in the late 1940s. It was, after all, a very complex system. That textbook would have looked intimidating, but it wouldn’t have been the final word in computing. For the most part the way we understand quantum computing today is at an ENIAC-like level, looking at the nuts-and-bolts of qubits and logic gates and linear algebra, and wondering what the higher-level understanding may be. The situation can be thought of as much like programming language design before the breakthroughs that led to languages such as Lisp and Haskell and Prolog and Smalltalk. That makes it a remarkable creative opportunity, a challenge for the decades and centuries ahead.

Speaking of nuts-and-bolts, let’s get back to the Hadamard gate. Here’s the circuit representation for the Hadamard gate. It looks just like the X X X gate in the circuit representation, except we change the gate label to H H H :

Just like the X X X gate, H H H has a matrix representation:

H = 1 2 [ 1 1 1 − 1 ] . H = \frac{1}{\sqrt 2} \left[ \begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array} \right]. H = 2 ​ 1 ​ [ 1 1 ​ 1 − 1 ​ ] .

To see that this matrix representation is correct, let’s check the action of the matrix on the ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ and the ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ states. Here we check for the ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ state:

1 2 [ 1 1 1 − 1 ] ∣ 0 ⟩ = 1 2 [ 1 1 1 − 1 ] [ 1 0 ] = 1 2 [ 1 1 ] = ∣ 0 ⟩ + ∣ 1 ⟩ 2 . \begin{aligned} \frac{1}{\sqrt 2} \left[ \begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array} \right] |0\rangle & = \frac{1}{\sqrt 2} \left[ \begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array} \right] \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] \\ & = \frac{1}{\sqrt 2} \left[ \begin{array}{c} 1 \\ 1 \end{array} \right] \\ & = \frac{|0\rangle+|1\rangle}{\sqrt 2}. \end{aligned} 2 ​ 1 ​ [ 1 1 ​ 1 − 1 ​ ] ∣ 0 ⟩ ​ = 2 ​ 1 ​ [ 1 1 ​ 1 − 1 ​ ] [ 1 0 ​ ] = 2 ​ 1 ​ [ 1 1 ​ ] = 2 ​ ∣ 0 ⟩ + ∣ 1 ⟩ ​ . ​

That is, this matrix acts the same way as the Hadamard gate on the ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ state. Now let’s check on the ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ state:

1 2 [ 1 1 1 − 1 ] ∣ 1 ⟩ = 1 2 [ 1 1 1 − 1 ] [ 0 1 ] = 1 2 [ 1 − 1 ] = ∣ 0 ⟩ − ∣ 1 ⟩ 2 . \begin{aligned} \frac{1}{\sqrt 2} \left[ \begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array} \right] |1\rangle & = \frac{1}{\sqrt 2} \left[ \begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array} \right] \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] \\ & = \frac{1}{\sqrt 2} \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] \\ & = \frac{|0\rangle-|1\rangle}{\sqrt 2}. \end{aligned} 2 ​ 1 ​ [ 1 1 ​ 1 − 1 ​ ] ∣ 1 ⟩ ​ = 2 ​ 1 ​ [ 1 1 ​ 1 − 1 ​ ] [ 0 1 ​ ] = 2 ​ 1 ​ [ 1 − 1 ​ ] = 2 ​ ∣ 0 ⟩ − ∣ 1 ⟩ ​ . ​

So the matrix acts the same way as the Hadamard gate on both the ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ and ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ states. By the linearity of matrix multiplication it follows that the matrix acts the same way as the Hadamard on all input states, and so they are the same operation.

What makes the Hadamard gate interesting as a quantum gate? What can we use it to do?

We don’t (quite) have enough background to give precise answers to these questions yet. But there is an analogy which gives insight.

Imagine you are living in North Africa, thousands of years in the past, and decide for some reason that you want to get over to the Iberian peninsula. If you don’t yet have boats or some other reliable method of moving across large bodies of water, you need to go all the way across Africa, past the Arabian peninsula, up and around through Europe, back to the Iberian peninsula:

Original photo source: Reto Stöckli, NASA Earth Observatory (2004).

Suppose, however, that you invent a new device, the boat, which expands the range of locations you can traverse. Then you can take a much more direct route over to the Iberian peninsula, greatly cutting down the time required:

What the Hadamard and similar gates do is expand the range of operations that it’s possible for a computer to perform. That expansion makes it possible for the computer to take shortcuts, as the computer “moves” in a way that’s not possible in a conventional classical computer. And, we hope, that may enable us to solve some computational problems faster.

Another helpful analogy is to the game of chess. Imagine you’re playing chess and the rules are changed in your favor, enabling your rook an expanded range of moves. That extra flexibility might enable you to achieve checkmate much faster because you can get to new positions much more quickly.

A similar thing is going on with the Hadamard gate. By expanding the range of states we can access (or, more precisely, the range of dynamical operations we can generate) beyond what’s possible on a classical computer, it becomes possible to take shortcuts in our computation.

We’ll see explicit examples in subsequent essays.

To get more familiar with the Hadamard gate, let’s analyze a simple circuit:

What’s this circuit do?

Before we compute, it’s worth pausing for a second to try guessing the result. The point of guessing isn’t to get it right – rather, it’s to challenge yourself to start coming up with heuristic mental models for thinking about what’s going on in quantum circuits. Those mental models likely won’t be very good at first, but that’s okay – if you keep doing this, they’ll get better.

Here’s one heuristic you can use to think about this circuit: you can think of H H H as sort of mixing the ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ and ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ states together. So if you apply H H H twice to ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ , perhaps it would thoroughly mix the ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ and ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ up together. What sort of state do you think would result, according to this heuristic? Do you believe the result? Why or why not? Can you think of other heuristics that might help you guess an answer?

Alright, let’s compute what actually happens. After we apply the first Hadamard to ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ we get

∣ 0 ⟩ + ∣ 1 ⟩ 2 . \frac{|0\rangle+|1\rangle}{\sqrt{2}}. 2 ​ ∣ 0 ⟩ + ∣ 1 ⟩ ​ .

Then we apply a second Hadamard gate. This takes the ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ term above to ∣ 0 ⟩ + ∣ 1 ⟩ 2 \frac{|0\rangle+|1\rangle}{\sqrt 2} 2 ​ ∣ 0 ⟩ + ∣ 1 ⟩ ​ and the ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ term to ∣ 0 ⟩ − ∣ 1 ⟩ 2 \frac{|0\rangle-|1\rangle}{\sqrt 2} 2 ​ ∣ 0 ⟩ − ∣ 1 ⟩ ​ , so the output is:

1 2 ( ∣ 0 ⟩ + ∣ 1 ⟩ 2 + ∣ 0 ⟩ − ∣ 1 ⟩ 2 ) . \frac{1}{\sqrt 2} \left( \frac{|0\rangle+|1\rangle}{\sqrt 2} + \frac{|0\rangle-|1\rangle}{\sqrt 2} \right). 2 ​ 1 ​ ( 2 ​ ∣ 0 ⟩ + ∣ 1 ⟩ ​ + 2 ​ ∣ 0 ⟩ − ∣ 1 ⟩ ​ ) .

If you look at the expression above, you’ll notice that the ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ terms cancel each other out, so you’re just left with the ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ terms. Collecting them up, we’re left with the ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ state, same as we started with:

1 2 ( ∣ 0 ⟩ 2 + ∣ 0 ⟩ 2 ) = ∣ 0 ⟩ . \frac{1}{\sqrt 2} \left(\frac{|0\rangle}{\sqrt 2}+ \frac{|0\rangle}{\sqrt 2} \right) = |0\rangle. 2 ​ 1 ​ ( 2 ​ ∣ 0 ⟩ ​ + 2 ​ ∣ 0 ⟩ ​ ) = ∣ 0 ⟩ .

In a similar fashion, after we run the ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ state through the first Hadamard gate, we get:

∣ 0 ⟩ − ∣ 1 ⟩ 2 . \frac{|0\rangle-|1\rangle}{\sqrt{2}}. 2 ​ ∣ 0 ⟩ − ∣ 1 ⟩ ​ .

Then we apply a second Hadamard gate to get:

1 2 ( ∣ 0 ⟩ + ∣ 1 ⟩ 2 − ∣ 0 ⟩ − ∣ 1 ⟩ 2 ) . \frac{1}{\sqrt 2} \left( \frac{|0\rangle+|1\rangle}{\sqrt 2} - \frac{|0\rangle-|1\rangle}{\sqrt 2} \right). 2 ​ 1 ​ ( 2 ​ ∣ 0 ⟩ + ∣ 1 ⟩ ​ − 2 ​ ∣ 0 ⟩ − ∣ 1 ⟩ ​ ) .

This time it’s the ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ terms which cancel out, and the ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ terms reinforce. When we collect up these terms, we see that the output is just the ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ state, same as we started with. And so both the ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ and ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ states are left unchanged by this quantum circuit, and the net effect of the circuit is exactly the same as a quantum wire:

There’s an alternate way of seeing this, which I’ll sketch out without working through in detail. It’s to note that if we input an arbitrary quantum state ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ to the circuit, then the output must be H H ∣ ψ ⟩ H H |\psi\rangle H H ∣ ψ ⟩ , i.e., the result of applying two H H H matrices to ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ . But if you just compute the matrix product H H H H H H it turns out to be the 2 × 2 2 \times 2 2 × 2 identity matrix, H H = I H H = I H H = I . And so the output from the circuit must be the same as the input, H H ∣ ψ ⟩ = I ∣ ψ ⟩ = ∣ ψ ⟩ HH|\psi\rangle = I |\psi\rangle = |\psi\rangle H H ∣ ψ ⟩ = I ∣ ψ ⟩ = ∣ ψ ⟩ , just as from a quantum wire.

Of course, this result violates our intuitive guess, which was that two Hadamards would thoroughly mix up the ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ and the ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ . It’s interesting to ponder what went wrong with our intuition, say by looking through the calculation for H H H acting twice on ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ . You see that after the second gate the ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ terms exactly cancel one another out, while the ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ terms reinforce one another.

This seems innocuous, almost like a mathematical accident. Still, I draw your attention to it because this type of cancellation or reinforcement is crucial in many algorithms for quantum computers. Without getting into details, the rough way many such algorithms work is to first use Hadamard gates to “spread out” in quantum states like ∣ 0 ⟩ + ∣ 1 ⟩ 2 \frac{|0\rangle+|1\rangle}{\sqrt 2} 2 ​ ∣ 0 ⟩ + ∣ 1 ⟩ ​ (or many-qubit analogs), i.e., in superpositions of multiple computational basis states. At the end of the algorithm they use clever patterns of cancellation and reinforcement to bring things back together again into one (or possibly a few, in the many-qubit case) computational basis state, containing the desired answer. That’s a somewhat vague and perhaps tantalizing description, but the point to take away is that the kind of cancellation-or-reinforcement we saw above is actually crucial in many quantum computations.

I’ll now pose a few simple exercises related to the Hadamard gate. Unlike the spaced-repetition questions, the point of the exercises below isn’t as an aid to memory, and so you won’t see these exercises repeatedly. Rather, they’re here because (should you choose to work through them) they will help you better understand the material of the essay. But they’ll only incidentally help with memorization. We’ll follow them with some spaced-repetition questions. Note that even if you don't work through the exercises, it's worth at least reading through them, since some of the results will be tested in the spaced-repetition questions.

Exercise: Verify that H H = I HH = I H H = I , where I I I is the 2 × 2 2 \times 2 2 × 2 identity matrix, I = [ 1 0 0 1 ] I = \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right] I = [ 1 0 ​ 0 1 ​ ] .

Exercise: Suppose that instead of H H H we’d defined a matrix J J J by:

J : = 1 2 [ 1 1 1 1 ] J := \frac{1}{\sqrt 2} \left[ \begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array} \right] J : = 2 ​ 1 ​ [ 1 1 ​ 1 1 ​ ]

At first, it might seem that J J J would make an interesting quantum gate, along lines similar to H H H . For instance, J ∣ 0 ⟩ = ∣ 0 ⟩ + ∣ 1 ⟩ 2 J|0\rangle = \frac{|0\rangle+|1\rangle}{\sqrt 2} J ∣ 0 ⟩ = 2 ​ ∣ 0 ⟩ + ∣ 1 ⟩ ​ , and J ∣ 1 ⟩ = ∣ 0 ⟩ + ∣ 1 ⟩ 2 J|1\rangle = \frac{|0\rangle+|1\rangle}{\sqrt 2} J ∣ 1 ⟩ = 2 ​ ∣ 0 ⟩ + ∣ 1 ⟩ ​ . These are both good, normalized quantum states. But what happens if we apply J J J to the quantum state ∣ 0 ⟩ − ∣ 1 ⟩ 2 \frac{|0\rangle-|1\rangle}{\sqrt 2} 2 ​ ∣ 0 ⟩ − ∣ 1 ⟩ ​ ? Why does this make J J J unsuitable for use as a quantum gate?

Exercise: Consider the quantum circuit:

Explain why the output from this circuit is X H ∣ ψ ⟩ XH|\psi\rangle X H ∣ ψ ⟩ , not H X ∣ ψ ⟩ HX|\psi\rangle H X ∣ ψ ⟩ , as you might naively assume if you wrote down gates in the order they occur in the circuit. This is a common gotcha to be aware of – it occurs because quantum gates compose left-to-right in the circuit representation, while matrix multiplications compose right-to-left.

Measuring a qubit

Suppose a (hypothetical!) quantum physicist named Alice prepares a qubit in her laboratory, in a quantum state α ∣ 0 ⟩ + β ∣ 1 ⟩ \alpha|0\rangle+\beta|1\rangle α ∣ 0 ⟩ + β ∣ 1 ⟩ . Then she gives her qubit to another quantum physicist, Bob, but doesn’t tell him the values of α \alpha α and β \beta β . Is there some way Bob can figure out α \alpha α and β \beta β ? That is, is there some experiment Bob can do to figure out the identity of the quantum state?

The surprising answer to this question turns out to be NO! There is, in fact, no way to figure out α \alpha α and β \beta β if they start out unknown. To put it a slightly different way, the quantum state of any system – whether it be a qubit or a some other system – is not directly observable.

I say this is surprising, because it’s very different from our usual everyday way of thinking about how the world works. If there’s something wrong with your car, a mechanic can use diagnostic tools to learn about the internal state of the engine. The better the diagnostic tools, the more they can learn. Of course, there may be parts of the engine that would be impractical to access – maybe they’d have to break a part, or use a microscope, for instance. But you’d probably be rather suspicious if the mechanic told you the laws of physics prohibited them from figuring out the internal state of the engine.

Similarly, when you first start learning about quantum circuits, it seems like we should be able to observe the amplitudes of a quantum state whenever we like. But that turns out to be prohibited by the laws of physics. Those amplitudes are better thought of as a kind of hidden information.

So, what can we figure out from the quantum state? Rather than somehow measuring α \alpha α and β \beta β , there are other ways of getting useful information out of a qubit. Let me describe an especially important process called measurement in the computational basis . This is a fundamental primitive in quantum computing: it’s the way we typically extract information from our quantum computers. I’ll explain now how it works for a single qubit, and later generalize to multi-qubit systems.

Suppose a qubit is in the state α ∣ 0 ⟩ + β ∣ 1 ⟩ \alpha|0\rangle+\beta|1\rangle α ∣ 0 ⟩ + β ∣ 1 ⟩ . When you measure this qubit in the computational basis it gives you a classical bit of information: it gives you the outcome 0 0 0 with probability ∣ α ∣ 2 |\alpha|^2 ∣ α ∣ 2 , and the outcome 1 1 1 with probability ∣ β ∣ 2 |\beta|^2 ∣ β ∣ 2 .

To think a little more concretely about this process, suppose your qubit is instantiated in some physical system. Perhaps it’s being stored in the state of an atom somehow. It doesn’t matter exactly what, but you have this qubit in your laboratory. And you have some measurement apparatus, probably something large and complicated, maybe involving lasers and microprocessors and a screen for readout of the measurement result. And this measurement apparatus interacts in some way with your qubit.

After the measurement interaction, your measurement apparatus registers an outcome. For instance, it might be that you get the outcome 0 0 0 . Or maybe instead you get the outcome 1 1 1 . The crucial fact is that the outcome is ordinary classical information – the stuff you already know how to think about – which you can then use to do other things, and to control other processes.

So the way a quantum computation works is that we manipulate a quantum state using a series of quantum gates, and then at the end of the computation (typically) we do a measurement to read out the result of the computation. If our quantum computer is just a single qubit, then that result will be a single classical bit. If, as is more usually the case, it’s multiple qubits, then the measurement result will be multiple classical bits.

A fundamental fact about this measurement process is that it disturbs the state of the quantum system. In particular, it doesn’t just leave the quantum state alone. After the measurement, if you get the outcome 0 0 0 then the state of the qubit afterwards (the “posterior state”) is the computational basis state ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ . On the other hand, if you get the outcome 1 1 1 then the posterior state of the qubit is the computational basis state ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ .

Summing all this up: if we measure a qubit with state α ∣ 0 ⟩ + β ∣ 1 ⟩ \alpha |0\rangle+\beta|1\rangle α ∣ 0 ⟩ + β ∣ 1 ⟩ in the computational basis, then the outcome is a classical bit: either 0 0 0 , with probability ∣ α ∣ 2 |\alpha|^2 ∣ α ∣ 2 , or 1 1 1 , with probability ∣ β ∣ 2 |\beta|^2 ∣ β ∣ 2 . The corresponding state of the qubit after the measurement is ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ or ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ .

A key point to note is that after the measurement, no matter what the outcome, α \alpha α and β \beta β are gone. No matter whether the posterior state is ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ or ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ , there is no trace of α \alpha α or β \beta β . And so you can’t get any more information about them. In that sense, α \alpha α and β \beta β are a kind of hidden information – the measurement doesn’t tell you what they were.

One reason this is important is because it means you can’t store an infinite amount of classical information in a qubit. After all, α \alpha α is a complex number, and you could imagine storing lots of classical bits in the binary expansion of the real component of α \alpha α . If there was some experimental way you could measure the value of α \alpha α exactly, then you could extract that classical information. But without a way of measuring α \alpha α that’s not possible.

I’ve been talking about measurement in the computational basis. In fact, there are other types of measurement you can do in quantum systems. But there’s a sense in which computational basis measurements turn out to be fundamental. The reason is that by combining computational basis measurements with quantum gates like the Hadamard and NOT (and other) gates, it’s possible to simulate arbitrary quantum measurements. So this is all you absolutely need to know about measurement, from an in-principle point of view.

It’s useful to have a way of denoting measurements in the quantum circuit model. Here’s a simple example:

It’s a single-qubit quantum circuit, with input the state ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ . A NOT gate is applied, followed by a Hadamard gate. The circuit finishes with a measurement in the computational basis, denoted by the slightly elongated semi-circle. The m m m is a classical bit denoting the measurement result – either 0 0 0 or 1 1 1 – and we use the double wire to indicate the classical bit m m m going off and being used to do something else.

Of course, I said above that after measurement the qubit is in either the ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ or the ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ state. You might think we’d draw a corresponding quantum wire coming out the other side of the measurement. But often in quantum circuits the qubit is discarded after measurement, and that’s assumed by this notation.

One final comment on measurement is that it’s connected to the normalization condition for quantum states that we discussed earlier. Suppose we have the quantum state:

α ∣ 0 ⟩ + β ∣ 1 ⟩ \alpha|0\rangle+\beta|1\rangle α ∣ 0 ⟩ + β ∣ 1 ⟩

Then the probability of the two possible measurement outcomes, 0 0 0 and 1 1 1 , must sum to 1 1 1 , and so we have:

∣ α ∣ 2 + ∣ β ∣ 2 = 1. |\alpha|^2+|\beta|^2 = 1. ∣ α ∣ 2 + ∣ β ∣ 2 = 1 .

This is exactly the normalization condition for quantum states – i.e., the quantum state must have length 1 1 1 . The origin of that constraint is really just the fact that measurement probabilities must add up to 1 1 1 .

Exercise: Suppose we’ve been given either the state ∣ 0 ⟩ + ∣ 1 ⟩ 2 \frac{|0\rangle+|1\rangle}{\sqrt 2} 2 ​ ∣ 0 ⟩ + ∣ 1 ⟩ ​ or the state ∣ 0 ⟩ − ∣ 1 ⟩ 2 \frac{|0\rangle-|1\rangle}{\sqrt 2} 2 ​ ∣ 0 ⟩ − ∣ 1 ⟩ ​ , but not told which state we’ve been given. We’d like to figure out which state we’ve been given. If we just do a computational basis measurement, then for both states we get outcome 0 0 0 with probability 1 2 \frac 12 2 1 ​ , and outcome 1 1 1 with probability 1 2 \frac 12 2 1 ​ . So we can’t distinguish the states directly using a computational basis measurement. But suppose instead we put the state into the following circuit:

Show that if the state ∣ 0 ⟩ + ∣ 1 ⟩ 2 \frac{|0\rangle+|1\rangle}{\sqrt 2} 2 ​ ∣ 0 ⟩ + ∣ 1 ⟩ ​ is input then the output is m = 0 m = 0 m = 0 with probability 1 1 1 , while if the state ∣ 0 ⟩ − ∣ 1 ⟩ 2 \frac{|0\rangle-|1\rangle}{\sqrt 2} 2 ​ ∣ 0 ⟩ − ∣ 1 ⟩ ​ is input then the output is m = 1 m = 1 m = 1 with probability 1 1 1 . Thus while these two states are indistinguishable if just measured in the computational basis, they can be distinguished with the help of a simple quantum circuit.

General single-qubit gates

So far we’ve learned about two quantum gates, the NOT and the Hadamard gate, and also about the measurement process that can be used to extract classical information from our quantum circuits. In this section, I return to quantum gates, and take a look at the most general single-qubit gate. To do that it helps to recall the matrix representations of the NOT and Hadamard gates:

X = [ 0 1 1 0 ] ;      H = 1 2 [ 1 1 1 − 1 ] \begin{aligned} X = \left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right]; \,\,\,\, H = \frac{1}{\sqrt 2} \left[ \begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array} \right] \end{aligned} X = [ 0 1 ​ 1 0 ​ ] ; H = 2 ​ 1 ​ [ 1 1 ​ 1 − 1 ​ ] ​

If the input to these gates is the quantum state ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ , then the output is X ∣ ψ ⟩ X|\psi\rangle X ∣ ψ ⟩ and H ∣ ψ ⟩ H|\psi\rangle H ∣ ψ ⟩ respectively.

