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**Quantum computing** is the exploitation of collective properties of quantum states, such as superposition and entanglement, to perform computation. The devices that perform quantum computations are known as **quantum computers**.^{ [1] }^{:I-5} They are believed to be able to solve certain computational problems, such as integer factorization (which underlies RSA encryption), substantially faster than classical computers. The study of quantum computing is a subfield of quantum information science. Expansion is expected in the next few years as the field shifts toward real-world use in pharmaceutical, data security and other applications.^{ [2] }

- Quantum circuit
- Definition
- Quantum algorithms
- Potential applications
- Cryptography
- Search problems
- Simulation of quantum systems
- Quantum annealing and adiabatic optimization
- Machine learning
- Computational biology
- Quantum supremacy
- Obstacles
- Quantum decoherence
- Developments
- Quantum computing models
- Physical realizations
- Relation to computability and complexity theory
- Computability theory
- Quantum complexity theory
- See also
- References
- Further reading
- Textbooks
- Academic papers
- External links

Quantum computing began in 1980 when physicist Paul Benioff proposed a quantum mechanical model of the Turing machine.^{ [3] } Richard Feynman and Yuri Manin later suggested that a quantum computer had the potential to simulate things a classical computer could not feasibly do.^{ [4] }^{ [5] } In 1994, Peter Shor developed a quantum algorithm for factoring integers with the potential to decrypt RSA-encrypted communications.^{ [6] } Despite ongoing experimental progress since the late 1990s, most researchers believe that "fault-tolerant quantum computing [is] still a rather distant dream."^{ [7] } In recent years, investment in quantum computing research has increased in the public and private sectors.^{ [8] }^{ [9] } On 23 October 2019, Google AI, in partnership with the U.S. National Aeronautics and Space Administration (NASA), claimed to have performed a quantum computation that was infeasible on any classical computer.^{ [10] }

There are several types of quantum computers (also known as quantum computing systems), including the quantum circuit model, quantum Turing machine, adiabatic quantum computer, one-way quantum computer, and various quantum cellular automata. The most widely used model is the quantum circuit, based on the quantum bit, or "qubit", which is somewhat analogous to the bit in classical computation. A qubit can be in a 1 or 0 quantum state, or in a superposition of the 1 and 0 states. When it is measured, however, it is always 0 or 1; the probability of either outcome depends on the qubit's quantum state immediately prior to measurement.

Efforts towards building a physical quantum computer focus on technologies such as transmons, ion traps and topological quantum computers, which aim to create high-quality qubits.^{ [1] }^{:2–13} These qubits may be designed differently, depending on the full quantum computer's computing model, whether quantum logic gates, quantum annealing, or adiabatic quantum computation. There are currently a number of significant obstacles to constructing useful quantum computers. It is particularly difficult to maintain qubits' quantum states, as they suffer from quantum decoherence and state fidelity. Quantum computers therefore require error correction.^{ [11] }^{ [12] }

Any computational problem that can be solved by a classical computer can also be solved by a quantum computer.^{ [13] } Conversely, any problem that can be solved by a quantum computer can also be solved by a classical computer, at least in principle given enough time. In other words, quantum computers obey the Church–Turing thesis. This means that while quantum computers provide no additional advantages over classical computers in terms of computability, quantum algorithms for certain problems have significantly lower time complexities than corresponding known classical algorithms. Notably, quantum computers are believed to be able to quickly solve certain problems that no classical computer could solve in any *feasible* amount of time—a feat known as "quantum supremacy." The study of the computational complexity of problems with respect to quantum computers is known as quantum complexity theory.

The prevailing model of quantum computation describes the computation in terms of a network of quantum logic gates.^{ [14] } This model can be thought of as an abstract linear-algebraic generalization of a classical circuit. Since this circuit model obeys quantum mechanics, a quantum computer capable of efficiently running these circuits is believed to be physically realizable.

A memory consisting of bits of information has possible states. A vector representing all memory states thus has entries (one for each state). This vector is viewed as a * probability vector * and represents the fact that the memory is to be found in a particular state.

