# Quantum computing

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Quantum computing is the study of a currently hypothetical model of computation. Whereas traditional models of computing such as the Turing machine or Lambda calculus rely on "classical" representations of computational memory, a quantum computation could transform the memory into a quantum superposition of possible classical states. A quantum computer is a device that could perform such computation. [1]

In computer science, and more specifically in computability theory and computational complexity theory, a model of computation is a model which describes how an output of a mathematical function is computed given an input. A model describes how units of computations, memories, and communications are organized. The computational complexity of an algorithm can be measured given a model of computation. Using a model allows studying the performance of algorithms independently of the variations that are specific to particular implementations and specific technology.

A Turing machine is a mathematical model of computation that defines an abstract machine, which manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, given any computer algorithm, a Turing machine capable of simulating that algorithm's logic can be constructed.

Lambda calculus is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation that can be used to simulate any Turing machine. It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics.

## Contents

Quantum computing began in the early 1980s when physicist Paul Benioff proposed a quantum mechanical model of the Turing machine. [2]  Richard Feynman and Yuri Manin later suggested that a quantum computer could perform simulations that are out of reach for classical computers. [3] [4] In 1994, Peter Shor developed a polynomial-time quantum algorithm for factoring integers. [5] This was a major breakthrough in the subject: an important method of asymmetric key exchange known as RSA is based on the belief that factoring integers is computationally difficult. The existence of a polynomial-time quantum algorithm proves that one of the most widely-used cryptographic protocols is vulnerable to an adversary who possesses a quantum computer.

Paul A. Benioff is an American physicist who helped pioneer the field of quantum computing. Benioff is best known for his research in quantum information theory during the 1970s and 80s that demonstrated the theoretical possibility of quantum computers by describing the first quantum mechanical model of a computer. In this work, Benioff showed that a computer could operate under the laws of quantum mechanics by describing a Schrödinger equation description of Turing machines. Benioff’s body of work in quantum information theory has continued on to the present day and has encompassed quantum computers, quantum robots, and the relationship between foundations in logic, math, and physics.

Richard Phillips "Dick" Feynman, ForMemRS was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics for which he proposed the parton model. For contributions to the development of quantum electrodynamics, Feynman received the Nobel Prize in Physics in 1965 jointly with Julian Schwinger and Shin'ichirō Tomonaga.

Yuri Ivanovitch Manin is a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics. Moreover, Manin was one of the first to propose the idea of a quantum computer in 1980 with his book "Computable and Uncomputable".

Experimental efforts towards building a quantum computer began after a slew of results known as fault-tolerance threshold theorems. These theorems proved that a quantum computation could be efficiently corrected against the effects of large classes of physically realistic noise models. One early result [6] demonstrated parts of Shor's algorithm in a liquid-state nuclear magnetic resonance experiment. Other notable experiments have been performed in superconducting systems, ion-traps, and photonic systems.

In quantum computing, the (quantum) threshold theorem, proved by Michael Ben-Or and Dorit Aharonov, states that a quantum computer with a physical error rate below a certain threshold can, through application of quantum error correction schemes, suppress the logical error rate to arbitrarily low levels. Current estimates put the threshold for the surface code on the order of 1%, though estimates range widely and are difficult to calculate due to the exponential difficulty of simulating large quantum systems. At a 0.1% probability of a depolarizing error, the surface code would require approximately 1,000-10,000 physical qubits per logical data qubit, though more pathological error types could change this figure drastically.

Nuclear magnetic resonance (NMR) is a physical observation in which nuclei in a strong constant magnetic field are perturbed by a weak oscillating magnetic field and respond by producing an electromagnetic signal with a frequency characteristic of the magnetic field at the nucleus. This process occurs near resonance, when the oscillation frequency matches the intrinsic frequency of the nuclei, which depends on the strength of the static magnetic field, the chemical environment, and the magnetic properties of the isotope involved; in practical applications with static magnetic fields up to ca. 20 tesla, the frequency is similar to VHF and UHF television broadcasts (60–1000 MHz). NMR results from specific magnetic properties of certain atomic nuclei. Nuclear magnetic resonance spectroscopy is widely used to determine the structure of organic molecules in solution and study molecular physics, crystals as well as non-crystalline materials. NMR is also routinely used in advanced medical imaging techniques, such as in magnetic resonance imaging (MRI).

Superconducting quantum computing is an implementation of a quantum computer in superconducting electronic circuits. Research in superconducting quantum computing is conducted by Google, IBM, BBN Technologies, Rigetti, and Intel. as of May 2016, up to nine fully controllable qubits are demonstrated in a 1D array, up to sixteen in a 2D architecture.

Despite rapid and impressive experimental progress, most researchers believe that "fault-tolerant quantum computing [is] still a rather distant dream". [7] As of September 2019, no scalable quantum computing hardware has been demonstrated. Nevertheless, there is an increasing amount of investment in quantum computing by governments, established companies, and start-ups. [8] Current research focusses on building and using near-term intermediate-scale devices [7] and demonstrating quantum supremacy [9] alongside the long-term goal of building and using a powerful and error-free quantum computer.

Quantum supremacy is the potential ability of quantum computing devices to solve problems that classical computers practically cannot. Quantum advantage is the potential to solve problems faster. In computational-complexity-theoretic terms, this generally means providing a superpolynomial speedup over the best known or possible classical algorithm. The term was originally popularized by John Preskill but the concept of a quantum computational advantage, specifically for simulating quantum systems, dates back to Yuri Manin's (1980) and Richard Feynman's (1981) proposals of quantum computing.

The field of quantum computing is closely related to quantum information science, which includes quantum cryptography and quantum communication.