A general single-qubit gate works similarly. In particular, a general single-qubit gate can be represented as a 2 × 2 2 \times 2 2 × 2 unitary matrix, U U U . (If you’re rusty, I’ll remind you what it means for a matrix to be unitary in a moment, and you can just think of it as a matrix for now.) If the input to the gate is the state ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ then the output from the gate is U ∣ ψ ⟩ U|\psi\rangle U ∣ ψ ⟩ . And so the NOT and Hadamard gates correspond to the special cases where U = X U = X U = X and U = H U = H U = H , respectively.

What does it mean for a matrix U U U to be unitary? It’s easiest to answer this question algebraically, where it simply means that U † U = I U^\dagger U = I U † U = I , that is, the adjoint of U U U , denoted U † U^\dagger U † , times U U U , is equal to the identity matrix. That adjoint is, recall, the complex transpose of U U U :

U † : = ( U T ) ∗ . U^\dagger := (U^T)^*. U † : = ( U T ) ∗ .

So for a 2 × 2 2 \times 2 2 × 2 matrix, the adjoint operation is just:

[ a b c d ] † = [ a ∗ c ∗ b ∗ d ∗ ] . \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right]^\dagger = \left[ \begin{array}{cc} a^* & c^* \\ b^* & d^* \end{array} \right]. [ a c ​ b d ​ ] † = [ a ∗ b ∗ ​ c ∗ d ∗ ​ ] .

(Note that the † \dagger † is also sometimes called the dagger operation , or Hermitian conjugation , or just the conjugation operation. We’ll use all three terms on occasion.)

There are a few basic questions you might ask: why are single-qubit gates described by unitary matrices? And how can we get an intuitive feel for what it means for a matrix to be unitary, anyway? While the equation U † U = I U^\dagger U = I U † U = I is easy to check algebraically, we’d like some intuition for what that equation means.

Another natural question is whether the NOT gate and the Hadamard gate are unitary? Of course, we’ll see that they are – I wouldn’t have described them as quantum gates if not – but we should go to the trouble of checking.

Yet another good question is whether there are useful examples of single-qubit gates that aren’t the NOT or Hadamard gates? The equation U † U = I U^\dagger U = I U † U = I is all very well, but it’d be nice to have more concrete examples than just an abstract equation.

We’ll answer all these questions over the next few sections.

Let’s start by checking the unitarity of the Hadamard gate. We start by computing the adjoint of H H H :

H † = ( ( 1 2 [ 1 1 1 − 1 ] ) T ) ∗ . H^\dagger = \left( \left( \frac{1}{\sqrt 2} \left[ \begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array} \right]\right)^T\right)^*. H † = ( ( 2 ​ 1 ​ [ 1 1 ​ 1 − 1 ​ ] ) T ) ∗ .

Note that taking the transpose doesn’t change the matrix, since it is already symmetric. And taking the complex conjugate doesn’t change anything, since all the entries are real. So we have H † = H H^\dagger = H H † = H , and thus H † H = H H H^\dagger H = HH H † H = H H . But we saw earlier in the essay that H H = I HH = I H H = I . So H H H is, indeed, unitary.

Exercise: Show that X X X is unitary.

Exercise: Show that the identity matrix I I I is unitary.

Exercise: Can you find an example of a 2 × 2 2 \times 2 2 × 2 matrix that is unitary, and is not I I I , X X X , or H H H ?

For flavor, let’s give a few more examples of single-qubit quantum gates. Earlier in the essay, I mentioned that the NOT gate X X X was introduced by the physicist Wolfgang Pauli in the early days of quantum mechanics. He introduced two other matrices, Y Y Y and Z Z Z , which are also useful quantum gates. The three gates, X , Y X, Y X , Y , and Z Z Z are known collectively as the Pauli matrices . The Y Y Y and Z Z Z gates will be useful extra tools in our toolkit of quantum gates; in terms of the earlier analogy they expand the repertoire of moves we have available to us. They’re crucial, for example, in protocols such as quantum teleportation and quantum error correction.

The Y Y Y gate is similar to the X X X gate, but instead of 1 1 1 s on the off-diagonal, it has i i i and − i -i − i , so it takes ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ to i ∣ 1 ⟩ i|1\rangle i ∣ 1 ⟩ and ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ to − i ∣ 0 ⟩ -i|0\rangle − i ∣ 0 ⟩ :

Y : = [ 0 − i i 0 ] . Y := \left[ \begin{array}{cc} 0 & -i \\ i & 0 \end{array} \right]. Y : = [ 0 i ​ − i 0 ​ ] .

The Z Z Z gate leaves ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ unchanged, and takes ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ to − ∣ 1 ⟩ -|1\rangle − ∣ 1 ⟩ :

Z : = [ 1 0 0 − 1 ] . Z := \left[ \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right]. Z : = [ 1 0 ​ 0 − 1 ​ ] .

Exercise: Show that the Y Y Y and Z Z Z matrices are unitary, and so legitimate quantum gates.

Another good example of a quantum gate is a rotation, the kind of matrix you’ve most likely been seeing since high school. It’s just the ordinary rotation of the 2 2 2 - dimensional plane by an angle θ \theta θ :

[ cos ⁡ ( θ ) − sin ⁡ ( θ ) sin ⁡ ( θ ) cos ⁡ ( θ ) ] . \left[ \begin{array}{cc} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{array} \right]. [ cos ( θ ) sin ( θ ) ​ − sin ( θ ) cos ( θ ) ​ ] .

You can easily check that this is a unitary matrix, and so it’s valid quantum gate. And sometimes it’s a very useful quantum gate!

That mostly wraps up our little library of single-qubit gates. In the next couple of sections we’ll build up some intuition about what unitarity means, but we won’t add any extra elements to our quantum computing model. In fact, at this point we know almost everything needed for quantum computation. There are just two extra elements needed: extending our description of single qubits to multiple qubits, and describing a simple two-qubit quantum gate. We’ll get to those things shortly – not surprisingly, they look much like what we’ve already seen.

I am, by the way, somewhat uncomfortable with some of the questions just asked. My personal experience is that spaced-repetition learning works well when learning facts I have a lot of context for, and care a lot about. It’s a rare person who finds the detailed entries in a unitary matrix fascinating! That’s not to say they aren’t – actually, they are , but you need a lot more context than I’ve provided to see why (for instance) that − i -i − i in the Y Y Y is just so darn interesting.

With that said, this essay is genuinely an experiment, and the questions above are included in that spirit. Maybe it will turn out that readers can use spaced-repetition to learn the entries of unitary matrices. And maybe they cannot. No matter which that’ll be a valuable thing to learn about the world, and to inform future experiments with spaced-repetition learning.

What does it mean for a matrix to be unitary?

Can we get an intuition for what it means for a matrix to be unitary? It turns out that unitary matrices preserve the length of their inputs . In other words, if we take any vector ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ and compute the length ∥ U ∣ ψ ⟩ ∥ \| U|\psi\rangle \| ∥ U ∣ ψ ⟩ ∥ it’s always equal to the length ∥ ∣ ψ ⟩ ∥ \||\psi\rangle\| ∥ ∣ ψ ⟩ ∥ of the original vector. In this, they’re much like rotations or reflections in ordinary (real) space, which also don’t change lengths. In a sense, the unitary matrices are a complex generalization of real rotations and reflections.

Up to now we’ve been dealing mostly with vectors which are states of qubits, i.e., normalized 2-dimensional vectors. But in fact the statement of the last paragraph is true for d × d d \times d d × d unitary matrices, too, i.e., ∥ U ∣ ψ ⟩ ∥ = ∥ ∣ ψ ⟩ ∥ \|U|\psi\rangle\| = \||\psi\rangle\| ∥ U ∣ ψ ⟩ ∥ = ∥ ∣ ψ ⟩ ∥ is true for any vector ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ in d d d dimensions.

The proof is actually pretty simple – mostly, we just compute ∥ U ∣ ψ ⟩ ∥ \|U|\psi\rangle \| ∥ U ∣ ψ ⟩ ∥ and use a little algebra to check that it’s equal to the length ∥ ∣ ψ ⟩ ∥ \||\psi\rangle\| ∥ ∣ ψ ⟩ ∥ . But before we get to the proof, observe that this result is good news for quantum gates. The reason is that in a quantum gate the input state will be normalized (have length 1 1 1 ) , since that’s a requirement for a quantum state. And we’d expect the corresponding output state to also be normalized, otherwise it wouldn’t be a legitimate quantum state. Fortunately, the length-preserving property of unitary matrices ensures the output state is properly normalized.

Recall also the larger story this is part of: not only are gates unitary, to ensure that states remain normalized; also, quantum states are normalized, since measurement probabilities must sum to 1 1 1 . It all fits together.

In fact it turns out that unitary matrices are the only matrices which preserve length in this way. And so a good way of thinking about unitary matrices is that they’re exactly the class of matrices which preserve length. That’s the geometric interpretation (and intuitive meaning) of the algebraic condition U † U = I U^\dagger U = I U † U = I .

Let’s prove now that unitary matrices really are length-preserving.

The proof is straightforward. The main thing is to compute ∥ U ∣ ψ ⟩ ∥ \|U|\psi\rangle\| ∥ U ∣ ψ ⟩ ∥ . Actually, it’s a little easier to compute the square of the length, ∥ U ∣ ψ ⟩ ∥ 2 \|U|\psi\rangle\|^2 ∥ U ∣ ψ ⟩ ∥ 2 . This is just the sum of the squares of the absolute values of the components of the vector U ∣ ψ ⟩ U|\psi\rangle U ∣ ψ ⟩ :

∥ U ∣ ψ ⟩ ∥ 2 = ∑ j ( U ψ ) j ∗ ( U ψ ) j \|U|\psi\rangle\|^2 = \sum_j (U\psi)_j^* (U\psi)_j ∥ U ∣ ψ ⟩ ∥ 2 = j ∑ ​ ( U ψ ) j ∗ ​ ( U ψ ) j ​

Note that I’ve used ( U ψ ) j (U\psi)_j ( U ψ ) j ​ to denote the j j j t h component of U ∣ ψ ⟩ U|\psi\rangle U ∣ ψ ⟩ , dropping the full ket notation, and just using ψ \psi ψ alone. To proceed, we’ll expand the equation above out, and look for opportunities to apply the unitarity of U U U . In particular, the component ( U ψ ) j (U\psi)_j ( U ψ ) j ​ is given by ∑ l U j l ψ l \sum_l U_{jl} \psi_l ∑ l ​ U j l ​ ψ l ​ and similarly for the complex conjugate term. So we can rewrite the above equation as:

∥ U ∣ ψ ⟩ ∥ 2 = ∑ j k l U j k ∗ ψ k ∗ U j l ψ l \|U|\psi\rangle\|^2 = \sum_{jkl} U_{jk}^*\psi_k^* U_{jl}\psi_l ∥ U ∣ ψ ⟩ ∥ 2 = j k l ∑ ​ U j k ∗ ​ ψ k ∗ ​ U j l ​ ψ l ​

To make use of the unitarity of U U U we’ll move the U U U terms together, and rewrite in terms of U † U^\dagger U † . This gives

∥ U ∣ ψ ⟩ ∥ 2 = ∑ j k l U k j † U j l ψ k ∗ ψ l , \|U|\psi\rangle\|^2 = \sum_{jkl} U_{kj}^\dagger U_{jl} \psi_k^* \psi_l, ∥ U ∣ ψ ⟩ ∥ 2 = j k l ∑ ​ U k j † ​ U j l ​ ψ k ∗ ​ ψ l ​ ,

where we interchanged the j k jk j k indices on the first U U U in order to rewrite it in terms of U † U^\dagger U † . The only place the j j j index appears anywhere in this equation is in the U k j † U j l U_{kj}^\dagger U_{jl} U k j † ​ U j l ​ term. We can therefore perform the sum over j j j and those terms become ( U † U ) k l (U^\dagger U)_{kl} ( U † U ) k l ​ , which is just the k l kl k l t h term in the identity matrix, since U † U = I U^\dagger U = I U † U = I . So the above equation becomes:

∥ U ∣ ψ ⟩ ∥ 2 = ∑ k l δ k l ψ k ∗ ψ l . \|U|\psi\rangle\|^2 = \sum_{kl} \delta_{kl} \psi_k^* \psi_l. ∥ U ∣ ψ ⟩ ∥ 2 = k l ∑ ​ δ k l ​ ψ k ∗ ​ ψ l ​ .

When we sum, the only time δ k l \delta_{kl} δ k l ​ is not equal to zero is when k = l k=l k = l , in which case it’s 1 1 1 . And so we can get rid of one of the summation indices, and the equation simplifies to:

∥ U ∣ ψ ⟩ ∥ 2 = ∑ k ψ k ∗ ψ k . \|U|\psi\rangle\|^2 = \sum_{k} \psi_k^* \psi_k. ∥ U ∣ ψ ⟩ ∥ 2 = k ∑ ​ ψ k ∗ ​ ψ k ​ .

The right-hand side is, of course, equal to the norm of ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ squared. And so we’ve shown ∥ U ∣ ψ ⟩ ∥ 2 = ∥ ∣ ψ ⟩ ∥ 2 \|U|\psi\rangle\|^2 = \||\psi\rangle\|^2 ∥ U ∣ ψ ⟩ ∥ 2 = ∥ ∣ ψ ⟩ ∥ 2 , and therefore ∥ U ∣ ψ ⟩ ∥ = ∥ ∣ ψ ⟩ ∥ \|U|\psi\rangle\| = \||\psi\rangle\| ∥ U ∣ ψ ⟩ ∥ = ∥ ∣ ψ ⟩ ∥ . That completes the proof that unitary matrices are length-preserving. QED

In fact, as I mentioned earlier, it’s also possible to prove that unitary matrices are the only matrices which preserve lengths in this way. Let me state that a little more precisely:

Theorem: Let M M M be a matrix. Then ∥ M ∣ ψ ⟩ ∥ = ∥ ∣ ψ ⟩ ∥ \|M|\psi\rangle \| = \||\psi\rangle\| ∥ M ∣ ψ ⟩ ∥ = ∥ ∣ ψ ⟩ ∥ for all vectors ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ if and only if M M M is unitary.

We’ve proved one half of this already; we’ll prove the other half in the next section. This theorem answers many of our questions from earlier: we see why quantum gates must be unitary, since they’re the only matrices which preserve normalization. Of course, it doesn’t completely answer the question, since it doesn’t tell us why quantum gates should be matrices (i.e., linear operations) in the first place. That fact we’re simply going to accept. In fact, it is possible to develop deeper levels of understanding of why that is true, but in this essay we’ll be satisfied with the partial explanation that unitary matrices are length-preserving.

Why are unitaries the only matrices which preserve length?

Alright, let’s prove the missing part from the last section: let’s show that unitaries are the only matrices which preserve length.

The proof is a little messy. But it turns out to be a good way to get familiar with a few extra pieces of standard quantum mechanical terminology and notation. I’ll be frank: while these pieces of terminology are extremely useful in quantum computing, we don’t strictly need them elsewhere in this essay (though we will in later essays). If you want to skip the section, or skim it, that’s okay – this is the best section of the essay to skip. But at some point you should come back and work through the material. And there is, in any case, a certain beauty to the proof.

We’ve been dealing with 2-dimensional vectors up to now, but what I’m about to say applies no matter how many dimensions we’re working in. So suppose we have a vector ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ which can be written in component form as:

∣ ψ ⟩ = [ a b ⋮ z ] . |\psi\rangle = \left[ \begin{array}{c} a \\ b \\ \vdots \\ z \end{array} \right]. ∣ ψ ⟩ = ⎣ ⎢ ⎢ ⎢ ⎡ ​ a b ⋮ z ​ ⎦ ⎥ ⎥ ⎥ ⎤ ​ .

We’re going to define a new object, also labeled with a ψ \psi ψ , but now with a bracket in the other direction:

⟨ ψ ∣ : = [ a ∗ b ∗ … z ∗ ] . \langle \psi| := [a^* b^* \dots z^*]. ⟨ ψ ∣ : = [ a ∗ b ∗ … z ∗ ] .

That is, ⟨ ψ ∣ \langle\psi| ⟨ ψ ∣ is a row vector, whose entries are the same as ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ , but complex conjugated. The vector ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ was called a ket, and (you’re going to groan) ⟨ ψ ∣ \langle \psi| ⟨ ψ ∣ is called a bra , making this the bra-ket , or bracket notation. Yes, theoretical physicists make dad jokes, too. These names were given by the theoretical physicist Paul Dirac in 1939, and it’s often called the Dirac bra-ket notation, or sometimes just the Dirac notation.

A key fact about the bra ⟨ ψ ∣ \langle \psi| ⟨ ψ ∣ is that it’s related to the ket ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ by the dagger operation:

∣ ψ ⟩ † = ⟨ ψ ∣ . |\psi\rangle^\dagger = \langle \psi|. ∣ ψ ⟩ † = ⟨ ψ ∣ .

It’s easy to see why this identity is true. Take the vector

∣ ψ ⟩ = [ a b ⋮ z ] , |\psi\rangle = \left[ \begin{array}{c} a \\ b \\ \vdots \\ z \end{array} \right], ∣ ψ ⟩ = ⎣ ⎢ ⎢ ⎢ ⎡ ​ a b ⋮ z ​ ⎦ ⎥ ⎥ ⎥ ⎤ ​ ,

and apply the dagger operation, which means taking the transpose, turning it into a row vector with entries a , b , … , z a, b, \ldots, z a , b , … , z , and then take the complex conjugate, giving us [ a ∗ b ∗ … z ∗ ] [a^* b^* \ldots z^*] [ a ∗ b ∗ … z ∗ ] , which is just the definition of ⟨ ψ ∣ \langle \psi| ⟨ ψ ∣ .

In a similar way we see that ⟨ ψ ∣ † = ∣ ψ ⟩ \langle \psi|^\dagger = |\psi\rangle ⟨ ψ ∣ † = ∣ ψ ⟩ .

Another useful identity expresses the length of ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ in terms of the Dirac notation:

∥ ∣ ψ ⟩ ∥ 2 = ⟨ ψ ∣ ∣ ψ ⟩ . \||\psi\rangle\|^2 = \langle \psi| |\psi\rangle. ∥ ∣ ψ ⟩ ∥ 2 = ⟨ ψ ∣ ∣ ψ ⟩ .

That is, the length squared is just equal to the product of the row vector ⟨ ψ ∣ \langle \psi| ⟨ ψ ∣ with the column vector ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ . To prove it just notice that the right-hand side is

[ a ∗ b ∗ … ] [ a b ⋮ ] = ∣ a ∣ 2 + ∣ b ∣ 2 + … [a^* b^* \ldots ] \left[ \begin{array}{c} a \\ b \\ \vdots \end{array} \right] = |a|^2 + |b|^2 + \ldots [ a ∗ b ∗ … ] ⎣ ⎢ ⎡ ​ a b ⋮ ​ ⎦ ⎥ ⎤ ​ = ∣ a ∣ 2 + ∣ b ∣ 2 + …

And that, of course, is ∥ ∣ ψ ⟩ ∥ 2 \||\psi\rangle\|^2 ∥ ∣ ψ ⟩ ∥ 2 , just as we wanted.

Physicists using the Dirac notation don’t usually write ⟨ ψ ∣ ∣ ψ ⟩ \langle \psi| |\psi\rangle ⟨ ψ ∣ ∣ ψ ⟩ . They simplify it slightly, omitting one of the vertical bars ∣ | ∣ in the middle, and just write it as:

⟨ ψ ∣ ψ ⟩ . \langle \psi| \psi \rangle. ⟨ ψ ∣ ψ ⟩ .

It’s only a slight simplification, but this omission of the extra bar turns out to make life considerably easier, and is well worth it. I’ve shown it above just for the special case of ⟨ ψ ∣ ψ ⟩ \langle \psi| \psi \rangle ⟨ ψ ∣ ψ ⟩ but the same omission of a vertical bar is done often in other contexts. In practice, it rarely causes confusion, although of course you do need to get used to it.

Another useful identity is that if M M M is a matrix and ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ is a ket, then

( M ∣ ψ ⟩ ) † = ⟨ ψ ∣ M † . (M|\psi\rangle)^\dagger = \langle \psi|M^\dagger. ( M ∣ ψ ⟩ ) † = ⟨ ψ ∣ M † .

If you think of ⟨ ψ ∣ \langle \psi| ⟨ ψ ∣ as the dagger of ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ , then what this equation means is that taking the dagger of the product M ∣ ψ ⟩ M|\psi\rangle M ∣ ψ ⟩ is the same as taking the product of the dagger of M M M with the dagger ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ , but reversing the order. This is a useful little mnemonic for remembering the above identity.

The identity is easy to prove. As a challenge to yourself you might want to stop right now and take a shot at proving it. Here’s the details if you get stuck.

Proof: The way to prove the identity is to apply the definitions. We’re going to look at the j j j t h component of the left-hand side, ( M ∣ ψ ⟩ ) j † (M|\psi\rangle)^\dagger_j ( M ∣ ψ ⟩ ) j † ​ , and we’ll show it’s equal to the j j j t h component of the right-hand side. By definition, the j j j t h row component ( M ∣ ψ ⟩ ) j † (M|\psi\rangle)^\dagger_j ( M ∣ ψ ⟩ ) j † ​ is equal to the complex conjugate of the j j j t h column component of M ∣ ψ ⟩ M|\psi\rangle M ∣ ψ ⟩ , i.e., ( M ∣ ψ ⟩ ) j ∗ (M|\psi\rangle)^*_j ( M ∣ ψ ⟩ ) j ∗ ​ . That column component is ∑ k M j k ∗ ψ k ∗ \sum_k M_{jk}^* \psi_k^* ∑ k ​ M j k ∗ ​ ψ k ∗ ​ . We can move the ψ \psi ψ terms to the left, and swap the indices on the M M M term to convert the ∗ * ∗ to a dagger, giving ∑ k ψ k ∗ M k j † \sum_k \psi_k^* M_{kj}^\dagger ∑ k ​ ψ k ∗ ​ M k j † ​ . That’s just the j j j t h component of the row vector ⟨ ψ ∣ M † \langle \psi|M^\dagger ⟨ ψ ∣ M † , as we set out to show. QED

With all these ideas in mind, here’s an exercise for you to work through, putting several of these ideas together:

Exercise: Show that for any matrix M M M and vector ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ , the following identity holds, expressing the length of M ∣ ψ ⟩ M|\psi\rangle M ∣ ψ ⟩ :

∥ M ∣ ψ ⟩ ∥ 2 = ⟨ ψ ∣ M † M ∣ ψ ⟩ . \|M|\psi\rangle\|^2 = \langle\psi|M^\dagger M |\psi\rangle. ∥ M ∣ ψ ⟩ ∥ 2 = ⟨ ψ ∣ M † M ∣ ψ ⟩ .