In the classical view, one entry would have a value of 1 (i.e. a 100% probability of being in this state) and all other entries would be zero. In quantum mechanics, probability vectors are generalized 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. This article focuses 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

A quantum memory may then be found in any quantum superposition of the two classical states and :

In general, the coefficients and are complex numbers. In this scenario, one qubit of information is said to be encoded into the quantum memory. The state is not itself a probability vector but can be connected with a probability vector via a measurement operation. If the quantum memory is measured to determine whether the state is or (this is known as a computational basis measurement), the zero state would be observed with probability and the one state with probability . The numbers and are called quantum amplitudes.

The state of this one-qubit quantum memory can be manipulated by applying quantum logic gates, analogous to how classical memory can be manipulated with classical logic gates. One important gate for both classical and quantum computation is the NOT gate, which can be represented by a matrix

Mathematically, the application of such a logic gate to a quantum state vector is modelled with matrix multiplication. Thus and .

The mathematics of single qubit gates can be extended to operate on multi-qubit quantum memories in two important ways. One way 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. These two choices can be illustrated using another example. The possible states of a two-qubit quantum memory are

The CNOT gate can then be represented using the following matrix:

As a mathematical consequence of this definition, , , , and . In other words, the CNOT applies a NOT gate ( from before) to the second qubit if and only if the first qubit is in the state . If the first qubit is , nothing is done to either qubit.

In summary, a quantum computation can be described as a network of quantum logic gates and measurements. However, any measurement can be deferred to the end of a quantum computation, though this deferment may come at a computational cost, so most quantum circuits depict a network consisting only of quantum logic gates and no measurements.

Any quantum computation (which is, in the above formalism, any unitary matrix over qubits) can be represented 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, since a computer that can run such circuits is a universal quantum computer. 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.

Progress in finding quantum algorithms typically focuses on this quantum circuit model, though exceptions like the quantum adiabatic algorithm exist. Quantum algorithms can be roughly categorized by the type of speedup achieved over corresponding classical algorithms.^{ [15] }

Quantum algorithms that offer more than a polynomial speedup over the best known classical algorithm include Shor's algorithm for factoring and the related quantum algorithms for computing discrete logarithms, solving Pell's equation, and more generally solving the hidden subgroup problem for abelian finite groups.^{ [15] } These algorithms depend on the primitive of the quantum Fourier transform. No mathematical proof has been found that shows that an equally fast classical algorithm cannot be discovered, although this is considered unlikely.^{ [16] } Certain oracle problems like Simon's problem and the Bernstein–Vazirani problem do give provable speedups, though this is in the quantum query model, which is a restricted model where lower bounds are much easier to prove, and doesn't necessarily translate to speedups for practical problems.

Other problems, including the simulation of quantum physical processes from chemistry and solid state physics, the approximation of certain Jones polynomials, and the quantum algorithm for linear systems of equations have quantum algorithms appearing to give super-polynomial speedups and are BQP-complete. Because these problems are BQP-complete, an equally fast classical algorithm for them would imply that *no quantum algorithm* gives a super-polynomial speedup, which is believed to be unlikely.^{ [17] }

Some quantum algorithms, like Grover's algorithm and amplitude amplification, give polynomial speedups over corresponding classical algorithms.^{ [15] } Though these algorithms give comparably modest quadratic speedup, they are widely applicable and thus give speedups for a wide range of problems.^{ [18] } Many examples of provable quantum speedups for query problems are related to Grover's algorithm, including Brassard, Høyer, and Tapp's algorithm for finding collisions in two-to-one functions,^{ [19] } which uses Grover's algorithm, and Farhi, Goldstone, and Gutmann's algorithm for evaluating NAND trees,^{ [20] } which is a variant of the search problem.

A notable application of quantum computation is for attacks on cryptographic systems that are currently in use. 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).^{ [21] } 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.

