Study of a model of computation

Quantum computing is the use of quantum phenomena such as superposition and entanglement to perform computation. Computers that perform quantum computations are known as quantum computers.[1]:I-5 Quantum computers 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.

Quantum computing began in the early 1980s, when physicist Paul Benioff proposed a quantum mechanical model of the Turing machine.[2]Richard FeynmanandYuri Maninlater suggested that a quantum computer had the potential to simulate things that a classical computer could not.[3][4] In 1994, Peter Shor developed a quantum algorithm for factoring integers that had the potential to decrypt RSA-encrypted communications.[5] Despite ongoing experimental progress since the late 1990s, most researchers believe that "fault-tolerant quantum computing [is] still a rather distant dream."[6] In recent years, investment into quantum computing research has increased in both the public and private sector.[7][8] 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 is infeasible on any classical computer.[9]

There are several models of quantum computers (or rather, 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. Quantum circuits are based on the quantum bit, or "qubit", which is somewhat analogous to the bit in classical computation. Qubits can be in a 1 or 0 quantum state, or they can be in a superposition of the 1 and 0 states. However, when qubits are measured the result of the measurement is always either a 0 or a 1; the probabilities of these two outcomes depend on the quantum state that the qubits were in immediately prior to the measurement.

Progress towards building a physical quantum computer focuses on technologies such as transmons, ion traps and topological quantum computers, which aim to create high-quality qubits.[1]:213 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 in the way of constructing useful quantum computers. In particular, it is difficult to maintain the quantum states of qubits as they suffer from quantum decoherence and state fidelity. Quantum computers therefore require error correction.[10][11]

Any computational problem that can be solved by a classical computer can also be solved by a quantum computer.[12] 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 ChurchTuring thesis. While this means that 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 timea 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.[13] 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 n {textstyle n} bits of information has 2 n {textstyle 2^{n}} possible states. A vector representing all memory states thus has 2 n {textstyle 2^{n}} 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

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

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

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

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

The mathematics of single qubit gates can be extended to operate on multiqubit 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

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

C N O T := ( 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 ) . {displaystyle CNOT:={begin{pmatrix}1&0&0&0\0&1&0&0\0&0&0&1\0&0&1&0end{pmatrix}}.}

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 n {displaystyle n} 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. The representation of multiple qubits can be shown as Qsphere.

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.[14]

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.[14] 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.[15] Certain oracle problems like Simon's problem and the BernsteinVazirani 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.[16]

Some quantum algorithms, like Grover's algorithm and amplitude amplification, give polynomial speedups over corresponding classical algorithms.[14] Though these algorithms give comparably modest quadratic speedup, they are widely applicable and thus give speedups for a wide range of problems.[17] Many examples of provable quantum speedups for query problems are related to Grover's algorithm, including Brassard, Hyer, and Tapp's algorithm for finding collisions in two-to-one functions,[18] which uses Grover's algorithm, and Farhi, Goldstone, and Gutmann's algorithm for evaluating NAND trees,[19] 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).[20] 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, DiffieHellman, and elliptic curve DiffieHellman 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. [21][22] 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.[21][23]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.[24] 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,[25] 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.[26]

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 n {displaystyle n} items in a database. This can be solved by Grover's algorithm using O ( n ) {displaystyle O({sqrt {n}})} queries to the database, quadratically fewer than than the ( n ) {displaystyle Omega (n)} 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]

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[27] 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.[citation needed] This application of quantum computing is a major interest of government agencies.[28]

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.[29] Quantum simulation could also be used to simulate the behavior of atoms and particles at unusual conditions such as the reactions inside a collider.[30] 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.[31]

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.[32][33] 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.[34][33]

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.[35]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.[36] Although skeptics like Gil Kalai doubt that quantum supremacy will ever be achieved,[37][38] in October 2019, a Sycamore processor created in conjunction with Google AI Quantum was reported to have achieved quantum supremacy,[39] with calculations more than 3,000,000 times as fast as those of Summit, generally considered the world's fastest computer.[40]Bill Unruh doubted the practicality of quantum computers in a paper published back in 1994.[41]Paul Davies argued that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the holographic principle.[42]

There are a number of technical challenges in building a large-scale quantum computer.[43] Physicist David DiVincenzo has listed the following requirements for a practical quantum computer:[44]

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 that are only made by the Japanese company Coax Co.[45]

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 had led to the development of quantum controllers which enable interfacing the qubit. Scaling these systems to support a growing number of qubits is an additional challenge in the scaling of quantum computers.[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 T2 (for NMR and MRI technology, also called the dephasing time), typically range between nanoseconds and seconds at low temperature.[46] Currently, some quantum computers require their qubits to be cooled to 20 millikelvins in order to prevent significant decoherence.[47] A 2020 study argues that ionizing radiation such as cosmic rays can nevertheless cause certain systems to decohere within milliseconds.[48]

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.[49]

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 103, 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.[50] With error correction, the figure would rise to about 107 bits. Computation time is about L2 or about 107 steps and at 1MHz, 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.[51][52]

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

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 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):

A 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.[80] 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 ChurchTuring thesis.[81]

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 many 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. However, the capacity of quantum computers to accelerate classical algorithms has rigid upper bounds, and the overwhelming majority of classical calculations cannot be accelerated by the use of quantum computers.[82]

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.[83] It is known that BPP {displaystyle subseteq } BQP and is widely suspected that BQP {displaystyle nsubseteq } BPP, which intuitively would mean that quantum computers are more powerful than classical computers in terms of time complexity.[84]

The exact relationship of BQP to P, NP, and PSPACE is not known. However, it is known that P {displaystyle subseteq } BQP {displaystyle subseteq } PSPACE; 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 NP {displaystyle nsubseteq } BQP; 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).[86]

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),[86] which is a subclass of PSPACE.

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 N {displaystyle N} -item database in at most O ( N 3 ) {displaystyle O({sqrt[{3}]{N}})} steps, a slight speedup over Grover's algorithm, which runs in O ( N ) {displaystyle O({sqrt {N}})} steps. Note, however, that neither search method would allow quantum computers to solve NP-complete problems in polynomial time.[87] 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.[88][89]

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Quantum computing - Wikipedia