Quantum information science is an area of study based on the idea that information science depends on quantum effects in physics. It includes theoretical issues in computational models as well as more experimental topics in quantum physics including what can and cannot be done with quantum information. The term quantum information theory is sometimes used, but it fails to encompass experimental research in the area and can be confused with a subfield of quantum information science that studies the processing of quantum information.

Quantum cryptography is the science of exploiting quantum mechanical properties to perform cryptographic tasks. The best known example of quantum cryptography is quantum key distribution which offers an information-theoretically secure solution to the key exchange problem. The advantage of quantum cryptography lies in the fact that it allows the completion of various cryptographic tasks that are proven or conjectured to be impossible using only classical communication. For example, it is impossible to copy data encoded in a quantum state. If one attempts to read the encoded data, the quantum state will be changed. This could be used to detect eavesdropping in quantum key distribution.

## Basic concept

In most [10] models of classical computation, the computer has access to memory. This is a system that can be found in one of a finite set of possible states, each of which is physically distinct. It is frequently convenient to represent the state of this memory as a string of symbols; most simply, as a string of the symbols 0 and 1. In this scenario, the fundamental unit of memory is called a bit and we can measure the "size" of the memory in terms of the number of bits needed to represent fully the state of the memory.

In computing, memory refers to a device that is used to store information for immediate use in a computer or related computer hardware device. It typically refers to semiconductor memory, specifically metal-oxide-semiconductor (MOS) memory, where data is stored within MOSFET memory cells on a silicon integrated circuit chip. The term "memory" is often synonymous with the term "primary storage". Computer memory operates at a high speed, for example random-access memory (RAM), as a distinction from storage that provides slow-to-access information but offers higher capacities. If needed, contents of the computer memory can be transferred to secondary storage; a very common way of doing this is through a memory management technique called "virtual memory". An archaic synonym for memory is store.

The bit is a basic unit of information in information theory, computing, and digital communications. The name is a portmanteau of binary digit.

If the memory obeys the laws of quantum physics, the state of the memory could be found in a quantum superposition of different possible "classical" states. If the classical states are to be represented as a string of bits, the quantum memory could be found in any superposition of the possible bit strings. In the quantum scenario, the fundamental unit of memory is called a qubit.

The defining property of a quantum computer is the ability to turn classical memory states into quantum memory states, and vice-versa. This is not possible with present-day computers because they are carefully designed to ensure that the memory never deviates from clearly defined informational states. To clarify this point, consider that information is normally transmitted through the computer as an electrical signal that could have one of two easily distinguished voltages. If the voltages were to become indistinct (in a classical or quantum sense), the computer would no longer operate correctly.

Of course, in the end we are classical beings and we can only observe classical states. That means the quantum computer must complete its task by returning to us a classical output. To produce these classical outputs, the quantum computer is obliged to measure parts of the memory at various times throughout the computation. The measurement process is inherently probabilistic, meaning that the output of a quantum algorithm is frequently random. The task of a quantum algorithm designer is to ensure that the randomness is tailored to the needs of the problem at hand. For example, if the quantum computer is searching a quantum database for one of several marked items, we can ask the quantum computer to return one of the marked items at random. The quantum computer succeeds in this task as long as it is unlikely to return an unmarked item.

## Quantum operations

The prevailing model of quantum computation describes the computation in terms of a network of quantum logic gates. What follows is a brief treatment of the subject based upon Chapter 4 of Nielsen and Chuang. [11]

We may represent the state of a computer memory as a vector whose length is equal to the number of possible states of the memory. So a memory consisting of ${\textstyle n}$bits of information has ${\textstyle 2^{n}}$ possible states, and the vector representing that memory state has ${\textstyle 2^{n}}$ entries. In the classical view, all but one of the entries of this vector would be zero and the remaining entry would be one. The vector should be viewed as a probability vector and represents the fact that the memory is to be found in a particular state with 100% probability (i.e. a probability of one).

In quantum mechanics, probability vectors are generalised to density operators. This is the technically rigorous mathematical foundation for quantum logic gates, but the intermediate quantum state vector formalism is usually introduced first because it is conceptually simpler. Here we focus only on the quantum state vector formalism for simplicity.

We begin by considering a simple memory consisting of only one bit. This memory may be found in one of two states: the zero state or the one state. We may represent the state of this memory using Dirac notation so that

${\displaystyle |0\rangle :={\begin{pmatrix}1\\0\end{pmatrix}};\quad |1\rangle :={\begin{pmatrix}0\\1\end{pmatrix}}}$

A quantum memory may then be found in any quantum superposition ${\textstyle |\psi \rangle }$ of the two classical states ${\textstyle |0\rangle }$ and ${\textstyle |1\rangle }$:

${\displaystyle |\psi \rangle :=\alpha \,|0\rangle +\beta \,|1\rangle ={\begin{pmatrix}\alpha \\\beta \end{pmatrix}};\quad |\alpha |^{2}+|\beta |^{2}=1.}$

In general, the coefficients ${\textstyle \alpha }$ and ${\textstyle \beta }$ are complex numbers. In this scenario, we say that one qubit of information can be encoded into the quantum memory. The state ${\textstyle |\psi \rangle }$ is not itself a probability vector but can be connected with a probability vector via a measurement operation. If we choose to measure the quantum memory to determine if the state is ${\textstyle |0\rangle }$ or ${\textstyle |1\rangle }$ (this is known as a computational basis measurement), we would observe the zero state with probability ${\textstyle |\alpha |^{2}}$ and the one state with probability ${\textstyle |\beta |^{2}}$. Please see the article on quantum amplitudes for further information.