You may be wondering why we care about all these identities? Of course, really I’m just frontloading them – they’re all identities we’re going to find useful in a moment in the proof. So in some sense they are a bit ad hoc in motivation. But an underlying theme is that they’re about relating lengths to matrix operations. And it’s not so surprising that’s of interest – we’re going to assume we have a length-preserving operation. It’s very convenient to be able to relate that property to familiar operations about matrix multiplication. That’s the essential source of the interest in the above identities.

Alright, we’ve been working through some heavy, detailed material. We’ve got just a little more background to get through, but it’s easier going. I’m going to introduce a unit vector, denoted ∣ e j ⟩ |e_j\rangle ∣ e j ​ ⟩ , meaning the vector with a 1 1 1 in the j j j t h component, and 0 0 0 s everywhere else. So, for instance, for a qubit:

∣ e 0 ⟩ = [ 1 0 ] ∣ e 1 ⟩ = [ 0 1 ] \begin{aligned} |e_0\rangle & = \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] \\ |e_1\rangle & = \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] \end{aligned} ∣ e 0 ​ ⟩ ∣ e 1 ​ ⟩ ​ = [ 1 0 ​ ] = [ 0 1 ​ ] ​

From elementary linear algebra, if M M M is a matrix, then M ∣ e k ⟩ M|e_k\rangle M ∣ e k ​ ⟩ is just the k k k t h column of M M M . (If you don’t recall that from elementary linear algebra, I encourage you to stop and figure out why it’s true.) And from that you can deduce easily that ⟨ e j ∣ M ∣ e k ⟩ \langle e_j|M|e_k\rangle ⟨ e j ​ ∣ M ∣ e k ​ ⟩ is the j k jk j k t h entry of M M M .

Exercise: If you’re not familiar with the proof, show that M ∣ e k ⟩ M|e_k\rangle M ∣ e k ​ ⟩ is the k k k t h column of the matrix M M M , and that ⟨ e j ∣ M ∣ e k ⟩ \langle e_j|M|e_k\rangle ⟨ e j ​ ∣ M ∣ e k ​ ⟩ is the j k jk j k t h entry of M M M .

Alright, that’s more than enough notational background! Let’s get to the main event. In particular, let’s recall the statement of the theorem we want to complete the proof of. Also recall that we proved the reverse implication in the last section, so we just need to prove the forward implication:

Theorem: Let M M M be a matrix. Then ∥ M ∣ ψ ⟩ ∥ = ∥ ∣ ψ ⟩ ∥ \|M|\psi\rangle \| = \| |\psi\rangle\| ∥ M ∣ ψ ⟩ ∥ = ∥ ∣ ψ ⟩ ∥ for all vectors ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ if and only if M M M is unitary.

Proof: We’ll assume M M M is length-preserving, and analyze the matrix M † M M^\dagger M M † M . Our goal is to show that this is the identity matrix, and thus M M M is unitary. To do this, we’re going to start by analyzing the diagonal elements ( M † M ) j j (M^\dagger M)_{jj} ( M † M ) j j ​ , and show that they’re all equal to 1 1 1 . Then we’ll turn our attention to the off-diagonal elements and show that they’re all equal to 0 0 0 .

To understand the diagonal elements, we know from earlier that: ( M † M ) j j = ⟨ e j ∣ M † M ∣ e j ⟩ = ∥ M ∣ e j ⟩ ∥ 2 (M^\dagger M)_{jj} = \langle e_j|M^\dagger M|e_j\rangle = \|M|e_j\rangle\|^2 ( M † M ) j j ​ = ⟨ e j ​ ∣ M † M ∣ e j ​ ⟩ = ∥ M ∣ e j ​ ⟩ ∥ 2 . But since M M M is length-preserving, the latter is just ∥ ∣ e j ⟩ ∥ 2 \||e_j\rangle\|^2 ∥ ∣ e j ​ ⟩ ∥ 2 , which is 1 1 1 . And so we conclude that all the diagonal elements are, indeed, 1 1 1 .

What about the off-diagonal elements, i.e., ( M † M ) j k (M^\dagger M)_{jk} ( M † M ) j k ​ where j ≠ k j \neq k j  ​ = k ? Can we show that these are all equal to 0 0 0 ? Well, what we’d like to do is somehow to relate ( M † M ) j k (M^\dagger M)_{jk} ( M † M ) j k ​ to the length of some vector M ∣ ψ ⟩ M|\psi\rangle M ∣ ψ ⟩ , and then use the length-preserving property. One idea is to try using ∣ ψ ⟩ = ∣ e j ⟩ + ∣ e k ⟩ |\psi\rangle = |e_j\rangle + |e_k\rangle ∣ ψ ⟩ = ∣ e j ​ ⟩ + ∣ e k ​ ⟩ , since that involves both the j j j t h and k k k t h directions. From the length-preserving property we have:

∥ M ∣ ψ ⟩ ∥ 2 = ∥ ∣ ψ ⟩ ∥ 2 = 1 2 + 1 2 = 2. \|M|\psi\rangle\|^2 = \| |\psi\rangle\|^2 = 1^2+1^2 = 2. ∥ M ∣ ψ ⟩ ∥ 2 = ∥ ∣ ψ ⟩ ∥ 2 = 1 2 + 1 2 = 2 .

We also have:

∥ M ∣ ψ ⟩ ∥ 2 = ⟨ ψ ∣ M † M ∣ ψ ⟩ = ⟨ e j ∣ M † M ∣ e j ⟩ + ⟨ e j ∣ M † M ∣ e k ⟩ = = + ⟨ e k ∣ M † M ∣ e j ⟩ + ⟨ e k ∣ M † M ∣ e k ⟩ = 1 + ⟨ e j ∣ M † M ∣ e k ⟩ + ⟨ e k ∣ M † M ∣ e j ⟩ + 1 \begin{aligned} \|M|\psi\rangle\|^2 & = \langle \psi| M^\dagger M |\psi\rangle \\ & = \langle e_j|M^\dagger M|e_j\rangle + \langle e_j|M^\dagger M|e_k\rangle \\ & \hphantom{ == } + \langle e_k|M^\dagger M|e_j\rangle + \langle e_k|M^\dagger M |e_k\rangle \\ & = 1 + \langle e_j|M^\dagger M|e_k\rangle + \langle e_k|M^\dagger M|e_j\rangle + 1 \end{aligned} ∥ M ∣ ψ ⟩ ∥ 2 ​ = ⟨ ψ ∣ M † M ∣ ψ ⟩ = ⟨ e j ​ ∣ M † M ∣ e j ​ ⟩ + ⟨ e j ​ ∣ M † M ∣ e k ​ ⟩ = = + ⟨ e k ​ ∣ M † M ∣ e j ​ ⟩ + ⟨ e k ​ ∣ M † M ∣ e k ​ ⟩ = 1 + ⟨ e j ​ ∣ M † M ∣ e k ​ ⟩ + ⟨ e k ​ ∣ M † M ∣ e j ​ ⟩ + 1 ​

Comparing the last two sets of equations, we have:

⟨ e j ∣ M † M ∣ e k ⟩ + ⟨ e k ∣ M † M ∣ e j ⟩ = 0. \langle e_j|M^\dagger M|e_k\rangle + \langle e_k|M^\dagger M|e_j\rangle = 0. ⟨ e j ​ ∣ M † M ∣ e k ​ ⟩ + ⟨ e k ​ ∣ M † M ∣ e j ​ ⟩ = 0 .

This is close to what we want, but isn’t quite right. It tells us that ( M † M ) j k + ( M † M ) k j = 0 (M^\dagger M)_{jk} + (M^\dagger M)_{kj} = 0 ( M † M ) j k ​ + ( M † M ) k j ​ = 0 . Can we do better? It’s tempting to go back and fiddle around and try to find some way of eliminating one or the other of those terms. But there’s no direct way to do it – at least, no direct way that I know of.

But what if we’d done something slightly different, and instead of using ∣ ψ ⟩ = ∣ e j ⟩ + ∣ e k ⟩ |\psi\rangle = |e_j\rangle+|e_k\rangle ∣ ψ ⟩ = ∣ e j ​ ⟩ + ∣ e k ​ ⟩ we’d used ∣ ψ ⟩ = ∣ e j ⟩ + i ∣ e k ⟩ |\psi\rangle = |e_j\rangle+i|e_k\rangle ∣ ψ ⟩ = ∣ e j ​ ⟩ + i ∣ e k ​ ⟩ ? It seems pretty plausible that following the same line of reasoning we’d get an equation involving ( M † M ) j k (M^\dagger M)_{jk} ( M † M ) j k ​ and ( M † M ) k j (M^\dagger M)_{kj} ( M † M ) k j ​ again. I won’t explicitly go through the steps – you can do that yourself – but if you do go through them you end up with the equation:

( M † M ) j k − ( M † M ) k j = 0. (M^\dagger M)_{jk} - (M^\dagger M)_{kj} = 0. ( M † M ) j k ​ − ( M † M ) k j ​ = 0 .

This is great! We can add it to the earlier equation to deduce that ( M † M ) j k = 0 (M^\dagger M)_{jk} = 0 ( M † M ) j k ​ = 0 whenever j ≠ k j \neq k j  ​ = k , and so we conclude that M † M = I M^\dagger M = I M † M = I , i.e., M M M is unitary. QED

The controlled-NOT gate

We’ve developed most of the ideas needed to do universal quantum computing. We understand qubits, quantum states, and have a repertoire of quantum gates. However, all our gates involve just a single qubit. To compute, we need some way for qubits to interact with one another. That is, we need quantum gates which involve two (or more) qubits.

An example of such a gate is the controlled-NOT (or CNOT) gate. In the quantum circuit language we have two wires, representing two qubits, and the following notation to represent the CNOT gate:

The wire with the small, filled dot on it (the top wire, in this example) is called the control qubit, for reasons which will become clear in a moment. And the wire with the larger, unfilled circle on it is called the target qubit.

Up to now I haven’t said what the possible states of a two-qubit system are, but you can probably guess. We now have four computational basis states, corresponding to the four possible states of a two-bit system: ∣ 00 ⟩ , ∣ 01 ⟩ , ∣ 10 ⟩ |00\rangle, |01\rangle, |10\rangle ∣ 0 0 ⟩ , ∣ 0 1 ⟩ , ∣ 1 0 ⟩ , and ∣ 11 ⟩ |11\rangle ∣ 1 1 ⟩ . And, for a two-qubit system, not only can we have those four states, we can also have superpositions (i.e., linear combinations) of them:

α ∣ 00 ⟩ + β ∣ 01 ⟩ + γ ∣ 10 ⟩ + δ ∣ 11 ⟩ \alpha|00\rangle+\beta|01\rangle+\gamma |10\rangle+\delta|11\rangle α ∣ 0 0 ⟩ + β ∣ 0 1 ⟩ + γ ∣ 1 0 ⟩ + δ ∣ 1 1 ⟩

Here, the amplitudes α , β , γ , δ \alpha, \beta, \gamma, \delta α , β , γ , δ are just complex numbers, and the sum of the squares of the absolute values is 1 1 1 , i.e, ∣ α ∣ 2 + ∣ β ∣ 2 + ∣ γ ∣ 2 + ∣ δ ∣ 2 = 1 |\alpha|^2+|\beta|^2+|\gamma|^2+|\delta|^2 = 1 ∣ α ∣ 2 + ∣ β ∣ 2 + ∣ γ ∣ 2 + ∣ δ ∣ 2 = 1 . This is the same kind of normalization condition as we had for a single qubit.

Now, the CNOT gate is, much like the quantum NOT gate, inspired directly by a classical gate. What it does is very simple. If the control qubit is set to 1 1 1 , as in the states ∣ 10 ⟩ |10\rangle ∣ 1 0 ⟩ and ∣ 11 ⟩ |11\rangle ∣ 1 1 ⟩ , then it flips (i.e., NOTs) the target qubit. And otherwise it does nothing. Writing out the action on all four computational basis states we have:

∣ 00 ⟩ → ∣ 00 ⟩ ∣ 01 ⟩ → ∣ 01 ⟩ ∣ 10 ⟩ → ∣ 11 ⟩ ∣ 11 ⟩ → ∣ 10 ⟩ \begin{aligned} |00\rangle & \rightarrow & |00\rangle \\ |01\rangle & \rightarrow & |01\rangle \\ |10\rangle & \rightarrow & |11\rangle \\ |11\rangle & \rightarrow & |10\rangle \end{aligned} ∣ 0 0 ⟩ ∣ 0 1 ⟩ ∣ 1 0 ⟩ ∣ 1 1 ⟩ ​ → → → → ​ ∣ 0 0 ⟩ ∣ 0 1 ⟩ ∣ 1 1 ⟩ ∣ 1 0 ⟩ ​

If you’re familiar with classical programming languages, then you can think of the CNOT as a very simple kind of if-then statement: if the control qubit is set, then NOT the target qubit. But while simple, it can be used as a building block to build up other, more complex kinds of conditional behavior.

There’s a way of summing up all four of the equations above in a single equation. Suppose x x x and y y y are classical bits, i.e., 0 0 0 or 1 1 1 . Then we can rewrite the equations above in a single equation as:

∣ x , y ⟩ → ∣ x , y ⊕ x ⟩ . \begin{aligned} |x, y\rangle & \rightarrow & |x, y\oplus x\rangle. \end{aligned} ∣ x , y ⟩ ​ → ​ ∣ x , y ⊕ x ⟩ . ​

Note the commas inserted to make this easier to read – this is pretty common in working with multi-qubit states.

The above equation makes clear that the CNOT leaves the control qubit x x x alone, but flips the target qubit y y y if x x x is set to 1 1 1 . Note that ⊕ \oplus ⊕ is addition modulo 2 2 2 , where 1 ⊕ 1 = 0 1 \oplus 1 = 0 1 ⊕ 1 = 0 , as we would expect from the fact that the CNOT takes ∣ 11 ⟩ |11\rangle ∣ 1 1 ⟩ to ∣ 10 ⟩ |10\rangle ∣ 1 0 ⟩ .

That’s all there is to the CNOT. It’s really a very simple idea and quantum gate. Note that it of course acts linearly on superpositions of computational basis states, as we expect for a quantum gate. So:

→ α ∣ 00 ⟩ + β ∣ 01 ⟩ + γ ∣ 10 ⟩ + δ ∣ 11 ⟩ → α ∣ 00 ⟩ + β ∣ 01 ⟩ + γ ∣ 11 ⟩ + δ ∣ 10 ⟩ \begin{aligned} & \hphantom{ \rightarrow } \alpha|00\rangle+\beta|01\rangle+\gamma |10\rangle+\delta|11\rangle \\ & \rightarrow \alpha|00\rangle+\beta|01\rangle+\gamma |11\rangle+\delta|10\rangle \end{aligned} ​ → α ∣ 0 0 ⟩ + β ∣ 0 1 ⟩ + γ ∣ 1 0 ⟩ + δ ∣ 1 1 ⟩ → α ∣ 0 0 ⟩ + β ∣ 0 1 ⟩ + γ ∣ 1 1 ⟩ + δ ∣ 1 0 ⟩ ​

And, though I won’t explicitly carry out the verification, the CNOT is unitary, and thus preserves the length of quantum states, as we expect.

Of course, the CNOT doesn’t just appear in two-qubit computations. It also appears in computations involving more qubits. Let’s suppose we have three qubits, for instance, and computational basis states such as ∣ 000 ⟩ , ∣ 001 ⟩ |000\rangle, |001\rangle ∣ 0 0 0 ⟩ , ∣ 0 0 1 ⟩ , and so on. Here’s a CNOT with the second qubit as the control qubit and the third qubit as the target:

What goes on? Well, we can write out what happens on an arbitrary computational basis state, ∣ x , y , z ⟩ |x, y, z\rangle ∣ x , y , z ⟩ , where x , y x, y x , y and z z z are all classical bits. Of course, the first bit x x x isn’t changed at all, since it’s not involved in the CNOT. The second bit y y y is the control bit, and so isn’t changed. But the third bit z z z is flipped if the control bit y y y is set to 1 1 1 . And so we can write the action of the CNOT as:

∣ x , y , z ⟩ → ∣ x , y , z ⊕ y ⟩ |x,y,z\rangle \rightarrow |x,y, z\oplus y\rangle ∣ x , y , z ⟩ → ∣ x , y , z ⊕ y ⟩

I’ve described the CNOT as a “classical” gate, but it can be combined with single-qubit gates to do non-classical things. Let me give you an explicit example. It’s another two-qubit computation. It starts with the ∣ 00 ⟩ |00\rangle ∣ 0 0 ⟩ computational basis state, we apply a Hadamard gate to the first qubit, and then do a CNOT:

Recall that for a single qubit the Hadamard gate takes ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ to an equal superposition ( ∣ 0 ⟩ + ∣ 1 ⟩ ) / 2 (|0\rangle+|1\rangle)/\sqrt{2} ( ∣ 0 ⟩ + ∣ 1 ⟩ ) / 2 ​ . For these two qubits it doesn’t affect the second qubit at all, and so it takes ∣ 00 ⟩ |00\rangle ∣ 0 0 ⟩ to ( ∣ 00 ⟩ + ∣ 10 ⟩ ) / 2 (|00\rangle+|10\rangle)/\sqrt{2} ( ∣ 0 0 ⟩ + ∣ 1 0 ⟩ ) / 2 ​ .

Next we apply the CNOT gate. This leaves the ∣ 00 ⟩ |00\rangle ∣ 0 0 ⟩ state unchanged, since the control bit is 0 0 0 . And it takes ∣ 10 ⟩ |10\rangle ∣ 1 0 ⟩ to ∣ 11 ⟩ |11\rangle ∣ 1 1 ⟩ , since the control bit is 1 1 1 . And so the output from the circuit is:

∣ 00 ⟩ + ∣ 11 ⟩ 2 . \frac{|00\rangle+|11\rangle}{\sqrt 2}. 2 ​ ∣ 0 0 ⟩ + ∣ 1 1 ⟩ ​ .

This output state is a highly non-classical state – it’s actually a type of state called an entangled state . There’s no obvious interpretation of this state as a classical state, unlike say a computational basis state such as ∣ 00 ⟩ |00\rangle ∣ 0 0 ⟩ . In fact, entangled states can be used to do all sorts of interesting information processing tasks, including quantum teleportation and fast quantum algorithms.

A point I glossed over above, but worth mentioning: in the circuit I drew ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ and ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ separately as input qubits. It’s conventional to do that kind of thing to denote a combined input of ∣ 00 ⟩ |00\rangle ∣ 0 0 ⟩ . More generally, people use ∣ 0 ⟩ ∣ 0 ⟩ |0\rangle|0\rangle ∣ 0 ⟩ ∣ 0 ⟩ interchangeably with ∣ 00 ⟩ |00\rangle ∣ 0 0 ⟩ , ∣ 0 ⟩ ∣ 1 ⟩ |0\rangle|1\rangle ∣ 0 ⟩ ∣ 1 ⟩ interchangeably with ∣ 01 ⟩ |01\rangle ∣ 0 1 ⟩ , and so on. Going back and forth takes a bit of getting used to, but everything works pretty much as you expect, and you just need a little practice before it seems quite natural.

More generally, if we have single-qubit states α ∣ 0 ⟩ + β ∣ 1 ⟩ \alpha|0\rangle+\beta|1\rangle α ∣ 0 ⟩ + β ∣ 1 ⟩ and γ ∣ 0 ⟩ + δ ∣ 1 ⟩ \gamma|0\rangle+\delta|1\rangle γ ∣ 0 ⟩ + δ ∣ 1 ⟩ , then the combined state when the two qubits are put together is just:

= ( α ∣ 0 ⟩ + β ∣ 1 ⟩ ) ( γ ∣ 0 ⟩ + δ ∣ 1 ⟩ ) = α γ ∣ 00 ⟩ + α δ ∣ 01 ⟩ + β γ ∣ 10 ⟩ + β δ ∣ 11 ⟩ . \begin{aligned} & \hphantom{ = } (\alpha|0\rangle+\beta|1\rangle)(\gamma|0\rangle+\delta|1\rangle) \\ & = \alpha\gamma|00\rangle+\alpha\delta |01\rangle+ \beta\gamma |10\rangle+ \beta\delta |11\rangle. \end{aligned} ​ = ( α ∣ 0 ⟩ + β ∣ 1 ⟩ ) ( γ ∣ 0 ⟩ + δ ∣ 1 ⟩ ) = α γ ∣ 0 0 ⟩ + α δ ∣ 0 1 ⟩ + β γ ∣ 1 0 ⟩ + β δ ∣ 1 1 ⟩ . ​

I said that the CNOT leaves the control qubit alone, and modifies the target qubit. That’s true in the computational basis. In fact, it’s actually possible for the target qubit to affect the control qubit. It’s worth taking a minute or two to at least understand (and, if you’re feeling energetic, attempting to solve) the following exercise:

Exercise: Can you find single-qubit states ∣ a ⟩ |a\rangle ∣ a ⟩ and ∣ b ⟩ |b\rangle ∣ b ⟩ so that applying a CNOT to the combined state ∣ a b ⟩ |ab\rangle ∣ a b ⟩ changes the first qubit, i.e., the control qubit?