Identifying cryptographic systems that may be secure against quantum algorithms is an actively researched topic under the field of * post-quantum cryptography *.^{ [22] }^{ [23] } 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.^{ [22] }^{ [24] } 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.^{ [25] } It has been proven that applying Grover's algorithm to break a symmetric (secret key) algorithm by brute force requires time equal to roughly 2^{n/2} invocations of the underlying cryptographic algorithm, compared with roughly 2^{n} in the classical case,^{ [26] } 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.^{ [27] }

The most well-known example of a problem admitting a polynomial quantum speedup is *unstructured search*, finding a marked item out of a list of items in a database. This can be solved by Grover's algorithm using queries to the database, quadratically fewer than the queries required for 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.

Problems that can be addressed with Grover's algorithm have the following properties:^{[ citation needed ]}

- There is no searchable structure in the collection of possible answers,
- The number of possible answers to check is the same as the number of inputs to the algorithm, and
- 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 scales 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^{ [28] } is Boolean satisfiability problem, where the *database* through which the algorithm iterates is that of all possible answers. An example and (possible) application of this is a password cracker that attempts to guess a password. Symmetric ciphers such as Triple DES and AES are particularly vulnerable to this kind of attack.^{[ citation needed ]} This application of quantum computing is a major interest of government agencies.^{ [29] }

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.^{ [30] } Quantum simulation could also be used to simulate the behavior of atoms and particles at unusual conditions such as the reactions inside a collider.^{ [31] } Quantum simulations might be used to predict future paths of particles and protons under superposition in the double-slit experiment.^{[ citation needed ]} About 2% of the annual global energy output is used for nitrogen fixation to produce ammonia for the Haber process in the agricultural fertilizer industry while naturally occurring organisms also produce ammonia. Quantum simulations might be used to understand this process increasing production.^{ [32] }

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.

Since quantum computers can produce outputs that classical computers cannot produce efficiently, and since quantum computation is fundamentally linear algebraic, some express hope in developing quantum algorithms that can speed up machine learning tasks.^{ [33] }^{ [34] } For example, the quantum algorithm for linear systems of equations, or "HHL Algorithm", named after its discoverers Harrow, Hassidim, and Lloyd, is believed to provide speedup over classical counterparts.^{ [35] }^{ [34] } Some research groups have recently explored the use of quantum annealing hardware for training Boltzmann machines and deep neural networks.^{ [36] }^{ [37] }

In the field of computational biology, computing has played a big role in solving many biological problems. One of the well known examples would be in computational genomics and how computing has drastically reduced the time to sequence a human genome. Given how computational biology is using generic data modeling and storage, its applications to computational biology are expected to arise as well.^{ [38] }

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.^{ [39] } 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.^{ [40] } Although skeptics like Gil Kalai doubt that quantum supremacy will ever be achieved,^{ [41] }^{ [42] } in October 2019, a Sycamore processor created in conjunction with Google AI Quantum was reported to have achieved quantum supremacy,^{ [43] } with calculations more than 3,000,000 times as fast as those of Summit, generally considered the world's fastest computer.^{ [44] } In December 2020, a group at USTC implemented a type of Boson sampling on 76 photons with a photonic quantum computer Jiuzhang to demonstrate quantum supremacy.^{ [45] }^{ [46] }^{ [47] } The authors claim that a classical contemporary supercomputer would require a computational time of 600 million years to generate the number of samples their quantum processor can generate in 20 seconds.^{ [48] } Bill Unruh doubted the practicality of quantum computers in a paper published back in 1994.^{ [49] } Paul Davies argued that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the holographic principle.^{ [50] }

There are a number of technical challenges in building a large-scale quantum computer.^{ [51] } Physicist David DiVincenzo has listed these requirements for a practical quantum computer:^{ [52] }

- Physically scalable 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

Sourcing parts for quantum computers is also very difficult. Many quantum computers, like those constructed by Google and IBM, need Helium-3, a nuclear research byproduct, and special superconducting cables made only by the Japanese company Coax Co.^{ [53] }

The control of multi-qubit systems requires the generation and coordination of a large number of electrical signals with tight and deterministic timing resolution. This has led to the development of quantum controllers which enable interfacing with the qubits. Scaling these systems to support a growing number of qubits is an additional challenge.^{[ citation needed ]}