To manipulate the state of this one-qubit quantum memory, we imagine applying quantum logic gates analogous to classical logic gates. One obvious gate is the NOT gate, which can be represented by a matrix

${\displaystyle X:={\begin{pmatrix}0&1\\1&0\end{pmatrix}}.}$

We can formally apply this logic gate to a quantum state vector through matrix multiplication. Thus we find ${\textstyle X|0\rangle =|1\rangle }$ and ${\textstyle X|1\rangle =|0\rangle }$ as expected. But this is not the only interesting single-qubit quantum logic gate. We might, for example, imagine applying one of the other two Pauli matrices.

We may imagine extending single qubit gates to operate on multiqubit quantum memories in two important ways. One way to operate a single qubit gate on a multiqubit memory is simply to select a qubit and apply that gate to the target qubit whilst leaving the remainder of the memory unaffected. Another way is to apply the gate to its target only if another part of the memory is in a desired state. We illustrate this with another example.

Consider a two-qubit quantum memory. Its possible states are

${\displaystyle |00\rangle :={\begin{pmatrix}1\\0\\0\\0\end{pmatrix}};\quad |01\rangle :={\begin{pmatrix}0\\1\\0\\0\end{pmatrix}};\quad |10\rangle :={\begin{pmatrix}0\\0\\1\\0\end{pmatrix}};\quad |11\rangle :={\begin{pmatrix}0\\0\\0\\1\end{pmatrix}}.}$

We may then define the CNOT gate as the following matrix:

${\displaystyle CNOT:={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{pmatrix}}.}$

It is easy to check that ${\textstyle CNOT|00\rangle =|00\rangle }$, ${\textstyle CNOT|01\rangle =|01\rangle }$, ${\textstyle CNOT|10\rangle =|11\rangle }$, and ${\textstyle CNOT|11\rangle =|10\rangle }$. In other words, the CNOT applies a NOT gate (${\textstyle X}$ from before) to the second qubit if and only if the first qubit is in the state ${\textstyle |1\rangle }$. If the first qubit is ${\textstyle |0\rangle }$, nothing is done to either qubit.

The preceding discussion is of course a very brief introduction to the concept of a quantum logic gate. Please see the article on quantum logic gates for further information.

To put the story together, we can describe a quantum computation as a network of quantum logic gates and measurements. One can always 'defer' a measurement to the end of a quantum computation, though this can come at a computational cost according to some cost models. Because of this possibility of deferring a measurement, most quantum circuits depict a network consisting only of quantum logic gates and no measurements. For more details on the sequences of operations used for various quantum algorithms, see universal quantum computer, Shor's algorithm, Grover's algorithm, Deutsch–Jozsa algorithm, amplitude amplification, quantum Fourier transform, quantum gate, quantum adiabatic algorithm and quantum error correction.

One can represent any quantum computation as a network of quantum logic gates from a fairly small family of gates. A choice of gate family that enables this construction is known as a universal gate set. One common such set includes all single-qubit gates as well as the CNOT gate from above. This means any quantum computation can be performed by executing a sequence of single-qubit gates together with CNOT gates. Though this gate set is infinite, it can be replaced with a finite gate set by appealing to the Solovay-Kitaev theorem.

## Potential

### Cryptography

Integer factorization, which underpins the security of public key cryptographic systems, is believed to be computationally infeasible with an ordinary computer for large integers if they are the product of few prime numbers (e.g., products of two 300-digit primes). [12] By comparison, a quantum computer could efficiently solve this problem using Shor's algorithm to find its factors. This ability would allow a quantum computer to break many of the cryptographic systems in use today, in the sense that there would be a polynomial time (in the number of digits of the integer) algorithm for solving the problem. In particular, most of the popular public key ciphers are based on the difficulty of factoring integers or the discrete logarithm problem, both of which can be solved by Shor's algorithm. In particular, the RSA, Diffie–Hellman, and elliptic curve Diffie–Hellman algorithms could be broken. These are used to protect secure Web pages, encrypted email, and many other types of data. Breaking these would have significant ramifications for electronic privacy and security.

However, other cryptographic algorithms do not appear to be broken by those algorithms. [13] [14] Some public-key algorithms are based on problems other than the integer factorization and discrete logarithm problems to which Shor's algorithm applies, like the McEliece cryptosystem based on a problem in coding theory. [13] [15] Lattice-based cryptosystems are also not known to be broken by quantum computers, and finding a polynomial time algorithm for solving the dihedral hidden subgroup problem, which would break many lattice based cryptosystems, is a well-studied open problem. [16] It has been proven that applying Grover's algorithm to break a symmetric (secret key) algorithm by brute force requires time equal to roughly 2n/2 invocations of the underlying cryptographic algorithm, compared with roughly 2n in the classical case, [17] meaning that symmetric key lengths are effectively halved: AES-256 would have the same security against an attack using Grover's algorithm that AES-128 has against classical brute-force search (see Key size). Quantum cryptography could potentially fulfill some of the functions of public key cryptography. Quantum-based cryptographic systems could, therefore, be more secure than traditional systems against quantum hacking. [18]

Besides factorization and discrete logarithms, quantum algorithms offering a more than polynomial speedup over the best known classical algorithm have been found for several problems, [19] including the simulation of quantum physical processes from chemistry and solid state physics, the approximation of Jones polynomials, and solving Pell's equation. No mathematical proof has been found that shows that an equally fast classical algorithm cannot be discovered, although this is considered unlikely. [20] However, quantum computers offer polynomial speedup for some problems. The most well-known example of this is quantum database search, which can be solved by Grover's algorithm using quadratically fewer queries to the database than that are required by classical algorithms. In this case, the advantage is not only provable but also optimal, it has been shown that Grover's algorithm gives the maximal possible probability of finding the desired element for any number of oracle lookups. Several other examples of provable quantum speedups for query problems have subsequently been discovered, such as for finding collisions in two-to-one functions and evaluating NAND trees.