Let me give you an example which solves the above exercise. Suppose we introduce single-qubit states ∣ + ⟩ |+\rangle ∣ + ⟩ and ∣ − ⟩ |-\rangle ∣ − ⟩ , defined by:

∣ + ⟩ : = ∣ 0 ⟩ + ∣ 1 ⟩ 2 ∣ − ⟩ : = ∣ 0 ⟩ − ∣ 1 ⟩ 2 \begin{aligned} |+\rangle & := \frac{|0\rangle+|1\rangle}{\sqrt 2} \\ |-\rangle & := \frac{|0\rangle-|1\rangle}{\sqrt 2} \end{aligned} ∣ + ⟩ ∣ − ⟩ ​ : = 2 ​ ∣ 0 ⟩ + ∣ 1 ⟩ ​ : = 2 ​ ∣ 0 ⟩ − ∣ 1 ⟩ ​ ​

A mnemonic for this notation is that these are both equal superpositions of ∣ 0 ⟩ |0\rangle ∣ 0 ⟩ and ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ , but the “+” or “-” corresponds to the sign of the amplitude for the ∣ 1 ⟩ |1\rangle ∣ 1 ⟩ state. The following circuit identity holds:

That is, if we put ∣ + − ⟩ |+-\rangle ∣ + − ⟩ in, then ∣ − − ⟩ |--\rangle ∣ − − ⟩ comes out, i.e., the target qubit isn’t changed, but the control qubit is! The proof is just to follow the definitions and the algebra through. It’s honestly not terribly enlightening to follow the algebra through at this point – it’s a lot easier once certain other facts are known to you. But, again, if you’re feeling enthusiastic it’s a good exercise to work through to become familiar with how things work.

It’s worth taking some caution from this example. It means that intuitions coming from the computational basis can sometimes be incomplete or misleading. The CNOT isn’t simply doing something to the target, conditional on the control. It’s also doing something to the control, conditional on the target.

Exercise: Show that the inverse of the CNOT gate is just the CNOT gate.

In natural science, Nature has given us a world and we’re just to discover its laws. In computers, we can stuff laws into it and create a world. — Alan Kay

Let’s return to the question we began with: is there a single computing device that can efficiently simulate any other physical system? At the moment, the best candidate humanity has for such a computing device is a quantum computer. As you’ve probably guessed, you make such a device by combining all the elements we’ve been discussing. In Part III of this essay we’ll discuss what a quantum computer is, why they’re useful, and whether they can be used to efficiently simulate any other physical system.

The quantum computing model

The theory of computation has traditionally been studied almost entirely in the abstract, as a topic in pure mathematics. This is to miss the point of it. Computers are physical objects, and computations are physical processes. What computers can or cannot compute is determined by the laws of physics alone, and not by pure mathematics. — David Deutsch

In a general quantum computation, you start out with many qubits – I’ll draw four here, but in general it might be many more (or less). You apply quantum gates of various kinds, in particular, single-qubit gates and CNOT gates. And at the end of the circuit you read out the result by measuring in the computational basis. Here’s what it all looks like:

I haven’t labeled the single-qubit gates explicitly, but they might be various things – Hadamard gates, NOT gates, rotations, and perhaps others. Note also that while the computation starts in the computational basis state ∣ 0000 ⟩ |0000\rangle ∣ 0 0 0 0 ⟩ , you can also start in some other computational basis state. I’ve just chosen ∣ 0000 ⟩ |0000\rangle ∣ 0 0 0 0 ⟩ for definiteness.

We can summarize the three steps in a quantum computation as follows:

• Start in a computational basis state.
• Apply a sequence of CNOT and single-qubit gates.
• To obtain the result, measure in the computational basis. The probability of any result, say 00 … 0 00\ldots 0 0 0 … 0 , is just the square of the absolute value of the corresponding amplitude.

That’s all a general quantum computation is! If you understand this model, you know what a quantum computer is. It's pretty simple, really – enough so that I've sometimes heard people say “Is that all there is to it?” But while the model is simple, it contains remarkable depths, and exploring it could occupy many lifetimes.

In practice, people sometimes introduce other ideas into the way they describe quantum computations. If you’re a programmer, you can think of this as like the way programming language designers introduce higher-level abstractions to help people design different kinds of programs. In principle, those abstractions can always be reduced down to the level of AND and NOT gates. And if you understand that level – AND and NOT, or the type of quantum circuit shown above – then you have a foundation for building an understanding of the other ideas.

That may leave you wondering: does that mean you also need to master all the higher-level abstractions? The answer is no!

There are several reasons for this. One reason is that, as we discussed earlier, humanity doesn’t yet know what the higher-level abstractions are. We’re still trying to figure them out. A second reason is that it seems likely that the list of higher-level abstractions is inexhaustible. My guess – and it’s just a guess – is that we will continue to discover beautiful new abstractions forever , in both classical and quantum computing.

A related idea is that there are models of quantum computation different to the quantum circuit model. Some are merely small variations on the model I’ve described. For instance, instead of only using CNOT gates, we might allow any two-qubit unitary gate to be used in the circuit. Or perhaps instead of using qubits, we might use some other type of basic quantum system – say, the qutrit , which has three computational basis states, ∣ 0 ⟩ , ∣ 1 ⟩ |0\rangle, |1\rangle ∣ 0 ⟩ , ∣ 1 ⟩ , and ∣ 2 ⟩ |2\rangle ∣ 2 ⟩ . It probably won’t surprise you that the resulting models of computation are essentially equivalent to the quantum circuit model I’ve described. By this, I mean they can simulate the quantum circuit model (and vice versa) using roughly comparable numbers of gates and other physical resources.

There are also much more exotic variations, ideas such as measurement-based quantum computation, topological quantum computation, and others. I won’t describe these in any detail here, but suffice to say that they appear superficially very different to the circuit model. Nonetheless, they’re all mathematically equivalent to one another, including to the quantum circuit model. Thus a quantum computation in any of those models can be translated into an equivalent in the quantum circuit model, with only a small overhead in the cost of computation. And vice versa.

You may wonder why people bother thinking about other models, if they’re mathematically equivalent to the quantum circuit model. The reason is that just because two models are mathematically equivalent doesn’t mean they’re psychologically equivalent. Different models of computation stimulate different ways of thinking, and give rise to different ideas. And so it’s valuable to have other equivalent models.

Exercise: We saw earlier that composition of quantum gates corresponds to matrix multiplication (in reverse order). Show that the product of two unitary matrices U U U and V V V is also unitary. As a consequence, the net effect of any quantum circuit (before measurement) is to effect a unitary operation on the state space of the system.

Let me conclude this section with a brief comment about a particular class of quantum gates. These are gates which are multiples of the identity matrix, I I I :

[ e i θ 0 0 e i θ ] = e i θ [ 1 0 0 1 ] = e i θ I . \begin{aligned} \left[ \begin{array}{cc} e^{i\theta} & 0 \\ 0 & e^{i\theta} \end{array} \right] = e^{i\theta} \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right] = e^{i \theta} I. \end{aligned} [ e i θ 0 ​ 0 e i θ ​ ] = e i θ [ 1 0 ​ 0 1 ​ ] = e i θ I . ​

When θ \theta θ is a real number this is a unitary matrix, and so a valid quantum gate. The effect of the gate is simply to multiply the state of the quantum computer by e i θ e^{i\theta} e i θ . The number e i θ e^{i\theta} e i θ is called a global phase factor . But even though the gate is valid, we rarely explicitly put such gates into quantum circuits. The reason is that such global phase factors have no impact on the results of the computation. To see why, imagine a quantum circuit which includes many such quantum gates, possibly for different values, θ 1 , θ 2 , … \theta_1, \theta_2, \ldots θ 1 ​ , θ 2 ​ , … . No matter where in the circuit the gates occur, the net effect is to multiply the state output from the circuit by an overall factor of e i θ 1 + i θ 2 + … e^{i \theta_1 + i\theta_2 + \ldots} e i θ 1 ​ + i θ 2 ​ + … . This doesn’t change the squared amplitudes of the computational basis states, and so has no impact on the measurement probabilities at the end of the computation. We could have left the gates out and it would make no difference.

Quantum computing people tend to be rather blase about such global phase factors. They’ll do things like not bother to distinguish between unitary gates such as X X X and − X -X − X , saying these gates are “the same up to a global phase factor”. They simply mean the − X -X − X gate is the same as doing the X X X gate, followed by the − I -I − I gate. Since the latter gate makes no difference to the output from the computation, it can safely be omitted. We’ll see examples in the next essay, about the quantum search algorithm, where at a couple of places it makes sense to multiply the quantum state by a global phase factor of − 1 -1 − 1 .

What are quantum computers good for?

Now we know what a quantum computer is, what are they good for? Earlier, I suggested quantum computers expand the range of operations available when computing. This is similar to the way boats expand the range of ways we can traverse space, so instead of having to get from point A to point B by land,

we can take a shortcut greatly cutting down the time required:

In order for the analogous story to hold for quantum computers, they need to be at least as capable as classical computers. Fortunately, it’s possible to convert any classical circuit into a quantum circuit. This requires some care. The obvious thing to do is to imagine that your classical circuit is expressed in terms of some standard universal set – say, the AND and NOT gates – and then to convert those gates into equivalent quantum gates.

This is easy to do with the NOT gate – we just turn it into an X X X gate. But what’s not so easy is the AND gate. You might wonder if there’s some quantum gate which takes two bits x x x and y y y as input, in the computational basis, ∣ x , y ⟩ |x,y\rangle ∣ x , y ⟩ , and then outputs a single qubit ∣ x ∧ y ⟩ |x \wedge y\rangle ∣ x ∧ y ⟩ , where x ∧ y x \wedge y x ∧ y is just the logical AND of the bits x x x and y y y .

Unfortunately, that “quantum gate” makes no sense at all! Not only is it not unitary, it’s not even close: a unitary gate with two qubits as input necessarily has two qubits as output. The “gate” I described has just a single qubit as output, so there’s no way it can be unitary.

You might wonder if instead there’s some way we can find a two-qubit quantum gate which has x ∧ y x \wedge y x ∧ y as one output, and something else as the other output. I won’t prove it, but it turns out that this is impossible. The proof actually isn’t all that hard – it’s a fun exercise to think through, if you want a challenge – but is more of a digression than I want to get into here.

That’s all rather disappointing. But there is a solution. It’s to use a three-qubit quantum gate called the Toffoli gate . The Toffoli gate is much like the CNOT gate, but instead of having a single control qubit, it has two control qubits, x x x and y y y , and a single target qubit, z z z . If both controls qubits are set, then the target is flipped. Otherwise, the target is left alone:

If the target starts out as z = 0 z = 0 z = 0 , then you can see that the target output is just x ∧ y x \wedge y x ∧ y , and so the Toffoli gate can be used to simulate a classical AND gate. So if we have any classical circuit of AND and NOT gates then there’s a corresponding quantum circuit involving the same number of X X X and Toffoli gates which computes the same function.

Exercise: What’s a quantum circuit that can compute the NAND gate? Recall that the NAND of two bits x x x and y y y is just the NOT of x ∧ y x \wedge y x ∧ y .

Exercise: Can you find a way of implementing a NAND gate using just a single Toffoli gate and no other quantum gates? Note that your answer here may be the same as your answer to the previous exercise, if you answered that exercise using just a single Toffoli gate and no other quantum gates.

A wrinkle in all this is that the Toffoli gate isn’t in our standard set of basic quantum gates. However, it’s possible to build the Toffoli gate up out of CNOT and single-qubit unitary gates. One way of doing the breakdown is shown below:

There are various slick ways of “explaining” why this circuit works, but I’ll let you in on a secret: much of the earliest work on this was done by pure brute force, people simply trying lots and lots of different ways of implementing the Toffoli gate (sometimes, using a computer to assist in doing the search). Frankly, I wouldn’t worry too much about why this circuit works, just take it for granted that it does. You can look it up if you ever need to, or dig down into why. Needing to know the circuit details actually isn’t all that common, so I wouldn’t suggest memorizing it – not until you have a good reason.

Exercise: Show that the inverse of the Toffoli gate is just the Toffoli gate.

No, really, what are quantum computers good for?

It’s comforting that we can always simulate a classical circuit – it means quantum computers aren’t slower than classical computers – but doesn’t answer the question of the last section: what problems are quantum computers good for? Can we find shortcuts that make them systematically faster than classical computers? It turns out there’s no general way known to do that. But there are some interesting classes of computation where quantum computers outperform classical.

Over the long term, I believe the most important use of quantum computers will be simulating other quantum systems. That may sound esoteric – why would anyone apart from a quantum physicist care about simulating quantum systems? But everybody in the future will (or, at least, will care about the consequences). The world is made up of quantum systems. Pharmaceutical companies employ thousands of chemists who synthesize molecules and characterize their properties. This is currently a very slow and painstaking process. In an ideal world they’d get the same information thousands or millions of times faster, by doing highly accurate computer simulations. And they’d get much more useful information, answering questions chemists can’t possibly hope to answer today. Unfortunately, classical computers are terrible at simulating quantum systems.

The reason classical computers are bad at simulating quantum systems isn’t difficult to understand. Suppose we have a molecule containing n n n atoms – for a small molecule, n n n may be 1 1 1 - 10 10 1 0 , for a complex molecule it may be hundreds or thousands or even more. And suppose we think of each atom as a qubit (not true, but go with it): to describe the system we’d need 2 n 2^n 2 n different amplitudes, one amplitude for each n n n - bit computational basis state, e.g., ∣ 010011 … ⟩ |010011\ldots\rangle ∣ 0 1 0 0 1 1 … ⟩ .

Of course, atoms aren’t qubits. They’re more complicated, and we need more amplitudes to describe them. Without getting into details, the rough scaling for an n n n - atom molecule is that we need k n k^n k n amplitudes, where k ≥ 2 k \geq 2 k ≥ 2 . The value of k k k depends upon context – which aspects of the atom’s behavior are important. For generic quantum simulations k k k may be in the hundreds or more.

That’s a lot of amplitudes! Even for comparatively simple atoms and small values of n n n , it means the number of amplitudes will be in the trillions. And it rises very rapidly, doubling or more for each extra atom. If k = 100 k=100 k = 1 0 0 , then even n = 10 n = 10 n = 1 0 atoms will require 100 million trillion amplitudes. That’s a lot of amplitudes for a pretty simple molecule.

The result is that simulating such systems is incredibly hard. Just storing the amplitudes requires mindboggling amounts of computer memory. Simulating how they change in time is even more challenging, involving immensely complicated updates to all the amplitudes.

Physicists and chemists have found some clever tricks for simplifying the situation. But even with those tricks simulating quantum systems on classical computers seems to be impractical, except for tiny molecules, or in special situations. The reason most educated people today don’t know simulating quantum systems is important is because classical computers are so bad at it that it’s never been practical to do. We’ve been living too early in history to understand how incredibly important quantum simulation really is.

That’s going to change over the coming century. Many of these problems will become vastly easier when we have scalable quantum computers, since quantum computers turn out to be fantastically well suited to simulating quantum systems. Instead of each extra simulated atom requiring a doubling (or more) in classical computer memory, a quantum computer will need just a small (and constant) number of extra qubits. One way of thinking of this is as a loose quantum corollary to Moore’s law:

The quantum corollary to Moore’s law: Assuming both quantum and classical computers double in capacity every few years, the size of the quantum system we can simulate scales linearly with time on the best available classical computers, and exponentially with time on the best available quantum computers.

In the long run, quantum computers will win, and win easily.

The punchline is that it’s reasonable to suspect that if we could simulate quantum systems easily, we could greatly speed up drug discovery, and the discovery of other new types of materials.

I will risk the ire of my (understandably) hype-averse colleagues and say bluntly what I believe the likely impact of quantum simulation will be: there’s at least a 50 percent chance quantum simulation will result in one or more multi-trillion dollar industries. And there’s at least a 30 percent chance it will completely change human civilization. The catch: I don’t mean in 5 years, or 10 years, or even 20 years. I’m talking more over 100 years. And I could be wrong.

What makes me suspect this may be so important?

For most of history we humans understood almost nothing about what matter is. That’s changed over the past century or so, as we’ve built an amazingly detailed understanding of matter. But while that understanding has grown, our ability to control matter has lagged. Essentially, we’ve relied on what nature accidentally provided for us. We’ve gotten somewhat better at doing things like synthesizing new chemical elements and new molecules, but our control is still very primitive.

We’re now in the early days of a transition where we go from having almost no control of matter to having almost complete control of matter. Matter will become programmable; it will be designable. This will be as big a transition in our understanding of matter as the move from mechanical computing devices to modern computers was for computing. What qualitatively new forms of matter will we create? I don’t know, but the ability to use quantum computers to simulate quantum systems will be an essential part of this burgeoning design science.

Alright, enough speculation.

Let me also briefly mention the sober-minded conventional answer given to the question “what are quantum computers good for?” That answer is to list various algorithmic problems that we have some evidence can be solved faster on a quantum computer than on a classical computer.

The most famous example is Peter Shor’s beautiful quantum factoring algorithm. To find the prime factors of an n n n - bit integer seems to be a very difficult problem on a classical computer. The best existing algorithms are incredibly computationally expensive, with a cost that rises exponentially with n n n . Even numbers with just a few hundred digits aren’t currently feasible to factor on classical computers. By contrast, Shor’s quantum factoring algorithm would make factoring into a comparatively easy task, if large-scale quantum computers can be built.

Factoring perhaps doesn’t seem like a very interesting application. But it turns out that the ability to factor lets you break some of the most widely-used encryption schemes, used by services such as Gmail and Amazon to keep your communications private. This ability to break encryption has made the world’s intelligence agencies very interested in factoring, and they’ve poured enormous sums of money into quantum computing research since the mid-1990s. Indeed, there’s a good (as yet unwritten) history book to be written about how the rise of quantum computing was caused by the interest of the world’s intelligence agencies in accessing humanity’s private thoughts.

There’s been surprisingly little public reflection about this on the part of the quantum computing community. I occasionally meet quantum computing researchers who complain in private about what they perceive as privacy violations by governments, and the dangers of surveillance states. But then some of those same people will take money to help those governments in their plans for surveillance, usually with some transparently self-serving justification about how they’re not really helping. One exception to this lack of public reflection is a brief discussion in Ronald de Wolf’s thoughtful essay The Potential Impact of Quantum Computers on Society (2017).

Are quantum computers really universal devices?

The eternal mystery of the world is its comprehensibility… The fact that it is comprehensible is a miracle. — Albert Einstein

At the beginning of this essay I asked whether there is any single universal computing device that can efficiently simulate any other physical system? We’ve learned that classical computers seem to have a lot of trouble efficiently simulating quantum systems.

What about quantum computers? While they can certainly simulate many quantum systems, does that mean they can be used to efficiently simulate any physical system?

This question is an open problem! We don’t yet know the answer.

Part of the trouble in answering the question is that humanity hasn’t yet discovered the final fundamental laws of physics. Modern physics is based on two astonishingly effective theories: Einstein’s general theory of relativity, which describes how gravitation works; and the standard model of particle physics, which explains how pretty much everything else (electromagnetism, the strong and weak nuclear forces) work.

Trouble is, we don’t yet have a good theory of quantum gravity which combines general relativity and the standard model. Without such a theory of quantum gravity we’re not able to answer the question of whether quantum computers can efficiently simulate any other physical system. Perhaps some future class of quantum gravitating computers, more powerful even than quantum computers, will be needed to simulate quantum gravity.

Let’s ask a slightly less ambitious question, which is whether we can use quantum computers to efficiently simulate general relativity and the standard model?

The standard model is an example of a particular type of quantum mechanical theory called a quantum field theory. John Preskill and his collaborators have written a series of papers For a review of progress see: John Preskill, Simulating quantum field theory with a quantum computer (2018). explaining how to use quantum computers to efficiently simulate quantum field theories. Those papers don’t yet simulate the full standard model, but they do make considerable progress. It remains an exciting open problem, albeit a problem where much encouraging progress has been made.

In the case of general relativity, as far as I know the problem remains open. Indeed, even stating what the problem means is not trivial. General relativity supports the existence of closed timelike curves, which can be used in some sense to send information back in time. This has interesting consequences for computation: there’s a way in which the computer can know the results of future computations. Unsurprisingly, this changes what is possible! Another complication is that when you talk about an “efficient simulation” in computation you mean the time and space overhead isn’t too large. But in general relativity even the basic units of space and time aren’t so clear. It’s hard to say what efficiency means. Finally, near singularities time and space get distorted in strange ways, again making it challenging to say exactly what it means to do an efficient computation.

There is, by the way, significant issue that I've been sweeping under the rug, and which may be bugging you: as I've explained it, a quantum computer isn't a single computing device at all, since there are many possible quantum circuits. This is okay, though, since there's a model known as the universal quantum Turing machine, which is a single computing device, and which can simulate any quantum circuit. So you should understand the discussion above as being implicitly about the universal quantum Turing machine. I won't explain the details of the universal quantum Turing machine in this essay, since in practice the quantum circuit model is far more commonly used. But if you're interested in the details, I recommend this paper by Bernstein and Vazirani .

The existence of universal computers is easy to take for granted. But there’s no a priori logical reason there should be a single machine that can efficiently simulate every other physical system. It’s like being able to use your car also as a surfboard, a supermarket trolley, and a rainforest In fact, there’s a sense in which this is possible, within limits: if you could rearrange protons, neutrons, and electrons arbitrarily well, you could turn a car into a surfboard, a supermarket trolley, or a (small) part of a rainforest. So matter does have intriguing universality properties. This is also remarkable, of course. . Yet the evidence so far suggests our universe does allow such universal machines. It’s a good problem – and, so far as I know, largely unexplored – to think about sets of laws of physics in which such a universal machine is not possible. That might sound pointless – why imagine other possible universes? Yet exploring such radical counterfactuals is often an excellent strategy for better understanding our own universe.

So there’s a very strange loop here. It’s that the laws of physics determine what kind of computations can be done. And yet the kind of computations which can be done seem to be powerful enough to describe the laws of physics. And that description can then be used to (efficiently!) simulate any physical system:

The top part of the loop is almost tautological. The bottom half of the loop is extraordinary. There’s no a priori reason the laws of physics should enable the existence of machines which can simulate physical systems. You might argue on anthropic grounds – we humans are here in the universe, and doing physics pretty successfully, so conditioned on that, it must be true. But that’s not a very satisfactory explanation of why. It remains a mystery. Einstein was right: the fact that the world is comprehensible at all is a miracle.

Final Reflections

We’ve worked through all the basics of the quantum computing model, but we haven’t yet used it in a full-on application – the sort of application which make people excited about quantum computing. But there will soon be available two considerably shorter(!) followup essays, explaining the quantum search algorithm and quantum teleportation. Someone who understands all three essays will have a good understanding of elementary quantum computing.