One of the greatest challenges involved with constructing quantum computers 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 *T*_{2} (for NMR and MRI technology, also called the *dephasing time*), typically range between nanoseconds and seconds at low temperature.^{ [54] } Currently, some quantum computers require their qubits to be cooled to 20 millikelvins in order to prevent significant decoherence.^{ [55] } A 2020 study argues that ionizing radiation such as cosmic rays can nevertheless cause certain systems to decohere within milliseconds.^{ [56] }

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.^{ [57] }

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 *L*^{2}, where *L* is the number of digits 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 10^{4} bits without error correction.^{ [58] } With error correction, the figure would rise to about 10^{7} bits. Computation time is about *L*^{2} or about 10^{7} 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.^{ [59] }^{ [60] }

Physicist Mikhail Dyakonov has expressed skepticism of quantum computing as follows:

- "So the number of continuous parameters describing the state of such a useful quantum computer at any given moment must be... about 10
^{300}... Could we ever learn to control the more than 10^{300}continuously variable parameters defining the quantum state of such a system? My answer is simple.*No, never.*"^{ [61] }^{ [62] }

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:

- Quantum gate array (computation decomposed into a sequence of few-qubit quantum gates)
- One-way quantum computer (computation decomposed into a sequence of one-qubit measurements applied to a highly entangled initial state or cluster state)
- Adiabatic quantum computer, based on quantum annealing (computation decomposed into a slow continuous transformation of an initial Hamiltonian into a final Hamiltonian, whose ground states contain the solution)
^{ [63] } - Topological quantum computer
^{ [64] }(computation decomposed into the braiding of anyons in a 2D lattice)

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

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

- Superconducting quantum computing
^{ [65] }^{ [66] }(qubit implemented by the state of small superconducting circuits [ Josephson junctions]) - Trapped ion quantum computer (qubit implemented by the internal state of trapped ions)
- Neutral atoms in optical lattices (qubit implemented by internal states of neutral atoms trapped in an optical lattice)
^{ [67] }^{ [68] } - Quantum dot computer, spin-based (e.g. the Loss-DiVincenzo quantum computer
^{ [69] }) (qubit given by the spin states of trapped electrons) - Quantum dot computer, spatial-based (qubit given by electron position in double quantum dot)
^{ [70] } - Quantum computing using engineered quantum wells, which could in principle enable the construction of quantum computers that operate at room temperature
^{ [71] }^{ [72] } - Coupled quantum wire (qubit implemented by a pair of quantum wires coupled by a quantum point contact)
^{ [73] }^{ [74] }^{ [75] } - Nuclear magnetic resonance quantum computer (NMRQC) implemented with the nuclear magnetic resonance of molecules in solution, where qubits are provided by nuclear spins within the dissolved molecule and probed with radio waves
- Solid-state NMR Kane quantum computers (qubit realized by the nuclear spin state of phosphorus donors in silicon)
- Electrons-on-helium quantum computers (qubit is the electron spin)
- Cavity quantum electrodynamics (CQED) (qubit provided by the internal state of trapped atoms coupled to high-finesse cavities)
- Molecular magnet
^{ [76] }(qubit given by spin states) - Fullerene-based ESR quantum computer (qubit based on the electronic spin of atoms or molecules encased in fullerenes)
^{ [77] } - Nonlinear optical quantum computer (qubits realized by processing states of different modes of light through both linear and nonlinear elements)
^{ [78] }^{ [79] } - Linear optical quantum computer (qubits realized by processing states of different modes of light through linear elements e.g. mirrors, beam splitters and phase shifters)
^{ [80] } - Diamond-based quantum computer
^{ [81] }^{ [82] }^{ [83] }(qubit realized by the electronic or nuclear spin of nitrogen-vacancy centers in diamond) - Bose-Einstein condensate-based quantum computer
^{ [84] } - Transistor-based quantum computer – string quantum computers with entrainment of positive holes using an electrostatic trap
- Rare-earth-metal-ion-doped inorganic crystal based quantum computers
^{ [85] }^{ [86] }(qubit realized by the internal electronic state of dopants in optical fibers) - Metallic-like carbon nanospheres-based quantum computers
^{ [87] }

The large number of candidates demonstrates that quantum computing, despite rapid progress, is still in its infancy.^{[ citation needed ]}

Any computational problem solvable by a classical computer is also solvable by a quantum computer.^{ [13] } Intuitively, this is because it is believed that all physical phenomena, including the operation of classical computers, can be described using quantum mechanics, which underlies the operation of quantum computers.