Problems that can be addressed with Grover's algorithm have the following properties:

1. There is no searchable structure in the collection of possible answers,
2. The number of possible answers to check is the same as the number of inputs to the algorithm, and
3. There exists a boolean function which evaluates each input and determines whether it is the correct answer

For problems with all these properties, the running time of Grover's algorithm on a quantum computer will scale as the square root of the number of inputs (or elements in the database), as opposed to the linear scaling of classical algorithms. A general class of problems to which Grover's algorithm can be applied [21] is Boolean satisfiability problem. In this instance, the database through which the algorithm is iterating is that of all possible answers. An example (and possible) application of this is a password cracker that attempts to guess the password or secret key for an encrypted file or system. Symmetric ciphers such as Triple DES and AES are particularly vulnerable to this kind of attack. [22] This application of quantum computing is a major interest of government agencies. [23]

### Quantum simulation

Since chemistry and nanotechnology rely on understanding quantum systems, and such systems are impossible to simulate in an efficient manner classically, many believe quantum simulation will be one of the most important applications of quantum computing. [24] Quantum simulation could also be used to simulate the behavior of atoms and particles at unusual conditions such as the reactions inside a collider. [25]

### Quantum annealing and adiabatic optimization

Quantum annealing or Adiabatic quantum computation relies on the adiabatic theorem to undertake calculations. A system is placed in the ground state for a simple Hamiltonian, which is slowly evolved to a more complicated Hamiltonian whose ground state represents the solution to the problem in question. The adiabatic theorem states that if the evolution is slow enough the system will stay in its ground state at all times through the process.

### Solving linear equations

The Quantum algorithm for linear systems of equations or "HHL Algorithm", named after its discoverers Harrow, Hassidim, and Lloyd, is expected to provide speedup over classical counterparts. [26]

### Quantum supremacy

John Preskill has introduced the term quantum supremacy to refer to the hypothetical speedup advantage that a quantum computer would have over a classical computer in a certain field. [27] Google announced in 2017 that it expected to achieve quantum supremacy by the end of the year though that did not happen. IBM said in 2018 that the best classical computers will be beaten on some practical task within about five years and views the quantum supremacy test only as a potential future benchmark. [28] Although skeptics like Gil Kalai doubt that quantum supremacy will ever be achieved, [29] [30] Google has been reported to have done so, with calculations more than 3,000,000 times as fast as those of Summit, generally considered the world's fastest computer. [31] Bill Unruh doubted the practicality of quantum computers in a paper published back in 1994. [32] Paul Davies argued that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the holographic principle. [33]

## Obstacles

There are a number of technical challenges in building a large-scale quantum computer, [34] and thus far quantum computers have yet to solve a problem faster than a classical computer. David DiVincenzo, of IBM, listed the following requirements for a practical quantum computer: [35]

• scalable physically to increase the number of qubits;
• qubits that can be initialized to arbitrary values;
• quantum gates that are faster than decoherence time;
• universal gate set;
• qubits that can be read easily.

### Quantum decoherence

One of the greatest challenges is controlling or removing quantum decoherence. This usually means isolating the system from its environment as interactions with the external world cause the system to decohere. However, other sources of decoherence also exist. Examples include the quantum gates, and the lattice vibrations and background thermonuclear spin of the physical system used to implement the qubits. Decoherence is irreversible, as it is effectively non-unitary, and is usually something that should be highly controlled, if not avoided. Decoherence times for candidate systems in particular, the transverse relaxation time T2 (for NMR and MRI technology, also called the dephasing time), typically range between nanoseconds and seconds at low temperature. [36] Currently, some quantum computers require their qubits to be cooled to 20 millikelvins in order to prevent significant decoherence. [37]

As a result, time-consuming tasks may render some quantum algorithms inoperable, as maintaining the state of qubits for a long enough duration will eventually corrupt the superpositions. [38]

These issues are more difficult for optical approaches as the timescales are orders of magnitude shorter and an often-cited approach to overcoming them is optical pulse shaping. Error rates are typically proportional to the ratio of operating time to decoherence time, hence any operation must be completed much more quickly than the decoherence time.

As described in the Quantum threshold theorem, if the error rate is small enough, it is thought to be possible to use quantum error correction to suppress errors and decoherence. This allows the total calculation time to be longer than the decoherence time if the error correction scheme can correct errors faster than decoherence introduces them. An often cited figure for the required error rate in each gate for fault-tolerant computation is 10−3, assuming the noise is depolarizing.

Meeting this scalability condition is possible for a wide range of systems. However, the use of error correction brings with it the cost of a greatly increased number of required qubits. The number required to factor integers using Shor's algorithm is still polynomial, and thought to be between L and L2, where L is the number of qubits in the number to be factored; error correction algorithms would inflate this figure by an additional factor of L. For a 1000-bit number, this implies a need for about 104 bits without error correction. [39] With error correction, the figure would rise to about 107 bits. Computation time is about L2 or about 107 steps and at 1 MHz, about 10 seconds.

A very different approach to the stability-decoherence problem is to create a topological quantum computer with anyons, quasi-particles used as threads and relying on braid theory to form stable logic gates. [40] [41]

## Developments

### Quantum computing models

There are a number of quantum computing models, distinguished by the basic elements in which the computation is decomposed. The four main models of practical importance are:

The quantum Turing machine is theoretically important but the direct implementation of this model is not pursued. All four models of computation have been shown to be equivalent; each can simulate the other with no more than polynomial overhead.

### Physical realizations

For physically implementing a quantum computer, many different candidates are being pursued, among them (distinguished by the physical system used to realize the qubits):

A large number of candidates demonstrates that the topic, in spite of rapid progress, is still in its infancy. There is also a vast amount of flexibility.