And what of the experimental mnemonic medium we’ve developed in this essay? Mastering new subjects requires internalizing the basic terminology and ideas of the subject. The mnemonic medium should radically speed up this memory step, converting it from a challenging obstruction into a routine step. Frankly, I believe it would accelerate human progress if all the deepest ideas of our civilization were available in a form like this. Perhaps some day.

With that said, memory is only part of what it means to understand. Is it possible to build powerful environments which enable deeper forms of understanding? That enable people to take action in new ways, to grow their sense of agency and of ability to contribute? This requires many things beyond memory: the ability to problem solve and to problem find; to connect with opportunities that genuinely matter, and to find pathways for meaningful contribution. We, the team at “Quantum Computing for the Very Curious”, believe there are many powerful and under-exploited patterns available to achieve these goals. In future projects we will explore and develop these patterns. Some of this exploration will continue to be done in the context of quantum computing. But we also hope to launch projects discussing some of humanity’s other great challenges, including optimistic, detailed visions of problems such as space travel, climate change, longevity, and the development of new forms of matter.

Thanks for reading this far. In a few days, you’ll receive a notification containing a link to your first review session. In that review session you’ll be retested on the material you’ve learned, helping you further commit it to memory. It should only take a few minutes. In subsequent days you’ll receive more notifications linking you to re-review, gradually working toward genuine long-term memory of all the core material in the essay.

Thanks for reading this far. If you’d like to remember the core ideas of this essay durably, please sign in below to set up an account. We’ll track your review schedule and send you occasional reminders containing links that will take you to the review experience.

Acknowledgments

Michael Nielsen is supported by Y Combinator Research . This essay is based in part on Michael’s earlier lecture series on Quantum Computing for the Determined .

Citing this work

In academic work, please cite this as:

Andy Matuschak and Michael A. Nielsen, “Quantum Computing for the Very Curious”, https://quantum.country/qcvc, San Francisco (2019).

Authors are listed in alphabetical order.

This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License . This means you’re free to copy, share, and build on this essay, but not to sell it. If you’re interested in commercial use, please contact us .

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What Is Quantum Computing?

When scientists want to do things like harness the power of molecules during photosynthesis , they won’t be able to do so using regular old computers. They need to use quantum computers , which are able to measure and observe quantum systems at the molecular level as well as solve the conditional probability of events. Basically, quantum computers can do billions of years worth of computing over the course of a weekend — and untangle some of the world’s most complex problems in the process.

Quantum computing is a process that uses the laws of quantum mechanics to solve problems too large or complex for traditional computers. Quantum computers rely on qubits to run and solve multidimensional quantum algorithms.

Indeed, quantum computing is vastly different from classical computing. Quantum physicist Shohini Ghose, of Wilfrid Laurier University, has likened the difference between quantum and classical computing to light bulbs and candles: “The light bulb isn’t just a better candle; it’s something completely different.”

Quantum computing solves mathematical problems and runs quantum models using the tenets of quantum theory. Some of the quantum systems it is used to model include photosynthesis, superconductivity and complex molecular formations.

To understand quantum computing and how it works, you first need to understand qubits, superposition, entanglement and quantum interference.

What Are Qubits?

Quantum bits, or qubits, are the basic unit of information in quantum computing. Sort of like a traditional binary bit in traditional computing. 

Qubits use superposition to be in multiple states at one time. Binary bits can only represent 0 or 1. Qubits can be 0 or 1, as well as any part of 0 and 1 in superposition of both states. 

What are qubits made of? The answer depends on the architecture of quantum systems, as some require extremely cold temperatures to function properly. Qubits can be made from trapped ions, photons, artificial or real atoms or quasiparticles, while binary bits are often silicon-based chips.

What Is Superposition?

To explain superposition, some people evoke Schrödinger’s cat , while others point to the moments a coin is in the air during a coin toss. 

Put simply, quantum superposition is a mode when quantum particles are a combination of all possible states. The particles continue to fluctuate and move while the quantum computer measures and observes each particle.

The more interesting fact about superposition — rather than the two-things-at-once point of focus — is the ability to look at quantum states in multiple ways, and ask it different questions, said John Donohue, scientific outreach manager at the University of Waterloo’s Institute for Quantum Computing. That is, rather than having to perform tasks sequentially, like a traditional computer, quantum computers can run vast numbers of parallel computations.

That’s about as simplified as we can get before trotting out equations. But the top-line takeaway is that that superposition is what lets a quantum computer “try all paths at once.”

What Is Entanglement?

Quantum particles are able to correspond measurements with one another, and when they are engaged in this state, it’s called entanglement. During entanglement, measurements from one qubit can be used to reach conclusions about other units. Entanglement helps quantum computers solve larger problems and calculate bigger stores of data and information.

What Is Quantum Interference?

As qubits experience superposition, they can also naturally experience quantum interference. This interference is the probability of qubits collapsing one way or another. Because of the possibility of interference, quantum computers work to reduce it and ensure accurate results.

Read Next 19 Quantum Computing Companies to Know

How Do Quantum Computers Work?

They use qubits and computational algorithms .

Quantum computers process information in a fundamentally different way than classical computers. Traditional computers operate on binary bits but quantum computers transmit information via qubits. The qubit’s ability to remain in superposition is the heart of quantum’s potential for exponentially greater computational power.

Quantum computers use a variety of algorithms to conduct measurements and observations. These algorithms are input by a user, the computer then creates a multidimensional space where patterns and individual data points are housed. For example, if a user wants to solve a protein folding problem to discover the least amount of energy to use, the quantum computer would measure the combinations of folds; this combination is the answer to the problem.

They Rely on Quantum-Specific Computer Infrastructure 

The physical build of a true quantum computer consists mainly of three parts. The first part is a traditional computer and infrastructure that runs programming and sends instructions to the qubits. The second part is a method to transfer signals from the computer to the qubits. Finally, there needs to be a storage unit for the qubits. This storage unit for qubits must be able to stabilize the qubits and certain needs or requirements have to be met. These can range from needing to be near zero degrees or the housing of a vacuum chamber.

They Require Physical Isolation and Cooling Mechanisms

Qubits, it turns out, are higher maintenance than even the most meltdown-prone rock star. Any number of simple actions or variables can send error-prone qubits falling into decoherence, or the loss of a quantum state. Things that can cause a quantum computer to crash include measuring qubits and running operations. In other words: using it. Even small vibrations and temperature shifts will cause qubits to decohere, too.

That’s why quantum computers are kept isolated, and the ones that run on superconducting circuits — the most prominent method, favored by Google and IBM — have to be kept at near-absolute zero (a cool -460 degrees Fahrenheit).

What Can Quantum Computing Solve?

Quantum computing can optimize problem solving by using quantum computers to run quantum-inspired algorithms. These optimizations can be applied to the science and industry fields because they rely heavily on factors like cost, quality and production time. With quantum computing, there will be new discoveries in how to manage air traffic control, package deliveries, energy storage and more.

Molecular Modeling and Simulations 

One quantum computing breakthrough came in 2017, when researchers at IBM modeled beryllium hydride, the largest molecule simulated on a quantum computer to date. Another key step arrived in 2019, when IonQ researchers used quantum computing to go bigger still, by simulating a water molecule .

Climate and Energy Optimization Problems

Some believe quantum computers can help combat climate change by improving carbon capture. Jeremy O’Brien, CEO of Palo Alto-based PsiQuantum, wrote that quantum simulation of larger molecules — if achieved — could help build a catalyst “for ‘scrubbing’ carbon dioxide directly from the atmosphere.”

Artificial Intelligence Breakthroughs

There’s also hope that large-scale quantum computers will help accelerate artificial intelligence technologies, and vice versa — although experts disagree on this point. “The reason there’s controversy is, things have to be redesigned in a quantum world,” said Rebecca Krauthamer, CEO of quantum computing consultancy Quantum Thought . “We can’t just translate [AI] algorithms from regular computers to quantum computers because the rules are completely different, at the most elemental level.”

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Quantum Computing Challenges 

Quantum noise disruptions.

Currently, we’re still in what’s known as the NISQ era — Noisy, Intermediate-Scale Quantum. Quantum noise refers to any disturbances that affect the state of qubits, which can disrupt superposition, entanglement and the overall accuracy of quantum systems. This noise can be caused by multiple factors like temperature, electromagnetic or mechanical fluctuations, making quantum computers incredibly difficult to keep in a proper quantum state. As such, NISQ computers can’t be trusted to make decisions of major commercial consequence, which means they’re currently used primarily for research and education.

Quantum Technology Is Difficult to Scale and Actualize  

While quantum computing has the potential to solve complex problems, its operational output and level of qubits required to actually complete these tasks are demanding, and the technology has yet to scale to be able to support these needs. 

With qubits in particular, the challenge is two-fold, according to Jonathan Carter, a scientist at Lawrence Berkeley National Laboratory. First, individual physical qubits need to have better fidelity. That would conceivably happen either through better engineering, discovering optimal circuit layout, and finding the optimal combination of components. Second, we have to arrange them to form logical qubits.

“Estimates range from hundreds to thousands to tens of thousands of physical qubits required to form one fault-tolerant qubit. I think it’s safe to say that none of the technology we have at the moment could scale out to those levels,” Carter said.

For example, researchers would need millions of qubits alone to compute “the chemical properties of a novel substance,” noted theoretical physicist Sabine Hossenfelder in the Guardian . Plus, the fragility of large-scale quantum systems make it difficult for current technologies to properly stabilize them long enough to even function. 

But the conceptual underpinning, at least, is there to overcome these hurdles. “A quantum computer knows quantum mechanics already, so I can essentially program in how another quantum system would work and use that to echo the other one,” explained Donohue.

Additionally, the challenges that quantum computing faces aren’t strictly hardware-related. The “magic” of quantum computing also resides in algorithmic advances, “not speed,” Greg Kuperberg, a mathematician at the University of California at Davis, is quick to underscore.

“If you come up with a new algorithm, for a question that it fits, things can be exponentially faster,” he said, using exponential literally, not metaphorically.

Quantum Computing Standards Are Still Being Developed

Another open question: Which method of quantum computing will become standard? While superconducting has borne the most fruit so far, researchers are exploring alternative methods that involve trapped ions, quantum annealing or so-called topological qubits. In Donohue’s view, it’s not necessarily a question of which technology is better so much as one of finding the best approach for different applications. For instance, superconducting chips naturally dovetail with the magnetic field technology that underpins neuroimaging.

Lack of Quantum Computing Expertise 

One roadblock for quantum computing, according to Krauthamer, is general lack of expertise. “There’s just not enough people working at the software level or at the algorithmic level in the field,” she said. Tech entrepreneur Jack Hidarity’s team set out to count the number of people working in quantum computing and found only about 800 to 850 people, according to Krauthamer. “That’s a bigger problem to focus on, even more than the hardware,” she said. “Because the people will [need to] bring that innovation.”

Why Quantum Computing Is Important

Quantum computers can review classical computer results .

Quantum computers’ research and development practicality is demonstrable, if decidedly small-scale. Donohue cites the molecular modeling of lithium hydrogen. That’s a small enough molecule that it can also be simulated using a supercomputer, but the quantum simulation provides an important opportunity to “check our answers” after a classical-computer simulation. 

These are generally still small problems that can be checked using classical simulation methods. “But it’s building toward things that will be difficult to check without actually building a large particle physics experiment, which can get very expensive,” Donohue said.

Quantum Computing May Transform Cryptography 

Quantum computers may have the potential to uproot some of our current systems. The cryptosystem known as RSA provides the safety structure for a host of privacy and communication protocols, from email to internet retail transactions. Current standards rely on the fact that no one has the computing power to test every possible way to de-scramble your data once encrypted , but a mature quantum computer could try every option within a matter of hours .

It should be stressed that quantum computers haven’t yet hit that level of maturity  — and won’t for some time — but if and when a large, stable device is built its unprecedented ability to factor large numbers would essentially leave the RSA cryptosystem in tatters. Thankfully, the technology is still a ways away — and the experts are on it.

“Don’t panic.” That’s what Mike Brown, CTO and co-founder of quantum-focused cryptography company ISARA Corporation , advises anxious prospective clients. The threat is far from imminent. “What we hear from the academic community and from companies like IBM and Microsoft is that a 2026-to-2030 timeframe is what we typically use from a planning perspective in terms of getting systems ready,” he said.

Cryptographers from ISARA are among several contingents that have taken part in the Post-Quantum Cryptography Standardization project , a contest of quantum-resistant encryption schemes. The aim is to standardize algorithms that can resist attacks levied by large-scale quantum computers. The competition was launched in 2016 by the National Institute of Standards and Technology, a federal agency that helps establish tech and science guidelines, and is now gearing up for its third round.

Indeed, the level of complexity and stability required of a quantum computer to launch the much-discussed RSA attack is extreme. Even granting that timelines in quantum computing — particularly in terms of scalability — are points of contention.

Read More Cryptographers Are Racing Against Quantum Computers

The Future of Quantum Computing

Quantum computers do exist, and they are being used right now. They are not, however, presently “solving” climate change, turbocharging financial forecasting probabilities or performing other similarly lofty tasks that get bandied about in reference to quantum computing’s potential. Quantum computing may have commercial applications related to those challenges, but that’s well down the road.

“The technology just isn’t quite there yet to provide a computational advantage over what could be done with other methods of computation at the moment,” said Dohonue. “Most [commercial] interest is from a long-term perspective. [Companies] are getting used to the technology so that when it does catch up — and that timeline is a subject of fierce debate — they’re ready for it.”

Though quantum computing still has a ways to go before a wide-scale commercial debut, curious minds can still get their hands dirty with the technology today. Users can operate small-scale quantum processors via the cloud through IBM’s online Q Experience and its open-source software Quiskit. Microsoft and Amazon both now have similar platforms, dubbed Azure Quantum and Amazon Braket . There are also over 60 algorithms listed and over 400 papers cited at Quantum Algorithm Zoo , an online catalog of quantum algorithms compiled by Microsoft quantum researcher Stephen Jordan. “That’s one of the cool things about quantum computing today,” said Krauthamer. “We can all get on and play with it.”

Frequently Asked Questions

What is quantum computing in simple terms.

Quantum computing refers to computing that operates off of the laws of quantum mechanics in order to solve problems faster than classical computers. Quantum computers use qubits to have information be in multiple states (such as 0 and 1) at once.

What can quantum computers do?

Quantum computers can run quantum algorithms to accelerate problem-solving processes. These processes may be applied to areas in medical research, financial modeling, AI and more to make decisions with increased accuracy and speed.

Do quantum computers exist now?

Quantum computers exist now, though they are mainly used in data centers, laboratories and universities for research and education purposes.

What is the main goal of quantum computing?

Quantum computing aims to speed up research and development initiatives as well as solve complex data or optimization problems that classical computers are unable to process.

Quantum Thought

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Home > Books > Quantum Computing and Communications

Introduction to Quantum Computing

Submitted: 23 August 2020 Reviewed: 18 September 2020 Published: 29 October 2020

DOI: 10.5772/intechopen.94103

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Quantum computing is a modern way of computing that is based on the science of quantum mechanics and its unbelievable phenomena. It is a beautiful combination of physics, mathematics, computer science and information theory. It provides high computational power, less energy consumption and exponential speed over classical computers by controlling the behavior of small physical objects i.e. microscopic particles like atoms, electrons, photons, etc. Here, we present an introduction to the fundamental concepts and some ideas of quantum computing. This paper starts with the origin of traditional computing and discusses all the improvements and transformations that have been done due to their limitations until now. Then it moves on to the basic working of quantum computing and the quantum properties it follows like superposition, entanglement and interference. To understand the full potentials and challenges of a practical quantum computer that can be launched commercially, the paper covers the architecture, hardware, software, design, types and algorithms that are specifically required by the quantum computers. It uncovers the capability of quantum computers that can impact our lives in various viewpoints like cyber security, traffic optimization, medicines, artificial intelligence and many more. At last, we concluded all the importance, advantages and disadvantages of quantum computers. Small-scale quantum computers are being developed recently. This development is heading towards a great future due to their high potential capabilities and advancements in ongoing research. Before focusing on the significances of a general-purpose quantum computer and exploring the power of the new arising technology, it is better to review the origin, potentials, and limitations of the existing traditional computing. This information helps us in understanding the possible challenges in developing exotic and competitive technology. It will also give us an insight into the ongoing progress in this field.

• quantum computing
• real-time systems
• program processors

Author Information

Surya teja marella *.

• University of Leicester, Leicester, UK

Hemanth Sai Kumar Parisa

*Address all correspondence to: [email protected]

1. Introduction

1.1 history of computing.

Evolution in one region of science and technology leads to the discovery of a new one. In less than a century, research and development of functional computing technologies have renovated science, technology, and nation massively. The first practical computer around the 20th century was not capable of doing mathematical computations, on its own. Practical devices need a solid physical implementation of theoretical concepts. Nowadays, computers are solving problems instantly and accurately provided the input is relevant, and a set of instructions given are favorable. It all started from World War II when Alan Turing created a real general-purpose computer with a storable program model and is known as the ‘Universal Turing Machine’. It was redesigned by Von Neumann and is now the most important architecture for almost every computer. The computers and their physical parts kept improving with time in terms of performance and their strengths. And gradually, the industry of computers became larger than the military department which initiated it. The advancement in control and understanding of humans over nature and physical systems has given us the latest electronic devices we are utilizing today [ 1 ].

2. A new kind of computing

Today’s computers are smaller, cheaper, faster, greatly efficient, and even more powerful as compared to early computers that used to be huge, costly, and more power-consuming. It becomes possible due to improvements in architecture, hardware components, and software running on them. Electronic circuits used in computers are getting smaller and smaller day by day. Transistors are small semiconductor devices that are used to amplify and also switch electric or electronic signals. They were used to be fabricated on a piece of silicon. The circuit was made by connecting these transistors together into a single silicon surface. The shape of circuits in an IC was printed together in all layers of silicon at the same time. This process takes the same amount of time even if the number of transistors in the circuit was increased. The cost of production of IC was decided by the size of silicon and not the number of transistors. This reduced the price of products due to which manufacturing and selling of IC increased and thus benefits and sales also. From the idea of connecting individual transistors to the collection of these transistors (Logic Gates) and finally, the collection of these Logic Gates used to get connected into a single integrated circuit (IC). Nowadays, a single IC can even integrate small computers onto it.

Gordon Moore, co-founder of Intel, in 1965, discovered that the number of transistors on a silicon microprocessor chip had made twice as much every year while the prices were reduced to half since their invention. This is known as Moore’s Law. Moore’s Law is considerable because it means that computers and their computing power get smaller and faster over time. Though this law is putting the brakes on now and consequently, the improvement in classical computers is not like before it used to be [ 2 ].

This leads to the idea of the smallest computer by reducing the size of the circuit up to the size of an atom. But then these circuits will not be able to act as a switch as electrons inside an atom can become invisible from one side of a barrier and appear on another side, i.e. they can exist in more than one place at the same time. This is due to the teleporting phenomena in quantum mechanics called “Quantum Tunneling”. It shows that the size of the circuits of the classical computer after 5–7 nanometers has reached their limit. The representation and processing of these computers can be illustrated by the law of classical physics that gives us an only deterministic justification of the Universe. But it fails to forecast all noticeable phenomena occurring in nature and this led to the discovery of quantum mechanics, the biggest changeover in physics. Thus, there is a need for new computing other than current classical computing to put its state into some physical information rather than a circuit. Since the quantum phenomena are bringing up more constraints on the design of the computers. It changes the basic building blocks of a computer that not only expects new type of hardware creation but also a new design, software, and layers of abstraction to facilitate the designers to create and exploit these systems even if their complexities scale over time. The design of the hardware components has to be governed by quantum properties [ 3 ].

Quantum Computing is a new kind of computing based on Quantum mechanics that deals with the physical world that is probabilistic and unpredictable in nature. Quantum mechanics being a more general model of physics than classical mechanics give rise to a more general model of computing- quantum computing that has more potential to solve problems that cannot be solved by classical ones. To store and manipulate the information, they use their own quantum bits also called ‘ Qubits ’ unlike other classical computers which are based on classical computing that uses binary bits 0 and 1 individually. The computers using such type of computing are known as ‘ Quantum Computers’ . In such small computers, circuits with transistors, logic gates, and Integrated Circuits are not possible. Hence, it uses the subatomic particles like atoms, electrons, photons, and ions as their bits along with their information of spins and states. They can be superposed and can give more combinations. Therefore, they can run in parallel using memory efficiently and hence is more powerful. Quantum computing is the only model that could disobey the Church-Turing thesis and thus quantum computers can perform exponentially faster than classical computers.

3. Need for quantum computers

Quantum computers can solve any computational problem that any classical computer can. According to the Church-Turing thesis, the converse is also true that classical computers can solve all the problems of quantum computers too. It means they provide no extra benefit over classical computers in terms of computability but there are some complex and impossible problems that cannot be solved by today’s conventional computers in a practical amount of time. It needs more computational power. Quantum computers can solve such problems in reasonably and exponentially lower time complexities, also known as “ Quantum Supremacy ” [ 4 ].

Peter Shor in 1993 showed that Quantum computers can help to solve these problems considerably more efficiently like in seconds without getting overheated. He developed algorithms for factoring large numbers quickly. Since their calculations are based on the probability of an atom’s state before it is actually known. These are having the potential to process data in an exponentially huge quantity. It also explains that a practical quantum computer could break the cryptographic secret codes. It can risk the security of encrypted data and communication. It can expose private and protected secret information. But the advantages of quantum computers are also kept in mind that is significantly more than its flaws. Hence, they are still needed and further research is going towards a brighter future.

4. Fundamentals of quantum computing

While designing the conventional computer, it was kept in mind that transistors’ performance especially when getting smaller, will be affected by noise if any type of quantum phenomenon takes place. They tried to avoid quantum phenomena completely for their circuits. But the quantum computer adapts a different technique instead of using classical bits and even works on the quantum phenomenon itself. It uses quantum bits that are analogous to classical bits and have two quantum states where it can be either 0 or 1 except it follows some quantum properties where it can have both values simultaneously leading to a concept of superposed bits.