Conversely, any problem solvable by a quantum computer is also solvable by a classical computer; or more formally, any quantum computer can be simulated by a Turing machine. In other words, quantum computers provide no additional power over classical computers in terms of computability. This means that quantum computers cannot solve undecidable problems like the halting problem and the existence of quantum computers does not disprove the Church–Turing thesis.^{ [88] }

As of yet, quantum computers do not satisfy the strong Church thesis. While hypothetical machines have been realized, a universal quantum computer has yet to be physically constructed. The strong version of Church's thesis requires a physical computer, and therefore there is no quantum computer that yet satisfies the strong Church thesis.

While quantum computers cannot solve any problems that classical computers cannot already solve, it is suspected that they can solve certain problems faster than classical computers. For instance, it is known that quantum computers can efficiently factor integers, while this is not believed to be the case for classical computers.

The class of problems that can be efficiently solved by a quantum computer with bounded error is called BQP, for "bounded error, quantum, polynomial time". More formally, BQP is the class of problems that can be solved by a polynomial-time quantum Turing machine with error probability of at most 1/3. As a class of probabilistic problems, BQP is the quantum counterpart to BPP ("bounded error, probabilistic, polynomial time"), the class of problems that can be solved by polynomial-time probabilistic Turing machines with bounded error.^{ [89] } It is known that BPPBQP and is widely suspected that BQPBPP, which intuitively would mean that quantum computers are more powerful than classical computers in terms of time complexity.^{ [90] }

The exact relationship of BQP to P, NP, and PSPACE is not known. However, it is known that PBQPPSPACE; that is, all problems that can be efficiently solved by a deterministic classical computer can also be efficiently solved by a quantum computer, and all problems that can be efficiently solved by a quantum computer can also be solved by a deterministic classical computer with polynomial space resources. It is further suspected that BQP is a strict superset of P, meaning there are problems that are efficiently solvable by quantum computers that are not efficiently solvable by deterministic classical computers. For instance, integer factorization and the discrete logarithm problem are known to be in BQP and are suspected to be outside of P. On the relationship of BQP to NP, little is known beyond the fact that some NP problems that are believed not to be in P are also in BQP (integer factorization and the discrete logarithm problem are both in NP, for example). It is suspected that NPBQP; that is, it is believed that there are efficiently checkable problems that are not efficiently solvable by a quantum computer. As a direct consequence of this belief, it is also suspected that BQP is disjoint from the class of NP-complete problems (if an NP-complete problem were in BQP, then it would follow from NP-hardness that all problems in NP are in BQP).^{ [91] }

The relationship of BQP to the basic classical complexity classes can be summarized as follows:

It is also known that BQP is contained in the complexity class * #P * (or more precisely in the associated class of decision problems *P ^{#P}*),

It has been speculated that further advances in physics could lead to even faster computers. For instance, it has been shown that a non-local hidden variable quantum computer based on Bohmian Mechanics could implement a search of an -item database in at most steps, a slight speedup over Grover's algorithm, which runs in steps. Note, however, that neither search method would allow quantum computers to solve NP-complete problems in polynomial time.^{ [92] } Theories of quantum gravity, such as M-theory and loop quantum gravity, may allow even faster computers to be built. However, defining computation in these theories is an open problem due to the problem of time; that is, within these physical theories there is currently no obvious way to describe what it means for an observer to submit input to a computer at one point in time and then receive output at a later point in time.^{ [93] }^{ [94] }