### Timeline

In 1980, Paul Benioff describes the first quantum mechanical model of a computer. In this work, Benioff showed that a computer could operate under the laws of quantum mechanics by describing a Schrodinger equation description of Turing machines, laying a foundation for further work in quantum computing. The paper [2] was submitted in June 1979 and published in April of 1980. Russian mathematician Yuri Manin then motivates the development of quantum computers. [60]

In 1981, at the First Conference on the Physics of Computation held at MIT and co-organized by MIT and IBM, Paul Benioff and Richard Feynman give talks on quantum computing. Benioff built on his earlier 1980 work showing that a computer can operate under the laws of quantum mechanics. The talk was titled “Quantum mechanical Hamiltonian models of discrete processes that erase their own histories: application to Turing machines”. In Feynman’s talk, he observed that it appeared to be impossible to efficiently simulate an evolution of a quantum system on a classical computer, and he proposed a basic model for a quantum computer. [61] Urging the world to build a quantum computer, he said, "Nature isn't classical, dammit, and if you want to make a simulation of nature, you'd better make it quantum mechanical, and by golly, it's a wonderful problem because it doesn't look so easy." [62]

In 1982, Paul Benioff further develops his original model of a quantum mechanical Turing machine. [63]

In 1984, IBM scientists Charles Bennett and Gilles Brassard published BB84, the world's first quantum cryptography protocol.

In 1985, David Deutsch describes the first universal quantum computer. Just as a Universal Turing machine can simulate any other Turing machine efficiently (Church-Turing thesis), so the universal quantum computer is able to simulate any other quantum computer with at most a polynomial slowdown.

In 1989, Bikas K. Chakrabarti & collaborators proposes the idea that quantum fluctuations could help explore rough energy landscapes by escaping from local minima of glassy systems having tall but thin barriers by tunneling (instead of climbing over using thermal excitations), suggesting the effectiveness of quantum annealing over classical simulated annealing. [64] [42]

In 1992, David Deutsch and Richard Jozsa propose a computational problem that can be solved efficiently with the determinist Deutsch–Jozsa algorithm on a quantum computer, but for which no deterministic classical algorithm is possible. This was perhaps the earliest result in the computational complexity of quantum computers, proving that they were capable of performing some well-defined computational task more efficiently than any classical computer.

In 1993, an international group of six scientists, including Charles Bennett, showed that perfect quantum teleportation is possible [65] in principle, but only if the original is destroyed.

In 1994, Peter Shor, at AT&T's Bell Labs, discovered an important quantum algorithm, which allows a quantum computer to factor large integers exponentially much faster than the best known classical algorithm. Shor's algorithm can theoretically break many of the Public-key cryptography systems in use today, [66] sparking a tremendous interest in quantum computers.

In 1996, the DiVincenzo's criteria are published, which are a list of conditions that are necessary for constructing a quantum computer, proposed by the theoretical physicist David P. DiVincenzo in his 2000 paper "The Physical Implementation of Quantum Computation".

In 2001, researchers demonstrated Shor's algorithm to factor 15 using a 7-qubit NMR computer. [67]

In 2005, researchers at the University of Michigan built a semiconductor chip ion trap. Such devices from standard lithography may point the way to scalable quantum computing. [68]

In 2009, researchers at Yale University created the first solid-state quantum processor. The 2-qubit superconducting chip had artificial atom qubits made of a billion aluminum atoms that acted like a single atom that could occupy two states. [69] [70]

A team at the University of Bristol also created a silicon chip based on quantum optics, able to run Shor's algorithm. [71] Further developments were made in 2010. [72] Springer publishes a journal, Quantum Information Processing, devoted to the subject. [73]

In February 2010, Digital Combinational Circuits like an adder, subtractor etc. are designed with the help of Symmetric Functions organized from different quantum gates. [74] [75]

In April 2011, a team of scientists from Australia and Japan made a breakthrough in quantum teleportation, successfully transferring a complex set of quantum data with full transmission integrity, without affecting the qubits' superpositions. [76] [77]

In 2011, D-Wave Systems announced the first commercial quantum annealer, the D-Wave One, claiming a 128-qubit processor. On 25 May 2011, Lockheed Martin agreed to purchase a D-Wave One system. [78] Lockheed and the University of Southern California (USC) will house the D-Wave One at the newly formed USC Lockheed Martin Quantum Computing Center. [79] D-Wave's engineers designed the chips with an empirical approach, focusing on solving particular problems. Investors liked this more than academics, who said D-Wave had not demonstrated that they really had a quantum computer. Criticism softened after a D-Wave paper in Nature that proved that the chips have some quantum properties. [80] [81] Two published papers have suggested that the D-Wave machine's operation can be explained classically, rather than requiring quantum models. [82] [83] Later work showed that classical models are insufficient when all available data is considered. [84] Experts remain divided on the ultimate classification of the D-Wave systems though their quantum behavior was established concretely with a demonstration of entanglement. [85]

During the same year, researchers at the University of Bristol created an all-bulk optics system that ran a version of Shor's algorithm to successfully factor 21. [86]

In September 2011, researchers proved quantum computers can be made with a Von Neumann architecture (separation of RAM). [87]

In November 2011, researchers factorized 143 using 4 qubits. [88]

In February 2012, IBM scientists said that they had made several breakthroughs in quantum computing with superconducting integrated circuits. [89]

In April 2012, a multinational team of researchers from the University of Southern California, the Delft University of Technology, the Iowa State University of Science and Technology, and the University of California, Santa Barbara constructed a 2-qubit quantum computer on a doped diamond crystal that can easily be scaled up and is functional at room temperature. Two logical qubit directions of electron spin and nitrogen kernels spin were used, with microwave pulses. This computer ran Grover's algorithm, generating the right answer on the first try in 95% of cases. [90]