5. Where the concept of bits came from?

Transistors are the fundamental construction blocks for an IC which are connected through wires in a circuit. They conduct electric signals between devices. The communication between transistors within an IC takes place through electric signals. The behavior of the signals is analog in nature. Therefore, their values are real numbers that change smoothly between 0 and 1. These electric signals can also interact with the environment resulting in noise. Therefore, a little change from 0 to 0.1 due to temperature or vibrations from the environment can drastically change the system’s behavior. There are two types of noise present in the environment. The first type of noise results from energy instabilities occurring suddenly within the object like temperature above absolute zero Kelvin. These are fundamental in nature. Other types of noise are the consequences of signal interactions. This type of noise could have corrected or designed. But neither of them got designed nor corrected or maybe left intentionally uncorrected at the hardware layer. They are systematic in nature [ 5 ].

To overcome these noises in analog circuits, the IC is built with transistors in such a way that it could work on digital signals (binary bits) instead of analog signals. These circuits are called ‘Logic Gates’. They perceive the electric signals containing values of real numbers as a binary digit or ‘bit’ of either 0 (low voltage) or 1 (high voltage). Registers are another type of Gate which stores a bit or the number of bits present in an input value to process further. Gates can remove noise from a signal by limiting the set of values a signal can hold. Constructing IC using logic gates rather than transistors simplifies the designing by creating a powerful circuit that is not sensitive to design and fabrication issues and facilitates abstraction to designers so that they can focus only on gate functions (Boolean functions) rather than circuit issues. Boolean functions are defined by the rules of Boolean algebra. They can use an automated design tool for mapping the required logic gates. A standard library containing a set of tested logic gates is integrated into the silicon chip design with the help of their manufacturing technology. Negligible error rates can be achieved using digital logic and standard libraries. This helps in making the design robust. Also, the data is encoded by adding some redundant bits in the memory using an error correction code. This code is checked at regular intervals to detect the error. It also helps in other traits of design like testing and debugging.

Quantum Bit or Qubit is the fundamental unit of quantum information that represents subatomic particles such as atoms, electrons, etc. as a computer’s memory while their control mechanisms work as a computer’s processor. It can take the value of 0, 1, or both simultaneously. It is a million times more powerful than today’s strongest supercomputers. Production and management of qubits are tremendous challenges in the field of engineering. They acquire both, digital as well as analog nature which gives the quantum computer their computational power. Their analog nature indicates that quantum gates have no noise limit and their digital nature provides a norm to recover from this serious weakness. Therefore, the approach of logic gates and abstractions created for classical computing is of no use in quantum computing. Quantum computing may adopt ideas only from classical computing. But this computing needs its own method to overcome the variations of processing and any type of noise. It also needs its own strategy to debug errors and handle defects in design.

Qubit has two quantum states similar to the classical binary states. The qubit can be in either state as well as in the superposed state of both states simultaneously. There is a representation of these quantum states also known as Dirac notation [ 6 ].

In this notation, the state label is kept between two symbols | and ⟩. Therefore, states are written as |0⟩ and |1⟩ which are literally having analog values and both are participating to give any value between 0 and 1 given that sum of probability of occurrence of each state must be 1. Thus any quantum bit wave function can be expressed as a two-state linear combination each with its own complex coefficient i.e. |w⟩ = x |0⟩ + y |1⟩ where x and y are coefficients of both the states. The probability of the state is directly proportional to the square of the magnitude of its coefficient. |x| 2 is the probability of identifying the qubit state 0 and |y| 2 is the probability of identifying the qubit state 1. These probabilities when summed up must give a total of 1 or say 100% mathematically, i.e. |x| 2  + |y| 2  = 1.

6. Properties of quantum computing

In quantum physics, the quantum object does not exist in an entirely determined state. It looks like a particle but behaves like a wave when not being observed. This dual nature of particles leads to interesting physical phenomena. The state of any quantum object is expressed as a sum of possible participating states or a wave-function. Such states are coherent due to the interference of all the participating states either in a constructive or a destructive manner. Observation of quantum objects when they interact with some larger physical system results in the extraction of information. Such observation of quantum objects is called quantum measurement. Measurement can also result in the loss of information by disrupting the quantum state. These are some of the properties of quantum objects. Quantum objects referred here are the qubits in the case of quantum computing. The progress of any quantum system is regulated by Schrodinger’s equation that tells us about the change in the wave-function of the system due to the energy environment. This environment is the system Hamiltonian which is a mathematical description of energies experiencing from all forces felt by all components of the system. To control any quantum system, there is a need to control this environment by isolating the system from the forces of the universe that cannot be controlled easily and by assigning energy within this isolated area only. A system cannot be completely isolated. However, energy and information exchanges can be minimized. This interaction with the outside environment can lead to loss of coherence and can result in “Decoherence” [ 7 ].

Superposition

Superposition in quantum mechanics states that any two quantum states can be summed up (superposed) resulting in another valid quantum state. It is a fundamental principle of quantum mechanics. Oppositely we can say that any quantum state is the sum of two or more than two other unique states.

Superposition in quantum computing refers to the ability of a quantum system where quantum particle or qubit can exist in two different positions or say, in multiple states at the same time. It provides high-speed parallel processing in an unbelievable way and is very different from their classical equivalents that have binary constraints. The quantum computer system holds the information that exists in two states simultaneously. Qubits are brought into a superposition by influencing them with the help of lasers so that it can simultaneously store 0 and 1 at the same time. In classical computing, if there are 2 bits, the total possible values after combining we get are 4, out of which only 1 value is possible at any instant. But on the other hand, if there are 2 qubits in the quantum computer. The total possible values after combination are 4 and all are possible at once. It looks like unthinkable because it is not like gravity that can be proved easily just by looking at the falling of an apple. The laws of classical physics fail here because superposition only exists in the territory of quantum particles.

Entanglement

Entanglement in quantum mechanics is a physical phenomenon where two or more quantum objects are inherently linked such that measurement of one rules the possible measurement of another. In other words, a pair or a group of particles interacts or share spatial locality such that the quantum state of each particle cannot be characterized independently of the other particle’s state in the same group even when they are separated by a large distance.

Entanglement is one of the important properties of quantum computing. It refers to the strong correlation existing between two quantum particles (physical properties of systems) or qubits. Qubits are linked together in a perfect instantaneous connection, even if they are isolated at any large distances such as located at the opposite ends of the Universe. They are entangled or defined with reference to each other. The fact is that the state of one particle influences the state of the other. It creates strong communication between qubits. Once they got entangled, they will stay connected even after separated at any distance. In classical computers, if bits are doubled, computational power also gets doubled. But in the case of Entanglement, adding extra bits to a quantum computer can increase its computational power exponentially. Quantum computer uses this property in a sort of quantum daisy chain.

Interference

The property of interference in quantum computers is similar to wave interference in classical physics. Wave interference happens when two waves interact with each other in the same medium. It forms a resultant wave with either their amplitudes added together when they are aligned in the same direction known as constructive interference or a resultant wave with their amplitudes canceled out when waves are in opposite direction known as destructive interference. The net wave can be bigger or smaller than the original wave depending on the type of interference. Since all subatomic particles along with light pose dual nature, i.e. particle and wave nature both. The quantum particle may experience interference. If each particle goes through both the slits (Young’s double-slit experiment) simultaneously due to superposition, they can cross its own path interfering with the path direction. The idea of interference allows us to intentionally bias the content of the qubit towards the needed state. However, it can also result in a quantum computer to combine its various computations into one making it more error-prone [ 9 ].

7. The topography of quantum technology

The quantum phenomena are not limited to just quantum computing but they apply to other technologies also including quantum information science, quantum communication, and quantum metrology. The progresses of all these technologies are mutually dependent on each other and can control as well as transform the entire quantum system. They share the same theory of physics, common hardware and related methods [ 10 ].

Quantum Information Science seeks the methods of encoding the information in a quantum system. It includes statistics of quantum mechanics along with their limitations. It provides a core for all other applications such as quantum computing, communications, networking, sensing and metrology.

Quantum Communication and networking concentrates on the conversation or exchange of information by encoding it into a quantum system to facilitate communication between quantum computers. Quantum cryptography is the subset of quantum communication in which quantum properties help to design the secure communication system.

Quantum sensing and metrology is the study and development of quantum systems. The drastic sensitivity of such a system to environmental nuisances can be utilized in order to measure important physical properties (e.g. electric and magnetic fields, temperature, etc.) more accurately than classical systems. Quantum sensors are based on qubits and are carried out using the experimental quantum systems.

Quantum computing is the central focus of this research which exploits the quantum mechanical properties of superposition, entanglement and interference to enact computations. In common, a quantum computer is a physical system that comprises a collection of qubits that must be isolated from the environment for their quantum state to stay coherent until it performs the computation. These qubits are organized and manipulated in order to enforce an algorithm and to achieve a result with high probability from the measurement of its final state.

8. The architecture of quantum computer

Application Layer- It is not a part of a quantum computer. It is used for representing a user interface, the operating system for a quantum computer, coding environment, etc. that are needed for formulating suitable quantum algorithms. It is hardware-independent.

Classical Layer- It optimizes and compiles the quantum algorithm into microinstructions. It also processes quantum-state measurement returned back from hardware in the below layers and gives it to a classical algorithm to produce results.

Digital Layer- It interprets microinstructions into signals (pulses) needed by qubit which act as quantum logic gates. It is the digital description of the required analog pulses in the below layers. It also gives quantum measurement as feedback to the above classical layer for merging the quantum outcomes to the final result.

Analog Layer- It creates voltage signals which are having a phase and amplitude modulations like in wave, for sending it to the below layer so that qubit operations can be executed.

Quantum Layer- It is integrated with the digital and the analog processing layer onto the same chip. It is used for holding qubits and is kept at room temperature (absolute). Error correction is handled here. This layer determines how well the computer performs.

Quantum Processing Unit (QPU) is made up of three layers including the digital processing layer, analog processing layer, and quantum processing layer. QPU and classical layer together constitute the Quantum Computer. Digital and Analog layers operate at room temperature.

The architecture of a practical quantum computer. It can be divided into five layers, each performing different types of processing [ 12 ].

9. Hardware and software of quantum computers

There should be an interface between the quantum computer and conventional computers for tasks related to data, networks, and users. In order to function usefully, the quantum qubit system needs organized control that can be managed by a conventional computer. The necessary hardware components for analog quantum computers are designed in 4 conceptual layers. First is the “quantum data plane” where qubit is present. Second is the “control and measurement plane” which is liable for performing operations and measurement on qubits as needed. The third is the “control processor plane” which defines the sequence of those operations and measurement outcomes to inform successive quantum operations required by the algorithm. And the last one is “host processor” which is a classical computer running a conventional operating system that handles user interfaces, network access, and big storage data structures. The processor is controlled using a high bandwidth connection that it provides [ 13 ].

A functional Quantum computer also requires software components in addition to the hardware. It is comparable to classical computers. Various new tools including programming languages are needed to substantiate quantum operations so that programmers can formulate algorithms, compilers that can map them to the hardware used by quantum computers and some other supports which can evaluate, optimize, debug and test programs. The programming language must be designed for any targeting quantum architecture. Some preparatory tools have been developed to support quantum computers and are accessible on the web [ 14 ]. These tools must be designed in an abstract way so that software developers can think more algorithmically without much concern for details of quantum mechanics. This software must be flexible enough to adapt to the changes in hardware and algorithms. This is one of the biggest challenges in quantum computing to develop complete software architecture. Other than programming languages, there must be simulation tools for modeling quantum operations and tracking quantum states and optimization tools for evaluating needed qubit resources so that it can perform different quantum algorithms in an efficient manner. The main goal is to minimize the number of qubits and the operations required for the hardware [ 15 ].

10. What is quantum algorithm?

An algorithm is a sequence of instructions or a set of rules to be followed to perform any task or calculation. It is a step-by-step process for solving a problem, especially by a computer. Any algorithm that can be executed on a quantum computer is called the Quantum algorithm. Generally, it is possible to execute all classical algorithms on quantum computers. However, the algorithms should contain at least one unique quantum step due to the property of either superposition or entanglement to be called a Quantum algorithm.

Quantum algorithms are characterized by a quantum circuit. A quantum circuit is a prototype for quantum computation that includes each step of the quantum algorithm as a quantum gate. A quantum gate is an operation that can be performed on any number of qubits. It changes the quantum state of the qubit. It can be divided into a single-qubit or multi-qubit gate, depending on the number of qubits on which it is applied at the same time. A quantum circuit is determined with qubit measurement [ 16 ].

An algorithm executing on a simulator rather than hardware is very profitable in terms of execution time by replacing the measurement overhead at the end of the algorithm. It is also known as simulation optimization. A quantum algorithm is always reversible when compared to the classical algorithm. It implies that if the measurement is not considered, a quantum circuit can be traversed back which can undo all the operations done by a forward traversing of the circuit. According to the undecidability problem, all problems that are unsolvable by a classical algorithm cannot be solved by quantum algorithms too. But these algorithms can solve problems significantly faster than classical algorithms. Some examples of the quantum algorithm are Shor’s algorithm and Grover’s algorithm. The Shor’s algorithm can do factorization of very large numbers in exponentially faster than best-known classical algorithms [ 17 ], whereas, Grover’s algorithm is used for searching large unordered list or unstructured databases that is four times faster than the classic algorithm [ 18 ].

Fourier transform-based quantum algorithms

Amplitude amplification-based quantum algorithms

Quantum walks based algorithm

BQP-complete problems

Hybrid quantum/classical algorithms

11. Design limitations of quantum computer

The exponential computing power of quantum computers can be accomplished by assessing and rectifying any kind of design limitation which helps to avoid their quality degradation. There are four major design limitations. The first limitation is that the number of coefficients in Dirac notation that defines the state of a quantum computer rise exponentially with the rise in the number of qubits, only when all the qubits get entangled with each other. To obtain the full potential of quantum computing, qubits must follow the property of entanglement where the state of any qubit must be linked with states of other qubits. It cannot be achieved directly since it is hard to generate a direct relation between qubits. But it can be decomposed into a number of simple fundamental operations directly aided by the hardware. One can also perform indirect coupling which is known to be an overhead in machines in classical computing and is crucial at the early stages of development especially when qubits and gate operations are confined.

The second limitation is that it is impossible to copy an entire quantum system because of a principle called a no-cloning principle [ 20 ]. There is a risk of deletion of arbitrary information from the original qubits since the state of qubits or set of qubits are moved to another set of qubits rather than being copied. The generation and storage of copies of intermediate states or partial outcomes in memory is a necessary aspect of classical computing. But quantum computers need a different strategy. There are quantum algorithms that help to access classical bits from the storage so that it can be known which bits are loaded and being queried into the memory of the quantum system to perform its task successfully.

The third limitation is due to the absence of noise protection of qubit operations. The small deformities in gate operations or input signals are collected over time disturbing the state of the system because they are not discarded by the fundamental gate operations. This can highly affect the calculation preciseness, measurements and coherence of the quantum systems and lessen the qubit operations integrity [ 21 ].

The final limitation is the incapability of the quantum machine to identify its full state even after it has finished its operation. Assume quantum computer has introduced an initial set of qubits with the superposition of all states combination. After applying a function to this state, the new quantum state will have information about the function value for each possible input and measuring this quantum system will not give this information. Therefore, a successful quantum algorithm can be achieved by manipulating the system in such a way so that states after finishing the operations have a higher probability of getting measured than any other probable result.

12. Approaches to quantum computing

If we can design each gate slightly different from others, then the generated electric signals on communicating with each other produce periodic noise in each other. Thus, the noise immunity of gates used will be adequate to cancel the impact of various noise origins. Therefore, the concluding system will produce the same outcome as the logical gate model, even with millions of gates operating in parallel. The goal of the design is to minimize the noise in qubit that can prevent the qubit state to pass through noisy channels. The qubit state can be changed by changing its physical energy environment.

Thus, it leads to 2 approaches to quantum computing. In the first approach, the energy environment representing Hamiltonian is frequently changed smoothly as qubits operations are analog in nature and smoothly changes from 0 to 1 which cannot be completely corrected. It initializes the quantum state and then uses Hamiltonian directly to develop the quantum state. This is known as ‘Analog Quantum Computing’. It includes quantum annealing, quantum simulation and adiabatic quantum computers.

The second approach is similar to the classical computer approach where the problem is decomposed into a sequence of fundamental operations or gates. These gates have adequately defined digital outcomes for some input states. The set of fundamental operations of quantum computing is different from that of classical computing. This approach is referred to as ‘Gate-based quantum computing’.

13. Different categories of quantum computer

13.1 analog quantum computer.

This type of system performs its operation by manipulating the analog values in the Hamiltonian representation. It does not use quantum gates. It includes quantum annealing, quantum simulation and adiabatic quantum computing . The quantum annealing is done using some initial set of qubits that gradually changes the energy encountered by the system until the problem parameters are defined by Hamiltonian. This is done in order to get the highest probability final state of the qubits that corresponds to the solution of that problem. The adiabatic quantum computer performs computation using some initial set of qubits in the Hamiltonian ground state and then Hamiltonian is changed slowly enough such that it stays in its ground state or lowest possible energy while the process takes place. It has processing power similar to a gate-based computer but still cannot perform full error correction.

Quantum Annealing

A basic rule of physics is that everything inclines towards a minimum energy state of a problem. This behavior is also true in the world of quantum physics. Quantum annealing is naturally used for real low-energy solutions such as optimization problems [ 22 ]. It is useful where the best solution is needed out of all possible solutions available. However, it is least powerful among all the types available. An example of this demonstrates an experiment to optimize traffic flows in a crowded city. Such an algorithm could successfully decrease traffic by choosing a convenient path. Volkswagen performs this with Google and D-wave system partnership. Such an experiment can be applied on a universal scale for all to get the cost-productive travel. This method can be applied to a collection of industry problems. For example, optimization of the flight route, petroleum price, weather and temperature information and passenger details, developing commercial aircraft.

Quantum Simulation

Adiabatic Quantum Computing

Adiabatic quantum computing is the most dominant, commonly applicable and hardest to create. A truly adiabatic quantum computer will use over a million of qubits. The maximum qubits we can access is less than 128 today. The basic idea behind this is that the machine can be directed at any complex calculation and obtain an immediate solution. This comprises analyzing the annealing equations, quantum phenomena simulation, etc. [ 24 ]. At least fifty unique algorithms other than Shor’s and Grover’s algorithm have been formulated to run on this quantum computer.

There is a possibility that quantum computers could revolutionize the area of artificial intelligence and machine learning. Some work has been done on algorithms that would operate as building blocks of machine learning but the hardware and software for quantum AI are still not practically accessible.

13.2 NISQ gate-based computer

NISQ stands for Noisy Intermediate-Scale Quantum. It is also known as the Digital NISQ computer. These type of systems are gate-based and operates on a collection of qubits without full error correction and cannot restrict all the errors. The computations must be designed in a way so that they remain practical on a quantum system with little noise and can be finished in fewer and sufficient steps so that Decoherence and gate errors do not hide the outcomes [ 25 ].

13.3 Gate-based quantum computer with full error correction

Such computers also perform gate-based operations on a set of qubits with the implementation of the Quantum Error Correction algorithm. It reduces or corrects the noise in the system occurring during the computation period. Errors may include inadequate signals, device forgery or undesired bonding of qubits to the environment or with each other. The error is reduced to such a limit that the system seems valid and precise for all computations. Such quantum computers can have various realizations and they must fulfill some conditions such as there must be an availability of a well-defined two-level system that can be used as qubits, a potential to initialize those qubits, a sufficiently extended amount of Decoherence time which can perform error correction and computation, quantum gates (a set of quantum operations) common for every quantum computation and a capability of measuring each quantum bit individually without bothering others [ 26 ]. The analog quantum computers and digital NISQ computers are in progress while the gate-based computers with full error corrections are much more difficult and demanding.

14. Advantages of quantum computing

According to researchers, quantum computers will be able to solve those complex mathematical problems that traditional computers find impossible to solve in a practical timeframe.

It provides that computing power which can sufficiently process excessively large amounts of data (2.5 Exabyte daily i.e. equal to 5 million laptops) created all around the world to extract meaning from it.

Due to the teleportation phenomenon known as ‘quantum tunneling,’ it can work in parallel and use less amount of electricity, hence, reducing the power consumption up to 100 to 1000 times.

A general quantum computer is “thousands of times” faster than any classical computer. For example, Google has made a quantum computer [ 27 ] that is 100 million times faster than any classical computer present in its lab.

It can solve complex problems without being overheated since for its stability it kept cold up to 0.2 Kelvin inside the quantum system.

It can easily solve optimization problems such as finding the best route and scheduling trains and flights. It would also be able to compute 1 trillion moves in chess per second. Quantum computers will be able to crack the highest security unbreakable encryption techniques. However, it would also build hack-proof alternates.

It can bring up revolution from drugs to petroleum industries. The invention of new drugs will become possible. The marketable algorithms of financial organizations can be improved. The field of artificial intelligence can be improved soon.

15. Disadvantages of quantum computing

Due to advancements in quantum computers, the security of the existing Internet of Things (IoT) would fall down. Cryptographic techniques, Databases of government and private large organizations, banks, and defense systems can be hacked. Considering these facts, quantum computers can be terrible for our future.

The Quantum Computer will work as a different device and cannot replace classical computers entirely. Since, classical computers are better at some chores than quantum computers like email, excel, etc.

It has not been invented completely yet as only parts are being implemented and people are still imagining how it would look.

It is very delicate and error-prone. Any kind of vibrations affects subatomic particles like atoms and electrons. Due to which noise, faults, and even failures are possible. It leads to “ Decoherence ” which is a loss of coherence in quantum.

Quantum processors are very unstable and are very hard to test even. For the stability of the quantum computer, it is kept at 0.2 Kelvin (absolute Kelvin) which is nearly below the universe temperature [ 28 ]. It is very hard to maintain and regulate such temperature. The main problem is to really develop it as a personal computer with the price range in the budget of consumers. They will be firstly accessible to large scale industry then come to retail markets.

16. Applications of quantum computing

Cryptography

Optimization Problems

Artificial Intelligence

It is an important utility in the field of quantum chemistry and material science [ 31 ]. This problem needs solving ground state energies of electrons and their wave functions, with or without the presence of some external electric or magnetic field. From the structure of atoms and electrons in chemistry to the rate at which chemical reactions are taking place, everything can be simulated very well. The classical computer when applied to this problem often fails to reach the level of precision needed to predict the rate of the chemical reaction.