- Chemical computer
- D-Wave Systems
- DNA computing
- Electronic quantum holography
- Intelligence Advanced Research Projects Activity
- Kane quantum computer
- List of emerging technologies
- List of quantum processors
- Magic state distillation
- Natural computing
- Photonic computing
- Post-quantum cryptography
- Quantum algorithm
- Quantum annealing
- Quantum bus
- Quantum cognition
- Quantum circuit
- Quantum complexity theory
- Quantum cryptography
- Quantum logic gate
- Quantum machine learning
- Quantum supremacy
- Quantum threshold theorem
- Quantum volume
- Rigetti Computing
- Supercomputer
- Superposition
- Theoretical computer science
- Timeline of quantum computing
- Topological quantum computer
- Valleytronics

In computational complexity theory, **bounded-error quantum polynomial time** (**BQP**) is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1/3 for all instances. It is the quantum analogue to the complexity class **BPP**.

In quantum computing, a **qubit** or **quantum bit** is the basic unit of quantum information—the quantum version of the classic 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 simultaneously, a property that is fundamental to quantum mechanics and quantum computing.

**Shor's algorithm** is a polynomial-time 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.

In quantum computing, **Grover's algorithm**, also known as the **quantum search algorithm**, refers to a quantum algorithm for unstructured search 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.

In logic circuits, the **Toffoli gate**, invented by Tommaso Toffoli, is a universal reversible logic gate, which means that any classical reversible circuit can be constructed from Toffoli gates. It is also known as the "controlled-controlled-not" gate, which describes its action. It has 3-bit inputs and outputs; if the first two bits are both set to 1, it inverts the third bit, otherwise all bits stay the same.

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.

In quantum computing and specifically the quantum circuit model of computation, a **quantum logic gate** is a basic quantum circuit operating on a small number of qubits. They are the building blocks of quantum circuits, like classical logic gates are for conventional digital circuits.

In computer 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 Bell 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 gate *S*; and therefore stabilizer circuits can be constructed using only these gates.

**D-Wave Systems Inc.** is a Canadian quantum computing company, based in Burnaby, British Columbia, Canada. D-Wave was the world's first company to sell computers to exploit quantum effects in their operation. D-Wave's early customers include Lockheed Martin, University of Southern California, Google/NASA and Los Alamos National Lab.

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.

**Adiabatic quantum computation** (**AQC**) is a form of quantum computing which relies on the adiabatic theorem to do calculations and is closely related to quantum annealing.

**Quantum complexity theory** is the subfield of computational complexity theory that deals with complexity classes defined using quantum computers, a computational model based on quantum mechanics. It studies the hardness of computational problems in relation to these complexity classes, as well as the relationship between quantum complexity classes and classical complexity classes.

**Linear Optical Quantum Computing** or **Linear Optics Quantum Computation** (**LOQC**) is a paradigm of quantum computation, allowing universal quantum computation. LOQC uses photons as information carriers, mainly uses linear optical elements, or optical instruments to process quantum information, and uses photon detectors and quantum memories to detect and store quantum information.

The **quantum algorithm for linear systems of equations**, also called **HHL algorithm**, designed by Aram Harrow, Avinatan Hassidim, and Seth Lloyd, is a quantum algorithm formulated in 2009 for solving linear systems. The algorithm estimates the result of a scalar measurement on the solution vector to a given linear system of equations.

**Quantum machine learning** is the integration of quantum algorithms within machine learning programs. The most common use of the term refers to machine learning algorithms for the analysis of classical data executed on a quantum computer, i.e. *quantum-enhanced machine learning*. While machine learning algorithms are used to compute immense quantities of data, quantum machine learning utilizes qubits and quantum operations or specialized quantum systems to improve computational speed and data storage done by algorithms in a program. This includes hybrid methods that involve both classical and quantum processing, where computationally difficult subroutines are outsourced to a quantum device. These routines can be more complex in nature and executed faster on a quantum computer. Furthermore, quantum algorithms can be used to analyze quantum states instead of classical data. Beyond quantum computing, the term "quantum machine learning" is also associated with classical machine learning methods applied to data generated from quantum experiments, such as learning the phase transitions of a quantum system or creating new quantum experiments. Quantum machine learning also extends to a branch of research that explores methodological and structural similarities between certain physical systems and learning systems, in particular neural networks. For example, some mathematical and numerical techniques from quantum physics are applicable to classical deep learning and vice versa. Furthermore, researchers investigate more abstract notions of learning theory with respect to quantum information, sometimes referred to as "quantum learning theory".