In September 2012, Australian researchers at the University of New South Wales said the world's first quantum computer was just 5 to 10 years away, after announcing a global breakthrough enabling the manufacture of its memory building blocks. A research team led by Australian engineers created the first working qubit based on a single atom in silicon, invoking the same technological platform that forms the building blocks of modern-day computers. [91] [92]

In October 2012, Nobel Prizes were awarded to David J. Wineland and Serge Haroche for their basic work on understanding the quantum world, which may help make quantum computing possible. [93] [94]

In November 2012, the first quantum teleportation from one macroscopic object to another was reported by scientists at the University of Science and Technology of China. [95] [96]

In December 2012, 1QBit, the first dedicated quantum computing software company, was founded in Vancouver, BC. [97] 1QBit is the first company to focus exclusively on commercializing software applications for commercially available quantum computers, including the D-Wave Two. 1QBit's research demonstrated the ability of superconducting quantum annealing processors to solve real-world problems. [98]

In February 2013, a new technique, boson sampling, was reported by two groups using photons in an optical lattice that is not a universal quantum computer, but may be good enough for practical problems. [99]

In May 2013, Google announced that it was launching the Quantum Artificial Intelligence Lab, hosted by NASA's Ames Research Center, with a 512-qubit D-Wave quantum computer. The Universities Space Research Association (USRA) will invite researchers to share time on it with the goal of studying quantum computing for machine learning. [100] Google added that they had "already developed some quantum machine learning algorithms" and had "learned some useful principles", such as that "best results" come from "mixing quantum and classical computing". [100]