It could also have commercial applications in areas such as medical and healthcare fields, chemical catalysts, storage of energy, pharmaceutical advancement and device displays.

17. Major challenges in quantum computing

The good news is that at any instant of time, the quantum state with the same number of quantum bits can stretch over all possible states as compared to classical computers and thus works in an exponentially massive space. However, to be able to use this space requires all qubits to remain interconnected. Even after such progress, improvements are still needed. The bad news is that making new and high-quality qubits does not guarantee the creation and efficient use of fault-tolerant quantum computers and is still having challenges in its path [ 32 ].

Qubits cannot naturally ignore the noise. Hence, the quantum system is more error-prone. It suffers from Decoherence . The biggest challenge is how it can handle any undesirable deviations or noise in quantum computers. Classical computers can produce clean noise-free outcomes by simply putting its state as off or ‘0’, which is not possible for quantum computers where errors occur in physical circuits. Qubits will gradually lose its information as well as interconnection (entanglement) between each other. The error rate is seen as a design parameter for such systems which should be improved in large qubit systems also. However, to make the qubits stable and error-free, they are being insulated from the outside environment in super-refrigerated fridges or vacuum chambers and accurately handled [ 33 ].

Qubits are neither completely binary nor digital. It is having analog properties also. Gate can reject noise by dealing with the input signal value of 0.8 and treating it as 1. But in the analog signal, every value between 0 and 1 is permitted since they have their meanings. Signals cannot be checked for any kind of noise or corruption. Since 0.8 can be 1 with some error or 0.8 without error. Presuming the error as 0 like Gates do or taking some noise value even if it was not present there can affect the adherence of the resulting quantum computation. Hence, there is a need for algorithms like quantum error correction similar to the logical error correction in classical computers. These algorithms can be run on a noisy gate-based quantum computer to eliminate the errors and noises present in them [ 34 ].

It is possible to employ a Quantum Error Correction algorithm on a quantum system. But quantum error correction requires dealing with the overhead such as a large number of qubits and their fundamental operations and generally needs more resources. Also, problems with large data inputs require a large amount of time to create the input quantum state that would monopolize the computation time lessening the quantum benefits.

Quantum algorithm development is another challenge since achieving quantum speedup expects entirely new types of algorithm design as the speed of computation depends on the design of the algorithm. The design of the algorithm should be corresponding to the number of qubits used.

Further development of software tools in addition to hardware, is required to create and debug quantum systems to help explain unknown issues and push towards designs.

Debugging quantum hardware and software is of utmost importance which depends on memory and intermediate machine states in classical computers. But in the case of quantum computing, states cannot be copied directly for later evaluation, and directly measuring intermediate state can bring it to halt . Hence, new strategies for debugging are essential for their development.

18. Importance of quantum computing

It is clearly possible to build a quantum computer that could perform computations that would run a lifetime on a classical computer. Practical applications of quantum computing need controlling the quantum phenomena and thus the quantum world to an exceptional level. This job requires substantial engineering and research to build, manage and employ a noiseless quantum system. The experiment with quantum supremacy is an important test of the theory of quantum mechanics that will help to improve the support of quantum theory and leads to unexpected discoveries. The development of aspects and components of quantum information technology and computing has already started to influence the area of physics. The quantum error correction theory to attain the fault-tolerant quantum system has proven important. The quantum information theory is practically useful to study physics and dynamics of multibody systems like a massive number of quantum subatomic particles and even in blackhole and related concepts. Advancement in this area is important for an accurate understanding of various physical structures. It has contributed to many other engineering fields like physics, mathematics, chemistry, computer science, material science, etc. It has also advanced classical computing. Strategies to develop a quantum computing algorithm have helped in improving the classical computing algorithm also. Research in the quantum algorithm has answered many questions in the computer science area. It can help to evaluate the safety of cryptographic systems, clarifying the limitations of physical computational and advancing computational methods. It will help to advance the human’s understanding of the universe. The qubits that are recently being used in quantum computing is also used for building sensors, precision clocks, and other applications. Quantum communication is used for communicating two quantum systems at distance. There is an increased risk of asymmetric cryptography as well as the entire security system. Hence, the actions are being taken towards new quantum cryptography. The development of quantum information, science, technology and computing is a global area now.

19. Future scope of quantum computing

A significant amount of struggle is remaining before a practical quantum computer can be launched. There are some future advancements that are needed. Some of the future needs are enabling a Quantum Error Correction algorithm that requires low overhead and decreases the error rates in qubits, developing more algorithms with lesser qubits for solving problems, reducing circuit thickness so that NISQ computers can be operated, the advancement of methods which can verify, debug, and simulate the quantum computers, scaling the number of qubits per processor in such a way so that error rate is maintained or can be improved if possible, interleaving of operations in a qubit, recognizing more algorithms that can reduce the computation time and creating input–output for the quantum processor.

Such ‘ Quantum games’ are predicted in the future that will give unexpected situations and results that a player can experience because quantum computers will take all the possible operations and throws them into the game randomly due to its quantum properties like superpositioning and entanglement of qubits. It will be a never-ending experience.

‘ Quantum computing in Cloud’ has the potential to overtake business initiatives like in other emerging technologies such as cryptography and artificial Intelligence. Since the classical simulation of fifty qubits is equal to the memory of one Petabyte that doubles with every single qubit added [ 35 ], the memory required should also be large enough to provide an environment for application development and testing for multiple developers to simulate quantum computers using suitable shared resources.

AI and machine learning problems could be solved in a practical amount of time that can be reduced from hundreds of thousands of years to seconds. Several quantum algorithms have been developed such as Grover’s algorithm for searching and Shor’s algorithm for factoring large numbers. More quantum algorithms are coming soon. Google has also declared that it would produce a workable quantum computer in the following 5 years with a 50-qubit quantum computer and will achieve quantum supremacy. IBM is also offering commercial quantum computers soon.

The progress of development in the field of quantum computers depends on many factors. Interest and financial support from the private sector can help developing commercial applications for NISQ computers. It depends on the progress of quantum algorithm development, availability of enough investment in the quantum technology field from government and the exchange of ideas within researchers, scientists and engineers [ 36 ]. To illuminate the limitations of quantum technology, a defensive result is also beneficial. It can help in overcoming those negative results which can lead to a new discovery.

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What does the future hold for quantum computing? Experts explain

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• A World Economic Forum issue briefing on quantum computing explored the technology, where it's come from and where it's heading.
• Experts from academia and industry discussed the challenges, next steps and where we might be in five years and beyond.
• There's a lot of uncertainty, but all participants stressed the need for collaboration.

Most of you are probably familiar with computers. But what about quantum computers?

Commercial quantum computers may still be a way off, but Goldman Sachs announced quantum algorithms could price financial instruments in the next five years.

At a World Economic Forum issue briefing, experts from industry and academia looked at the technology - what it is, where it's come from and where it's heading.

Taking part were Dario Gil , Senior Vice President and Director Research, IBM; Freeke Heijman , Founding Director at Quantum Delta, NL; Jeremy Jurgens , Managing Director, World Economic Forum; Robert Hackett , Senior Writer, Fortune ; John Preskill , Amazon Scholar, Amazon Web Services (AWS) and Richard P. Feynman Professor of Theoretical Physics, Caltech.

Have you read?

Issue briefing: what’s next for quantum computing, is your cybersecurity ready to take the quantum leap.

Here are some of the key talking points. You can watch the full session here .

How is a quantum computer different from a regular computer?

"When many particles interact with one another, according to the principles of quantum mechanics, then it turns out it's extraordinarily complex to describe what those particles are doing using ordinary language or classical computers," explained John Preskill .

"And so, if we can control how those particles interact, then we should be able to perform information processing tasks that wouldn't otherwise be possible."

What could we do with quantum computers?

There's lots we still don't know though, said Preskill, including the areas where quantum computers will have a big advantage.

At the moment, the most important applications will be in reference to how matter behaves, chemistry and the discovery of materials, he explained. But, it will still take time to understand how we scale this up, he added.

"It's not that we understand all the science now, and we just need to put the resources together to do the engineering. We're going to need a lot of innovation to get to quantum computers that can solve those very impactful problems."

Equally, though, Preskill emphasized that it's important to understand we have "very limited ability at this stage to imagine the applications of quantum computing down the road in the near term".

Next-generation technologies such as AI, ubiquitous connectivity and quantum computing have the potential to generate new risks for the world, and at this stage, their full impact is not well understood.

There is an urgent need for collective action, policy intervention and improved accountability for government and business in order to avert a potential cyber pandemic.

The Forum's Centre for Cybersecurity launched the Future Series: Cybercrime 2025 initiative to identify what approaches are required to manage cyber risks in the face of the major technology trends taking place in the near future.

Find out more on how the Forum is leading over 150 global experts from business, government and research institutions, and how to get involved, in our impact story .

Where do things stand with the technology and where are we headed next?

About five years ago, we were able to build a small quantum computer and make it available through the cloud, explained Dario Gil . And, as a result, we started exposing a growing community to the concept.

Since that point, the number of systems available has grown and "what we're starting to see is a broad set of institutions getting interested in the topic".

We need to work to deliver better and better quantum computers and systems every year, he said. "But, it's very important to bring people along on the journey." We can't just say 'one day we'll have a magical computer that does all these things', we need to come together and work through it together, he urged.

Just a few years ago, we used machines with just 5 cubits. But, IBM hopes to build a machine with over 1,000 cubits by 2023, Gil said.

But, the number of cubits alone doesn't tell you the power of a quantum computer, he added. "The capacity of the machine, how many circuits you can run, the quality with which you can run these circuits. All of those are extraordinarily important."

What other factors do we need to consider beyond just the technology?

"What we're looking at is not only the technological roadmap, but an ecosystem approach," explained Freeke Heijman . "How do you get the right talent, how to educate the right people to move into the field?"

Investment and capital are also important - and the opportunity exists for start-ups, she believes. Equally, collaboration will be key. "We're working very closely in a European context," she said. "For instance, we just signed a memo with the French."

This Memorandum of Understanding will enable collaboration, but it's vital it's not just about high-level policy but concrete action, Heijman emphasized. For example, there's now a website where you can see all the relevant available jobs in the Netherlands and France.

"We would just like to see as much openness in the ecosystem as possible so that people can connect and build new technology together."

Of course, it's still competitive, she said. But, because the industry is still in its early days, there's already a lot of collaboration and exchanges of ideas and people.

"We're all looking forward to much better performance, which we're hoping to achieve both through improvement in the hardware and through software methods for mitigating the imperfections of the hardware," added Preskill.

"We all want to see those innovations happen and that'll bring us closer to the day when quantum computing can have a broad impact on humanity."

The sentiment was echoed by Gill. "I think one of the things that unites all of us is we want to see a quantum industry thrive and be successful... I think we're all excited about the historical moment that we're witnessing," he said.

How can we enable this collaboration in quantum computing?

There are still a lot of unknowns in the field, explained Jeremy Jurgens . That's why this collaboration is so important. The World Economic Forum has established the Quantum Computing Network, which facilitates collaboration with companies like Amazon, IBM and Microsoft, as well as academic institutions and national governments.

And this extends beyond the technology itself. What are the skills we'll need? How can we make sure the benefits are extended across society?

"We need to both harness the opportunities in front of us... as we also manage the risks and prepare society more broadly for the significance of the transition," said Jurgens.

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The world is on the cusp of another computer revolution. It will be driven by the convergence of powerful technologies: high performance computing, AI, and quantum computing.

Quantum computing is not simply a faster way of doing what today’s computers do – it is a fundamentally different approach that promises to solve problems that classical computing can never realistically solve. It holds the promise to help humanity confront many important challenges, from solving long-standing questions in science to overcoming obstacles in improving industrial efficiency. Working in conjunction with classical computers and cloud-based architectures, quantum computers could even be the answer to problems we haven’t yet dreamed of. The opportunities for society and the economy are potentially limitless.

Quantum computing can help to expedite the response to future pandemics, ongoing health crises, and the proliferation of debilitating diseases affecting millions worldwide through vastly improved chemical simulations in drug discovery and development. It will be able to improve the simulation accuracy of computational fluid dynamics, allowing for lower-cost approaches to improved manufacturing design processes. And it could aid in optimizing portfolio investment strategies, using advanced modeling techniques that can better analyze the behavior of complicated financial markets.

But quantum computers also pose a challenge for one important area of digital life: encryption.

The Crypto Conundrum

As detailed in a recent essay from IBM Research, advances in quantum computing will eventually present a significant information security challenge. The world is already heavily reliant on cryptography to protect data and critical infrastructure, and as we transition into an era in which quantum computers become more ubiquitous, digital platforms being designed and deployed today can increasingly be vulnerable without the concurrent development and adoption of quantum-safe encryption.

The world is still a long way off from quantum computers that could break today’s widely-used cryptography. And we already know how to perform encryption that is resistant to a quantum computer’s attack. Yet such foundational quantum-safe algorithms are only the start. Many industry security standards and protocols need to be updated for these new algorithms, and advances in quantum computing will need to coincide with advances in quantum-safe cryptography to ensure data is secured now from future threats.

Preparing for Tomorrow by Future-Proofing in the Present

To prepare for what comes next, policymakers and industry need to look to mitigate against these risks by future-proofing in the present. We must act now.

IBM is. Our Researchers are developing practical cryptographic solutions that are resistant to the threats posed by quantum computers. We have found a number of cryptographic schemes that are currently thought to be quantum-safe. These include lattice-based cryptography, hash trees, multivariate equations, and super-singular isogeny elliptic curves.

The key advantage of such quantum-safe schemes is the absence of exploitable structure in the mathematical problem an attacker needs to solve in order to break the encryption. Certain such quantum-safe schemes (e.g., supersingular isogeny) are future-proofed against particularly patient attackers who store their victims’ encrypted messages today to decrypt them with more powerful methods in the future. Other schemes (e.g., lattice cryptography) can enable game-changing technologies like fully homomorphic encryption, in which data can be directly computed upon in its encrypted form, stymieing a common strategy of attackers today to loiter in a victim’s computer system until sensitive data has to be taken out of encryption to perform computations upon it.

To advance these and other innovative new methods for securing data in an age of quantum computing, we are collaborating with academic institutions – such as the University of Waterloo and the University of Toronto – to advance the science behind these techniques. IBM is also engaging in global efforts to standardize quantum-safe cryptography. The most notable of these is the NIST PQC process. IBM has submitted a number of algorithms to the NIST PQC process and is working closely with other industry leaders in standard development organizations, such as ETSI, ISO and ANSI. But governments have a role to play here too.

To supplement private industry’s engagement in standards development, governments need to accelerate investments in, and promote the adoption of, quantum-safe cryptographic schemes that can safeguard data now and long into the future.

Quantum Readiness

Forward leaning companies and governments are preparing for a quantum computing future and positioning themselves ready to capture the many benefits of this technology. Yet, more can and should be done and collaboration is key. Governments, researchers, academics, and industry will need to work together on policies to accelerate the adoption of new educational curricula, fund R&D, create new talent pipelines, and more.

As governments look to lead in quantum computing, policymakers should consider the following recommendations:

• Governments should recommend adoption of quantum-safe cryptography now to address future threats to data being encrypted today. 
• Standard development organizations and their members should accelerate efforts around new quantum-safe cryptographic schemes and prioritize workstreams to establish a quantum-safe infrastructure.   Government agencies, such as NIST, should take the lead in convening industry to agree on quantum-safe cryptographic standards, working with international partners.
• Government agencies should accelerate the development of quantum computers through significant, sustained, and focused long-term investment in quantum information science to secure their nation’s position at the forefront of the quantum computing race. Fundamental research in quantum theory, hardware, and software includes the development of novel qubits, methods to improve the quality of qubits and their performance in quantum circuits, techniques to mitigate and correct errors, development of optimized quantum circuits and compilation schemes, and components to enable the development of advanced scaling technologies.
• Government agencies should support the rapid deployment of advanced, reliable quantum systems by being an early adopter to help drive developments and to enable an ecosystem of research, software and algorithm development, and commercialization. It is essential to promote industry uptake and experimentation and user-level ecosystem building in parallel to the efforts to advance the development of the hardware and systems themselves.
• Governments must foster a collaborative framework involving national laboratories, academia, industry and international partners to advance the technology and build a competitive edge. Government, academia, and industry must work together to advance the fundamental science and execute on an efficient and aggressive development roadmap with meaningful, well-defined metrics.
• Governments should help build a robust enabling technology ecosystem and supply chain for the quantum industry and promote education and training of the necessary workforce to make the industry sustainable. Examples of initiatives to build up quantum industry supply chains include the Quantum Economic Development Consortium (QED-C) in the United States, and IBM’s collaboration with Germany’s Fraunhofer-Gesellschaft, Europe’s leading organization for applied research.
• Standard development organizations should prioritize the updating of system-relevant industry standards such as critical infrastructure and financial industry standards. The efforts should include updates to ISO 27001, COBIT, NIST SP 800-53, ANSI/ISA-62443, and standards developed by the Council on Cybersecurity Critical Security Controls.

-Ryan Hagemann, co-Director, IBM Policy Lab – Washington, DC

-Zaira Nazario, Technical Lead, Quantum Theory, Algorithms, and Applications, IBM Quantum

About IBM Policy Lab The IBM Policy Lab is a forum providing policymakers with a vision and actionable recommendations to harness the benefits of innovation while ensuring trust in a world being reshaped by data. As businesses and governments break new ground and deploy technologies that are positively transforming our world, we work collaboratively on public policies to meet the challenges of tomorrow.

Sign up for the IBM Policy Lab newsletter for our latest updates:

Media Contact: Jordan Humphreys [email protected]

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Home — Essay Samples — Information Science and Technology — Information Technology — Quantum Computing: Beyond the Limits of Traditional Computers

Quantum Computing: Beyond The Limits of Traditional Computers

• Categories: Information Technology Quantum Mechanics

About this sample

Words: 527 |

Published: Feb 12, 2019

Words: 527 | Page: 1 | 3 min read

Works Cited

• Preskill, J. (2018). Quantum Computing in the NISQ era and beyond. Quantum, 2, 79.
• Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press.
• Arute, F., Arya, K., Babbush, R., Bacon, D., Bardin, J. C., Barends, R., ... & Boixo, S. (2019). Quantum supremacy using a programmable superconducting processor. Nature, 574(7779), 505-510.
• Kitaev, A. Y. (2002). Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1), 2-30.
• Ladd, T. D., Jelezko, F., Laflamme, R., Nakamura, Y., Monroe, C., & O’Brien, J. L. (2010). Quantum computers. Nature, 464(7285), 45-53.
• Shor, P. W. (1994). Algorithms for quantum computation: discrete logarithms and factoring. In Proceedings 35th Annual Symposium on Foundations of Computer Science (pp. 124-134). IEEE.
• Steane, A. M. (1998). Quantum computing. Reports on Progress in Physics, 61(2), 117.
• Vandersypen, L. M., & Chuang, I. L. (2005). NMR techniques for quantum control and computation. Reviews of Modern Physics , 76(4), 1037.
• Montanaro, A. (2016). Quantum algorithms: an overview. npj Quantum Information, 2(1), 1-14.

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Opinion: The quantum computing race is on

A lthough scientists don’t completely understand the underlying science of quantum mechanics , a global geopolitical, military and commercial race is on for the development of affordable and reliable quantum computers and associated software. How long it will take and how much it will cost is far from clear.  

Over the next few years, the world will learn whether practical (not experimental) quantum computing is another “technology that is always 5 years away” or a revolution in high-end computing. Many governments and businesses fear they cannot be left behind, because quantum computing technology is so powerful. Thus, the race is on for a practical quantum computer with software. 

For the average person, there are few areas of science more confusing than quantum physics. No less than Albert Einstein once disputed some theoretical foundations of quantum mechanics; quantum notions like an object can be in two places or be in two different states at the same time do not make sense. But by the 1960s, quantum physics was established as a discipline, and by the 1980s it was commonly recognized that quantum mechanics could someday be used to operate a computer — and that quantum computers could be far more powerful than the transistor/chip-based computers that became widespread after World War II.  

By the 1990s, most scientists involved had concluded that a quantum-based computer could perform in some respects far faster than any conventional computer, even with the processing speed of conventional computers increasing. This led to the development of important algorithms and software platforms designed to harness the raw power of a quantum computer, and then to significant efforts in the 2000s to actually build a functioning and reliable quantum computer. Finally, all of this led to questions as to how such computers might actually be used and what tasks a quantum computer could perform that a conventional supercomputer could not.  

The results of these millennial investigations drew global attention to what until then had been largely theoretical. The speed of quantum computers could allow discoveries in fields in which the variables are so vast that the process would be out of the reach of conventional supercomputers. These include such fields as pharmaceuticals, patterns of climate, chemical compounds and new materials . Perhaps more important, the speed of quantum computing could allow virtually any cryptographic secret code to be broken, and it could supercharge the ability of artificial intelligence (and its robot progeny) to do almost anything imaginable at lightening speeds.  

During the 2010s, big tech companies like Google and IBM variously claimed that their experimental quantum computers could perform functions in minutes that it would take a supercomputer decades to perform. (It’s important to note, however, that the overwhelming number of calculations that any person, enterprise or government performs every day do not require the vast speed of a quantum computer; powerful chip-based computers will surely continue to serve almost every common computing function for the foreseeable future.) And so, by the 2020s, the commercial, geopolitical and military races to develop a working and reliable quantum computer and software for very special high-end functions had begun. 

Building a reliable quantum computer takes a lot of infrastructure, material and expert staff. For example, many prototypes rely on stable temperatures of near absolute zero, along with a variety of highly educated and experienced scientists. Although large companies like Google, IBM, Fujitsu, Amazon and Microsoft began investing in efforts to build practical quantum computers during the 2010s, start-up investments in quantum computing only began to take off in 2021-2022, at what McKinsey reports to be almost $2.5 billion per year (dropping to around $1.7 billion in 2023.)  

More important have been recent quantum investments by governments. Depending on the country involved, much government spending on quantum computer R&D is contracted out to technology companies — thus government and industry often have symbiotic quantum R&D efforts. Governments’ spending on quantum computing is driven by a combination of an economic fear of being left behind by other economic powers (EU, U.S., Japan, India and China) and a geopolitical fear of being overpowered by a military rival (U.S., China, Russia, NATO countries). 