In quantum computing, **quantum supremacy** or **quantum advantage** is the goal of demonstrating that a programmable quantum device can solve a problem that no classical computer can solve in any feasible amount of time. Conceptually, quantum supremacy involves both the engineering task of building a powerful quantum computer and the computational-complexity-theoretic task of finding a problem that can be solved by that quantum computer and has a superpolynomial speedup over the best known or possible classical algorithm for that task. The term was coined by John Preskill in 2012, 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. Examples of proposals to demonstrate quantum supremacy include the boson sampling proposal of Aaronson and Arkhipov, D-Wave's specialized frustrated cluster loop problems, and sampling the output of random quantum circuits.

**Quil** is a quantum instruction set architecture that first introduced a shared quantum/classical memory model. It was introduced by Robert Smith, Michael Curtis, and William Zeng in *A Practical Quantum Instruction Set Architecture*. Many quantum algorithms require a shared memory architecture. Quil is being developed for the superconducting quantum processors developed by Rigetti Computing through the Forest quantum programming API. A Python library called `pyQuil`

was introduced to develop Quil programs with higher level constructs. A Quil backend is also supported by other quantum programming environments.

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.

**Quantum gate teleportation** is a variant of the one-way quantum computer where quantum gates are applied to quantum states via quantum teleportation. It proves that universal quantum computation can be performed using only single qubit gates, Bell measurements and GHZ states. This makes it convenient for implementations where gates can't be applied directly; like in Linear Optical Quantum Computing where two qubit gates are difficult to implement. For example, the figure on the right shows how a CNOT gate can be applied to states and with a state ( is created before the dashed lines). Since universal fault tolerant quantum computation can be implemented with this protocol, quantum gate teleportation can be used to circumvent the Eastin-Knill theorem. Quantum gate teleportation has been experimentally demonstrated in Linear optical quantum computing, Superconducting quantum computing, and Ion trap quantum computing.

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*Elements of Quantum Computing: History, Theories and Engineering Applications*. Springer International Publishing. ISBN 978-3-319-08284-4. - Benenti, Giuliano (2004).
*Principles of Quantum Computation and Information Volume 1*. New Jersey: World Scientific. ISBN 978-981-238-830-8. OCLC 179950736. - Stolze, Joachim; Suter, Dieter (2004).
*Quantum Computing*. Wiley-VCH. ISBN 978-3-527-40438-4. - Wichert, Andreas (2014).
*Principles of Quantum Artificial Intelligence*. World Scientific Publishing Co. ISBN 978-981-4566-74-2. - Hiroshi, Imai; Masahito, Hayashi (2006).
*Quantum Computation and Information*. Berlin: Springer. ISBN 978-3-540-33132-2. - Jaeger, Gregg (2006).
*Quantum Information: An Overview*. Berlin: Springer. ISBN 978-0-387-35725-6. OCLC 255569451.

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Wikimedia Commons has media related to .Quantum computer |

- Stanford Encyclopedia of Philosophy: "Quantum Computing" by Amit Hagar and Michael E. Cuffaro.
- "Quantum computation, theory of",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - Quantum computing for the very curious by Andy Matuschak and Michael Nielsen
- Quantum Computing Made Easy on Satalia blog

- Lectures

- Quantum computing for the determined – 22 video lectures by Michael Nielsen
- Video Lectures by David Deutsch
- Lectures at the Institut Henri Poincaré (slides and videos)
- Online lecture on An Introduction to Quantum Computing, Edward Gerjuoy (2008)
- Lomonaco, Sam. Four Lectures on Quantum Computing given at Oxford University in July 2006

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