In early 2014, based on documents provided by former NSA contractor Edward Snowden, it was reported that the U.S. National Security Agency (NSA) is running a $79.7 million research program titled "Penetrating Hard Targets", to develop a quantum computer capable of breaking vulnerable encryption. [101] In 2014, a group of researchers from ETH Zürich, USC, Google, and Microsoft reported a definition of quantum speedup, and were not able to measure quantum speedup with the D-Wave Two device, but did not explicitly rule it out. [102] [103] In 2014, researchers at University of New South Wales used silicon as a protectant shell around qubits, making them more accurate, increasing the length of time they will hold information, and possibly making quantum computers easier to build. [104] In April 2015, IBM scientists claimed two critical advances towards the realization of a practical quantum computer, claiming the ability to detect and measure both kinds of quantum errors simultaneously, as well as a new, square quantum bit circuit design that could scale to larger dimensions. [105] In October 2015, QuTech successfully conducted the Loophole-free Bell inequality violation test using electron spins separated by 1.3 kilometres. [106] In October 2015, researchers at the University of New South Wales built a quantum logic gate in silicon for the first time. [107] In December 2015, NASA publicly displayed the world's first fully operational quantum computer made by D-Wave Systems at the Quantum Artificial Intelligence Lab at its Ames Research Center. The device was purchased in 2013 via a partnership with Google and Universities Space Research Association. The presence and use of quantum effects in the D-Wave quantum processing unit is more widely accepted. [108] In some tests, it can be shown that the D-Wave quantum annealing processor outperforms Selby’s algorithm. [109] Only two of these computers have been made so far. In May 2016, IBM Research announced [110] that for the first time ever it is making quantum computing available to members of the public via the cloud, who can access and run experiments on IBM’s quantum processor, calling the service the IBM Quantum Experience. The quantum processor is composed of five superconducting qubits and is housed at IBM's Thomas J. Watson Research Center. In August 2016, scientists at the University of Maryland successfully built the first reprogrammable quantum computer. [111] In October 2016, the University of Basel described a variant of the electron-hole based quantum computer, which instead of manipulating electron spins, uses electron holes in a semiconductor at low (mK) temperatures, which are much less vulnerable to decoherence. This has been dubbed the "positronic" quantum computer, as the quasi-particle behaves as if it has a positive electrical charge. [112] In March 2017, IBM announced an industry-first initiative, called IBM Q, to build commercially available universal quantum computing systems. The company also released a new API for the IBM Quantum Experience that enables developers and programmers to begin building interfaces between its existing 5-qubit cloud-based quantum computer and classical computers, without needing a deep background in quantum physics. In May 2017, IBM announced [113] that it had successfully built and tested its most powerful universal quantum computing processors. The first is a 16-qubit processor that will allow for more complex experimentation than the previously available 5-qubit processor. The second is IBM's first prototype commercial processor with 17 qubits, and leverages significant materials, device, and architecture improvements to make it the most powerful quantum processor created to date by IBM. In July 2017, a group of U.S. researchers announced a quantum simulator with 51 qubits. The announcement was made by Mikhail Lukin of Harvard University at the International Conference on Quantum Technologies in Moscow. [114] A quantum simulator differs from a computer. Lukin’s simulator was designed to solve one equation. Solving a different equation would require building a new system, whereas a computer can solve many different equations. In September 2017, IBM Research scientists used a 7-qubit device to model beryllium hydride molecule, the largest molecule to date by a quantum computer. [115] The results were published as the cover story in the peer-reviewed journal Nature. In October 2017, IBM Research scientists successfully "broke the 49-qubit simulation barrier" and simulated 49- and 56-qubit short-depth circuits, using the Lawrence Livermore National Laboratory's Vulcan supercomputer, and the University of Illinois' Cyclops Tensor Framework (originally developed at the University of California). [116] In November 2017, the University of Sydney research team successfully made a microwave circulator, an important quantum computer part, that was 1000 times smaller than a conventional circulator, by using topological insulators to slow down the speed of light in a material. [117] In December 2017, IBM announced [118] its first IBM Q Network clients. The companies, universities, and labs that will explore practical business and science quantum applications, using IBM Q 20-qubit commercial systems, include: JPMorgan Chase, Daimler AG, Samsung, JSR Corporation, Barclays, Hitachi Metals, Honda, Nagase, Keio University, Oak Ridge National Lab, Oxford University and University of Melbourne. In December 2017, Microsoft released a preview version of a "Quantum Development Kit", [119] which includes a programming language, Q# that can be used to write programs that are run on an emulated quantum computer. In 2017, D-Wave was reported to be selling a 2,000-qubit quantum computer. [120] In late 2017 and early 2018, IBM, [121] Intel, [122] and Google [123] each reported testing quantum processors containing 50, 49, and 72 qubits, respectively, all realized using superconducting circuits. By number of qubits, these circuits are approaching the range in which simulating their quantum dynamics is expected to become prohibitive on classical computers, although it has been argued that further improvements in error rates are needed to put classical simulation out of reach. [124] In February 2018, scientists reported, for the first time, the discovery of a new form of light, which may involve polaritons, that could be useful in the development of quantum computers. [125] [126] In February 2018, QuTech reported successfully testing a silicon-based two-spin-qubits quantum processor. [127] In June 2018, Intel began testing a silicon-based spin-qubit processor, manufactured in the company's D1D Fab in Oregon. [128] In July 2018, a team led by the University of Sydney achieved the world's first multi-qubit demonstration of a quantum chemistry calculation performed on a system of trapped ions, one of the leading hardware platforms in the race to develop a universal quantum computer. [129] In December 2018, IonQ reported that its machine could be built as large as 160 qubits. [130] In January 2019, IBM launched IBM Q System One, its first integrated quantum computing system for commercial use. [131] [132] IBM Q System One is designed by industrial design company Map Project Office and interior design company Universal Design Studio. [133] In March 2019, a group of Russian scientists used the open-access IBM quantum computer to demonstrate a protocol for the complex conjugation of the probability amplitudes needed for time reversal of a physical process, [134] in this case, for an electron scattered on a two-level impurity, a two-qubit experiment. However, for the three-qubit experiment, the amplitude fell below 50% (failure of time reversal, due to its increased complexity). [135] In September 2019 Google AI Quantum and NASA published a paper [136] [137] "Quantum supremacy using a programmable superconducting processor" and supplementary material [138] which was later removed from NASA. ## Relation to computational complexity theory The class of problems that can be efficiently solved by quantum computers is called BQP, for "bounded error, quantum, polynomial time". Quantum computers only run probabilistic algorithms, so BQP on quantum computers is the counterpart of BPP ("bounded error, probabilistic, polynomial time") on classical computers. It is defined as the set of problems solvable with a polynomial-time algorithm, whose probability of error is bounded away from one half. [140] A quantum computer is said to "solve" a problem if, for every instance, its answer will be right with high probability. If that solution runs in polynomial time, then that problem is in BQP. BQP is contained in the complexity class #P (or more precisely in the associated class of decision problems P#P), [141] which is a subclass of PSPACE. BQP is suspected to be disjoint from NP-complete and a strict superset of P, but that is not known. Both integer factorization and discrete log are in BQP. Both of these problems are NP problems suspected to be outside BPP, and hence outside P. Both are suspected to not be NP-complete. There is a common misconception that quantum computers can solve NP-complete problems in polynomial time. That is not known to be true, and is generally suspected to be false. [141] The capacity of a quantum computer to accelerate classical algorithms has rigid limits—upper bounds of quantum computation's complexity. The overwhelming part of classical calculations cannot be accelerated on a quantum computer. [142] A similar fact prevails for particular computational tasks, like the search problem, for which Grover's algorithm is optimal. [143] Bohmian Mechanics is a non-local hidden variable interpretation of quantum mechanics. It has been