By far, the largest government efforts to develop reliable quantum computing services are by China and the U.S. , which have respectively committed to invest around $15 billion and $5 billion (the U.S. estimate excludes major associated private-sector investments.) China has established a multibillion-dollar Quantum Lab, while the US announced its National Quantum Initiative , which brings together billions in U.S. military, scientific and civil quantum efforts. Not far behind is the combination of the EU’s billion-dollar Flagship quantum Initiative and over $8 billion in commitments to associated quantum research by Germany, France and Holland. Finally, individual billion-dollar-plus quantum computing R&D commitments have been made by the UK, India, Japan, Russia, South Korea and Canada.  

While PR announcements about quantum developments are often issued by governments, big tech and start-ups, few, if any, claim that a practical/reliable quantum computer exists . Moreover, although important advances in quantum computing are likely to take place in secret within various nations’ military/intelligence programs, the usefulness of a secret military quantum computer designed to protect military secrets for eventual civil or commercial uses is unclear (secret military technologies sometimes take decades to find civil and commercial applications.)  

The development of reliable, quantum computers and supporting software will probably take years, and their eventual usefulness is likely to be tightly focused on secret cryptography or such mega-tasks as the development of new materials, drugs and chemicals or the analysis of financial or scientific data. Whether politicians, investors and the media have the patience to see this epic effort through to its conclusion is probably the largest, among many, quantum computing uncertainties.   

  Roger Cochetti has served as a senior executive with COMSAT, IBM, VeriSign and CompTIA. A former U.S. government official, he has helped found a number of nonprofits in the tech sector and is the author of textbooks on the history of satellite communications.  

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Quantum computing has a hype problem

Quantum computing startups are all the rage, but it’s unclear if they’ll be able to produce anything of use in the near future.

• Sankar Das Sarma archive page

As a buzzword, quantum computing probably ranks only below AI in terms of hype. Large tech companies such as Alphabet, Amazon, and Microsoft now have substantial research and development efforts in quantum computing. A host of startups have sprung up as well, some boasting staggering valuations. IonQ, for example, was valued at $2 billion when it went public in October through a special-purpose acquisition company. Much of this commercial activity has happened with baffling speed over the past three years.  

I am as pro-quantum-computing as one can be: I’ve published more than 100 technical papers on the subject, and many of my PhD students and postdoctoral fellows are now well-known quantum computing practitioners all over the world. But I’m disturbed by some of the quantum computing hype I see these days, particularly when it comes to claims about how it will be commercialized.

Established applications for quantum computers do exist. The best known is Peter Shor’s 1994 theoretical demonstration that a quantum computer can solve the hard problem of finding the prime factors of large numbers exponentially faster than all classical schemes. Prime factorization is at the heart of breaking the universally used RSA-based cryptography, so Shor’s factorization scheme immediately attracted the attention of national governments everywhere, leading to considerable quantum-computing research funding.   

The only problem? Actually making a quantum computer that could do it. That depends on implementing an idea pioneered by Shor and others called quantum-error correction, a process to compensate for the fact that quantum states disappear quickly because of environmental noise (a phenomenon called “decoherence”). In 1994, scientists thought that such error correction would be easy because physics allows it. But in practice, it is extremely difficult. 

The most advanced quantum computers today have dozens of decohering (or “noisy”) physical qubits. Building a quantum computer that could crack RSA codes out of such components would require many millions if not billions of qubits. Only tens of thousands of these would be used for computation—so-called logical qubits; the rest would be needed for error correction, compensating for decoherence. 

The qubit systems we have today are a tremendous scientific achievement, but they take us no closer to having a quantum computer that can solve a problem that anybody cares about. It is akin to trying to make today’s best smartphones using vacuum tubes from the early 1900s. You can put 100 tubes together and establish the principle that if you could somehow get 10 billion of them to work together in a coherent, seamless manner, you could achieve all kinds of miracles. What, however, is missing is the breakthrough of integrated circuits and CPUs leading to smartphones—it took 60 years of very difficult engineering to go from the invention of transistors to the smartphone with no new physics involved in the process. 

There are in fact ideas, and I played some role in developing the theories for these ideas, for bypassing quantum error correction by using far-more-stable qubits, in an approach called topological quantum computing. Microsoft is working on this approach . But it turns out that developing topological quantum-computing hardware is also a huge challenge. It is unclear whether extensive quantum error correction or topological quantum computing (or something else, like a hybrid between the two) will be the eventual winner. 

Physicists are smart as we all know (disclosure: I am a physicist), and some physicists are also very good at coming up with substantive-sounding acronyms that stick. The great difficulty in getting rid of decoherence has led to the impressive acronym NISQ for “noisy intermediate scale quantum” computer—for the idea that small collections of noisy physical qubits could do something useful and better than a classical computer can. I am not sure what this object is: How noisy? How many qubits? Why is this a computer? What worthy problems can such a NISQ machine solve?

A recent laboratory experiment at Google has observed some predicted aspects of quantum dynamics (dubbed “time crystals”) using 20 noisy superconducting qubits. The experiment was an impressive showcase of electronic control techniques, but it showed no computing advantage over conventional computers, which can readily simulate time crystals with a similar number of virtual qubits. It also did not reveal anything about the fundamental physics of time crystals. Other NISQ triumphs are recent experiments simulating random quantum circuits , again a highly specialized task of no commercial value whatsoever. 

Using NISQ is surely an excellent new fundamental research idea—it could help physics research in fundamental areas such as quantum dynamics. But despite a constant drumbeat of NISQ hype coming from various quantum computing startups, the commercialization potential is far from clear. I have seen vague claims about how NISQ could be used for fast optimization or even for AI training. I am no expert in optimization or AI, but I have asked the experts, and they are equally mystified. I have asked researchers involved in various startups how NISQ would optimize any hard task involving real-world applications, and I interpret their convoluted answers as basically saying that since we do not quite understand how classical machine learning and AI really work, it is possible that NISQ could do this even faster. Maybe, but this is hoping for the best, not technology. 

There are proposals to use small-scale quantum computers for drug design, as a way to quickly calculate molecular structure, which is a baffling application given that quantum chemistry is a minuscule part of the whole process. Equally perplexing are claims that near-term quantum computers will help in finance. No technical papers convincingly demonstrate that small quantum computers, let alone NISQ machines, can lead to significant optimization in algorithmic trading or risk evaluation or arbitrage or hedging or targeting and prediction or asset trading or risk profiling. This however has not prevented several investment banks from jumping on the quantum-computing bandwagon. 

A real quantum computer will have applications unimaginable today, just as when the first transistor was made in 1947, nobody could foresee how it would ultimately lead to smartphones and laptops. I am all for hope and am a big believer in quantum computing as a potentially disruptive technology, but to claim that it would start producing millions of dollars of profit for real companies selling services or products in the near future is very perplexing to me. How? 

Quantum computing is indeed one of the most important developments not only in physics, but in all of science. But “entanglement” and “superposition” are not magic wands that we can shake and expect to transform technology in the near future. Quantum mechanics is indeed weird and counterintuitive, but that by itself does not guarantee revenue and profit.

A decade and more ago, I was often asked when I thought a real quantum computer would be built. (It is interesting that I no longer face this question as quantum-computing hype has apparently convinced people that these systems already exist or are just around the corner).  My unequivocal answer was always that I do not know. Predicting the future of technology is impossible—it happens when it happens. One might try to draw an analogy with the past. It took the aviation industry more than 60 years to go from the Wright brothers to jumbo jets carrying hundreds of passengers thousands of miles. The immediate question is where quantum computing development, as it stands today, should be placed on that timeline. Is it with the Wright brothers in 1903? The first jet planes around 1940? Or maybe we’re still way back in the early 16 th century, with Leonardo da Vinci’s flying machine? I do not know. Neither does anybody else.

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What Is Quantum Computing?

• How It Works
• Uses and Benefits
• Limitations
• Quantum vs. Classical
• In Development
• Quantum Computing FAQs

The Bottom Line

• Other Technologies

Quantum Computing: Definition, How It's Used, and Example

Investopedia / Joules Garcia

Quantum computing is an area of computer science that uses the principles of quantum theory. Quantum theory explains the behavior of energy and material on the atomic and subatomic levels.

Quantum computing uses subatomic particles, such as electrons or photons. Quantum bits, or qubits, allow these particles to exist in more than one state (i.e., 1 and 0) at the same time.

Theoretically, linked qubits can "exploit the interference between their wave-like quantum states to perform calculations that might otherwise take millions of years."

Classical computers today employ a stream of electrical impulses (1 and 0) in a binary manner to encode information in bits. This restricts their processing ability, compared to quantum computing.

Key Takeaways

• Quantum computing uses phenomena in quantum physics to create new ways of computing.
• Quantum computing involves qubits. 
• Unlike a normal computer bit, which can be either 0 or 1, a qubit can exist in a multidimensional state.
• The power of quantum computers grows exponentially with more qubits.
• Classical computers that add more bits can increase power only linearly.

Understanding Quantum Computing

The field of quantum computing emerged in the 1980s. It was discovered that certain computational problems could be tackled more efficiently with quantum algorithms than with their classical counterparts.

Quantum computing has the capability to sift through huge numbers of possibilities and extract potential solutions to complex problems and challenges. Where classical computers store information as bits with either 0s or 1s, quantum computers use qubits. Qubits carry information in a quantum state that engages 0 and 1 in a multidimensional way.

Such massive computing potential and the projected market size for its use have attracted the attention of some of the most prominent companies. These include IBM, Microsoft, Google, D-Waves Systems, Alibaba, Nokia, Intel, Airbus, HP, Toshiba, Mitsubishi, SK Telecom, NEC, Raytheon, Lockheed Martin, Rigetti, Biogen, Volkswagen, and Amgen. 

Uses and Benefits of Quantum Computing

Quantum computing could contribute greatly to the fields of security, finance , military affairs and intelligence, drug design and discovery, aerospace designing, utilities (nuclear fusion), polymer design, machine learning , artificial intelligence (AI), Big Data search, and digital manufacturing. 

Quantum computers could be used to improve the secure sharing of information. Or to improve radars and their ability to detect missiles and aircraft. Another area where quantum computing is expected to help is the environment and keeping water clean with chemical sensors.

Here are some potential benefits of quantum computing:

• Financial institutions may be able to use quantum computing to design more effective and efficient investment portfolios for retail and institutional clients. They could focus on creating better trading simulators and improve fraud detection.
• The healthcare industry could use quantum computing to develop new drugs and genetically-targeted medical care. It could also power more advanced DNA research.
• For stronger online security, quantum computing can help design better data encryption and ways to use light signals to detect intruders in the system.
• Quantum computing can be used to design more efficient, safer aircraft and traffic planning systems.

Percentage of large companies planing to create initiatives around quantum computing by 2025, according to research by Gartner.

Features of Quantum Computing

Superposition and entanglement are two features of quantum physics on which quantum computing is based. They empower quantum computers to handle operations at speeds exponentially higher than conventional computers and with much less energy consumption.

Superposition

According to IBM,  it's what a qubit can do rather than what it is that's remarkable. A qubit places the quantum information that it contains into a state of superposition. This refers to a combination of all possible configurations of the qubit. "Groups of qubits in superposition can create complex, multidimensional computational spaces. Complex problems can be represented in new ways in these spaces."

Entanglement

Entanglement is integral to quantum computing power. Pairs of qubits can be made to become entangled. This means that the two qubits then exist in a single state. In such a state, changing one qubit directly affects the other in a manner that's predictable.

Quantum algorithms are designed to take advantage of this relationship to solve complex problems. While doubling the number of bits in a classical computer doubles its processing power, adding qubits results in an exponential upswing in computing power and ability.

Decoherence

Decoherence occurs when the quantum behavior of qubits decays. The quantum state can be disturbed instantly by vibrations or temperature changes. This can cause qubits to fall out of superposition and cause errors to appear in computing. It's important that qubits be protected from such interference by, for instance, supercooled refridgerators, insulation, and vacuum chambers.

Limitations of Quantum Computing

Quantum computing offers enormous potential for developments and problem-solving in many industries. However, currently, it has its limitations.

• Decoherence, or decay, can be caused by the slightest disturbance in the qubit environment. This results in the collapse of computations or errors to them. As noted above, a quantum computer must be protected from all external interference during the computing stage.
• Error correction during the computing stage hasn't been perfected. That makes computations potentially unreliable. Since qubits aren't digital bits of data, they can't benefit from conventional error correction solutions used by classical computers.
• Retrieving computational results can corrupt the data. Developments such as a particular database search algorithm that ensures that the act of measurement will cause the quantum state to decohere into the correct answer hold promise.
• Security and quantum cryptography is not yet fully developed.
• A lack of qubits prevents quantum computers from living up to their potential for impactful use. Researchers have yet to produce more than 128, as of 2019.

According to global energy leader Iberdola, "quantum computers must have almost no atmospheric pressure, an ambient temperature close to absolute zero (-273°C) and insulation from the earth's magnetic field to prevent the atoms from moving, colliding with each other, or interacting with the environment."

"In addition, these systems only operate for very short intervals of time, so that the information becomes damaged and cannot be stored, making it even more difficult to recover the data."

Quantum Computer vs. Classical Computer

Quantum computers have a more basic structure than classical computers. They have no memory or processor. All a quantum computer uses is a set of superconducting qubits.

Quantum computers and classical computers process information differently. A quantum computer uses qubits to run multidimensional quantum algorithms. Their processing power increases exponentially as qubits are added. A classical processor uses bits to operate various programs. Their power increases linearly as more bits are added. Classical computers have much less computing power.

Classical computers are best for everyday tasks and have low error rates. Quantum computers are ideal for a higher level of task, e.g., running simulations, analyzing data (such as for chemical or drug trials), creating energy-efficient batteries. They can also have high error rates.

Classical computers don't need extra-special care. They may use a basic internal fan to keep from overheating. Quantum processors need to be protected from the slightest vibrations and must be kept extremely cold. Super-cooled superfluids must be used for that purpose.

Quantum computers are more expensive and difficult to build than classical computers.

In 2019, Google demonstrated a quantum computer that can solve a problem in minutes that would take a classical computer 10,000 years.

Quantum Computers In Development

Google is spending billions of dollars to build its quantum computer by 2029. The company opened a campus in California called Google AI to help it meet this goal. Once developed, Google could launch a quantum computing service via the cloud.

IBM plans to have a 1,000-qubit quantum computer in place by 2023. For now, IBM allows access to its machines for those research organizations, universities, and laboratories that are part of its Quantum Network.  

Microsoft offers companies access to quantum technology via the Azure Quantum platform.

There’s interest in quantum computing and its technology from financial services firms such as JPMorgan Chase and Visa.

What Is Quantum Computing in Simplest Terms?

Quantum computing relates to computing made by a quantum computer. Compared to traditional computing done by a classical computer, a quantum computer should be able to store much more information and operate with more efficient algorithms. This translates to solving extremely complex tasks faster.

How Hard Is It to Build a Quantum Computer?

Building a quantum computer takes a long time and is vastly expensive. Google has been working on building a quantum computer for years and has spent billions of dollars. It expects to have its quantum computer ready by 2029. In November 2022, Google announced a 433-qubit system, followed a year later by IBM announcing its Condor, a 1,121 superconducting qubit quantum processor. At that time, IBM put out its 2033 roadmap, saying it was aiming at Blue Jay, a system capable of executing one billion gates across 2,000 qubits by 2033.

How Much Does a Quantum Computer Cost?

A quantum computer cost billions to build. However, in 2020, China-based Shenzhen SpinQ Technology planned to sell a $5,000 desktop quantum computer to consumers for schools and colleges. The previous year, it had began selling a quantum computer for $50,000.

How Fast Is a Quantum Computer?

A quantum computer is many times faster than a classical computer or a supercomputer. Google’s quantum computer in development, Sycamore, is said to have performed a calculation in 200 seconds, compared to the 10,000 years that one of the world’s fastest computers, IBM's Summit, would take to solve it. IBM disputed Google's claim, saying its supercomputer could solve the calculation in 2.5 days. Even so, that's 1,000 times slower than Google's quantum machine.

Quantum computing is very different from classical computing. It uses qubits, which can be 1 or 0 at the same time. Classical computers use bits, which can only be 1 or 0.

As a result, quantum computing is much faster and more powerful. It is expected to be used to solve a variety of extremely complex, worthwhile tasks.

While it has its limitations at this time, it is poised to be put to work by many high-powered companies in myriad industries.

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What is a quantum computer?

Alexandra L.

Checked : Karen F. , Kamye B.

Latest Update 19 Jan, 2024

Table of content

How a quantum computer works (overlapping states and entanglement)

Quantum computing: potential applications of the quantum computer.

The definition of the quantum computer is quite simple. It is a computer that exploits the laws of physics and quantum mechanics for   data processing   using the qubit as a fundamental unit. Unlike electronic calculation, at the base of computers as we have always known them, whose fundamental unit is the bit!

In particular, quantum bits have some properties that derive from the laws of quantum physics such as:

• The superposition   of states (they can be 0 and 1 at the same time) due to which parallel rather than sequential calculations can be made as happens today with the computational capacity of "traditional" computers.
• The entanglement   that is the correlation (the bond) that exists between one qubit and another, a very important aspect because it has a strong acceleration in the calculation process derives due to the influence that one qubit can produce on another even if they have distance.
• Quantum interference:   It is, in fact, the effect of the first principle (the superposition of states); quantum interference allows you to "control" the measurement of qubits based on the wave nature of the particles. The interference represents the superposition of two or more waves that depend on whether there is an overlap or not between grows and bellies. For instance, higher and lower parts of the wave - constructive interference can occur. When crests or bellies coincide and form a wave, which is the sum of the overlapping waves, or destructive interference when overlapping are the crest of a wave and belly of another, in this case, the two waves cancel each other out.

To understand how we got to the quantum computer, we have to go back to the miniaturization of circuits and Moore's Law. From the 1960s onwards, there has been a progressive increase in the computing power of computers, an increase that has gone hand in hand with the miniaturization of the electronic circuits from which it derives the famous Moore's Law. According to this law, “the complexity of a microcircuit, measured with the number of transistors in a chip (processor), and the relative calculation speed doubles every 18 months ".

Following this law - which over time has become a real measurement parameter and also guide of objectives for processor manufacturers - we have come to have integrated microchips, i.e., processors that integrate a CPU, a GPU, and a Digital Signal inside them processing, within our smartphones.

However, a threshold that today has reached the limits of quantum mechanics, making it very complex (almost impossible) to continue the path of miniaturization, together with the increase in the density of transistors. Limit that has actually opened the way to a paradigm shift trying to exploit the laws of physics and quantum mechanics to achieve a computing power higher than that of computers based on electronic calculation without necessarily thinking about the miniaturization of circuits.

The information units that encode two states open and closed (whose values are 1 and 0) of a switch, exploit those that are called   qubits.   The units of quantum information that are coded not by 1 or 0 but by the quantum state in which a particle or atom is found, which can have both the value 1 and the value 0 at the same time. Moreover, in a variety of combinations that produce different quantum states (a particle can be 70% in state 1 and 30% in state 0, or 40% and 60%, or 15 and 85).

A condition that takes on an incredible meaning when you think of mathematical progression such as 2 qubits can have 4 states simultaneously. For example, a pair of qubits can be in any quantum superposition of 4 states), 3 qubits can be in any 8 state superpositions. And, eight strings of three different bits: 000, 001, 010, 011, 100, 101, 110 and 111), 4 qubits in overlapping 16 states, 8 qubits of 256 states and so on. In a quantum computer, the   n   qubits can be in any superposition up to   2 to ‘n’   different states.

In fact, atomic and subatomic particles can exist in an overlap of quantum states, a situation that greatly expands the possibilities of encoding information by opening the possibility of exploiting this processing capacity for the resolution of extremely complex problems, such as those underlying the   Artificial intelligence .

The critical issues that have so far slowed down the race to develop these systems are related to the controlled manipulation of atoms and particles. It is possible with a few qubits but for complex processing hundreds and thousands of qubits are needed. Their connection and communication, as well as the development of algorithms are suitable for the quantum computer.

The functioning of the quantum computer, as mentioned in the first paragraph of this service) is based on two laws of quantum mechanics:

• The superposition principle from which derives, as we have seen, the possibility for the particles to be simultaneously in several different states. The superposition of states, in quantum physics, represents the simultaneous existence of all possible states of a particle or physical entity before its measurement. Only with the measurement, it is possible to define precisely the property of the qubit, and this is one of the most critical aspects that have not yet made the quantum computer available on a large scale. The particles are unstable, and their measurement is very complex, to which it must be added that the instability of the particles generates heat, which, to date, can only be controlled with advanced cooling systems.
• The quantum correlation (entanglement): It expresses the constraint, the correlation precisely that exists between two particles or two qubits.

According to this principle, it is possible to know the state of a particle (or a qubit) by measuring the other with which it has the constraint.

According to Gartner analysts, applications for quantum computing will be restricted and targeted, as the general-purpose quantum computer - most likely - will fail to be economically accessible on a large scale (at least not in the short term).

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However, technology has the potential to revolutionize certain sectors. Quantum calculation could allow discoveries and be applied in many sectors:

• Machine-learning:   improved machine learning due to a faster forecasting structure (due to parallel calculation). Examples include quantum Boltzmann machines, semi-supervised learning, unsupervised learning, and deep learning.
• Artificial intelligence:   faster calculations could improve the perception, understanding, and diagnosis of circuit faults / binary classifiers.
• Chemistry:   New fertilizers, catalysts, battery chemicals will bring enormous improvements in the use of resources;
• Biochemistry:   New drugs, customized drugs, personalized medicine.
• Finance:   the quantum calculation could allow the so-called faster and more complex "Monte Carlo simulations"; for example in the field of trading, optimization of "trajectories," market instability, price optimization, and hedging strategies.
• Medicine and health:   DNA gene sequencing, such as optimization of radiation therapy treatment/brain tumor detection, could be done in seconds rather than hours or weeks.
• Materials:   super-resistant materials; anti-corrosive paints, lubricants, semiconductors, the research could be greatly accelerated due to super-fast calculations.
• Computer science:   faster multidimensional search functions, for example, query optimization, mathematical calculations, and simulations.

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Using AI to predict grade point average from college application essays

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A Pithy YouTube Celebrity’s Plea: Buy This Video Game

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Bigmode made only one major request during the development cycle, Leah Gastrow said: Put more animals into the well.

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