shown that a non-local hidden variable quantum computer could implement a search of an N-item database at most in ${\displaystyle O({\sqrt[{3}]{N}})}$ steps. This is slightly faster than the ${\displaystyle O({\sqrt {N}})}$ steps taken by Grover's algorithm. Neither search method will allow quantum computers to solve NP-Complete problems in polynomial time. [144] Although quantum computers may be faster than classical computers for some problem types, those described above cannot solve any problem that classical computers cannot already solve. A Turing machine can simulate these quantum computers, so such a quantum computer could never solve an undecidable problem like the halting problem. The existence of "standard" quantum computers does not disprove the Church–Turing thesis. [145] It has been speculated that theories of quantum gravity, such as M-theory or loop quantum gravity, may allow even faster computers to be built. Currently, defining computation in such theories is an open problem due to the problem of time , i.e., there currently exists no obvious way to describe what it means for an observer to submit input to a computer and later receive output. [146] [78] ## See also ## Related Research Articles In quantum computing, a qubit or quantum bit is the basic unit of quantum information—the quantum version of the classical binary bit physically realized with a two-state device. A qubit is a two-state quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include: the spin of the electron in which the two levels can be taken as spin up and spin down; or the polarization of a single photon in which the two states can be taken to be the vertical polarization and the horizontal polarization. In a classical system, a bit would have to be in one state or the other. However, quantum mechanics allows the qubit to be in a coherent superposition of both states/levels simultaneously, a property which is fundamental to quantum mechanics and quantum computing. Shor's algorithm is a quantum computer algorithm for integer factorization. Informally, it solves the following problem: Given an integer , find its prime factors. It was invented in 1994 by the American mathematician Peter Shor. Grover's algorithm is a quantum algorithm that finds with high probability the unique input to a black box function that produces a particular output value, using just evaluations of the function, where is the size of the function's domain. It was devised by Lov Grover in 1996. This is a timeline of quantum computing. In quantum computing, a quantum algorithm is an algorithm which runs on a realistic model of quantum computation, the most commonly used model being the quantum circuit model of computation. A classical algorithm is a finite sequence of instructions, or a step-by-step procedure for solving a problem, where each step or instruction can be performed on a classical computer. Similarly, a quantum algorithm is a step-by-step procedure, where each of the steps can be performed on a quantum computer. Although all classical algorithms can also be performed on a quantum computer, the term quantum algorithm is usually used for those algorithms which seem inherently quantum, or use some essential feature of quantum computation such as quantum superposition or quantum entanglement. A trapped ion quantum computer is one proposed approach to a large-scale quantum computer. Ions, or charged atomic particles, can be confined and suspended in free space using electromagnetic fields. Qubits are stored in stable electronic states of each ion, and quantum information can be transferred through the collective quantized motion of the ions in a shared trap. Lasers are applied to induce coupling between the qubit states or coupling between the internal qubit states and the external motional states. Quantum programming is the process of assembling sequences of instructions, called quantum programs, that are capable of running on a quantum computer. Quantum programming languages help express quantum algorithms using high-level constructs. Quantum networks form an important element of quantum computing and quantum communication systems. Quantum networks facilitate the transmission of information in the form of quantum bits, also called qubits, between physically separated quantum processors. A quantum processor is a small quantum computer being able to perform quantum logic gates on a certain number of qubits. Quantum networks work in a similar way to classical networks. The main difference, as will be detailed more in later paragraphs, is that quantum networking like quantum computing is better at solving certain problems, such as modeling quantum systems. In computing science, the controlled NOT gate is a quantum logic gate that is an essential component in the construction of a gate-based quantum computer. It can be used to entangle and disentangle EPR states. Any quantum circuit can be simulated to an arbitrary degree of accuracy using a combination of CNOT gates and single qubit rotations. In quantum computing, the Gottesman–Knill theorem is a theoretical result by Daniel Gottesman and Emanuel Knill that states that stabilizer circuits, circuits that only consist of gates from the normalizer of the qubit Pauli group, also called Clifford group, can be perfectly simulated in polynomial time on a probabilistic classical computer. The Clifford group can be generated solely by using CNOT, Hadamard, and phase gates; and therefore stabilizer circuits can be constructed using only these gates. The one-way or measurement based quantum computer (MBQC) is a method of quantum computing that first prepares an entangled resource state, usually a cluster state or graph state, then performs single qubit measurements on it. It is "one-way" because the resource state is destroyed by the measurements. In classical computation, any memory bit can be turned on or off at will, requiring no prior knowledge or extra gadgetry. However, this is not the case in quantum computing or classical reversible computing. In these models of computing, all operations on computer memory must be reversible, and toggling a bit on or off would lose the information about the initial value of that bit. For this reason, in a quantum algorithm there is no way to deterministically put bits in a specific prescribed state unless one is given access to bits whose original state is known in advance. Such bits, whose values are known a priori, are known as ancilla bits in a quantum or reversible computing task. David P. DiVincenzo is an American theoretical physicist. He is the director of the Institute of Theoretical Nanoelectronics at the Peter Grünberg Institute in Jülich and Professor at the Institute for Quantum Information at RWTH Aachen University. With Daniel Loss, he proposed the Loss-DiVincenzo quantum computer in 1997, which would use electron spins in quantum dots as qubits. The DiVincenzo criteria are conditions necessary for constructing a quantum computer, conditions proposed in 2000 by the theoretical physicist David P. DiVincenzo, as being those necessary to construct such a computer—a computer first proposed by mathematician Yuri Manin, in 1980, and physicist Richard Feynman, in 1982—as a means to efficiently simulate quantum systems, such as in solving the quantum many-body problem. The IBM Q Experience is an online platform that gives users in the general public access to a set of IBM's prototype quantum processors via the Cloud, an online internet forum for discussing quantum computing relevant topics, a set of tutorials on how to program the IBM Q devices, and other educational material about quantum computing. It is an example of cloud based quantum computing. As of May 2018, there are three processors on the IBM Q Experience: two 5-qubit processors and a 16-qubit processor. This service can be used to run algorithms and experiments, and explore tutorials and simulations around what might be possible with quantum computing. The site also provides an easily discoverable list of research papers published using the IBM Q Experience as an experimentation platform. Continuous-variable quantum information is the area of quantum information science that makes use of physical observables, like the strength of an electromagnetic field, whose numerical values belong to continuous intervals. One primary application is quantum computing. In a sense, continuous-variable quantum computation is "analog", while quantum computation using qubits is "digital." In more technical terms, the former makes use of Hilbert spaces that are infinite-dimensional, while the Hilbert spaces for systems comprising collections of qubits are finite-dimensional. One motivation for studying continuous-variable quantum computation is to understand what resources are necessary to make quantum computers more powerful than classical ones. The Mølmer–Sørensen gate is a two qubit gate used in quantum computing. It was proposed by Klaus Mølmer and Anders Sørensen. In quantum computing, a qubit is a unit of information analogous to a bit in classical computing, but it is affected by quantum mechanical properties such as superposition and entanglement which allow qubits to be in some ways more powerful than classical bits for some tasks. Qubits are used in quantum circuits and quantum algorithms composed of quantum logic gates to solve computational problems, where they are used for input/output and intermediate computations. ## References 1. The National Academies of Sciences, Engineering, and Medicine (2019). Grumbling, Emily; Horowitz, Mark (eds.). Quantum Computing : Progress and Prospects (2018). Washington, DC: National Academies Press. p. 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