begin quote from:https://en.wikipedia.org/wiki/Quantum_computing

# Quantum computing

A **quantum computer** is a computer that exploits quantum mechanical phenomena. On small scales, physical matter exhibits properties of both particles and waves, and quantum computing leverages this behavior using specialized hardware. Classical physics cannot explain the operation of these quantum devices, and a scalable quantum computer could perform some calculations exponentially faster than any modern "classical" computer. In particular, a large-scale quantum computer could break widely used encryption schemes and aid physicists in performing physical simulations; however, the current state of the art is largely experimental and impractical, with several obstacles to useful applications.

The basic unit of information in quantum computing, the qubit (or "quantum bit"), serves the same function as the bit in classical computing. However, unlike a classical bit, which can be in one of two states (a binary), a qubit can exist in a superposition of its two "basis" states, which loosely means that it is in both states simultaneously. When measuring a qubit, the result is a probabilistic output of a classical bit. If a quantum computer manipulates the qubit in a particular way, wave interference effects can amplify the desired measurement results. The design of quantum algorithms involves creating procedures that allow a quantum computer to perform calculations efficiently and quickly.

Physically engineering high-quality qubits has proven challenging. If a physical qubit is not sufficiently isolated from its environment, it suffers from quantum decoherence, introducing noise into calculations. National governments have invested heavily in experimental research that aims to develop scalable qubits with longer coherence times and lower error rates. Two of the most promising technologies are superconductors (which isolate an electrical current by eliminating electrical resistance) and ion traps (which confine a single atomic particle using electromagnetic fields).

In principle, a classical computer can solve the same
computational problems as a quantum computer, given enough time. Quantum
advantage comes in the form of time complexity rather than computability, and quantum complexity theory
shows that some quantum algorithms are exponentially more efficient
than the best known classical algorithms. A large-scale quantum computer
could in theory solve computational problems unsolvable by a classical
computer in any reasonable amount of time. While claims of such *quantum supremacy* have drawn significant attention to the discipline, near-term practical use cases remain limited.

## History

For many years, the fields of quantum mechanics and computer science formed distinct academic communities.^{[2]} Modern quantum theory developed in the 1920s to explain the wave–particle duality observed at atomic scales,^{[3]} and digital computers emerged in the following decades to replace human computers for tedious calculations.^{[4]} Both disciplines had practical applications during World War II; computers played a major role in wartime cryptography,^{[5]} and quantum physics was essential for the nuclear physics used in the Manhattan Project.^{[6]}

As physicists applied quantum mechanical models to computational problems and swapped digital bits for qubits, the fields of quantum mechanics and computer science began to converge.
In 1980, Paul Benioff introduced the quantum Turing machine, which uses quantum theory to describe a simplified computer.^{[7]}
When digital computers became faster, physicists faced an exponential increase in overhead when simulating quantum dynamics,^{[8]} prompting Yuri Manin and Richard Feynman to independently suggest that hardware based on quantum phenomena might be more efficient for computer simulation.^{[9]}^{[10]}^{[11]}
In a 1984 paper, Charles Bennett and Gilles Brassard applied quantum theory to cryptography protocols and demonstrated that quantum key distribution could enhance information security.^{[12]}^{[13]}

Quantum algorithms then emerged for solving oracle problems, such as Deutsch's algorithm in 1985,^{[14]} the Bernstein–Vazirani algorithm in 1993,^{[15]} and Simon's algorithm in 1994.^{[16]}
These algorithms did not solve practical problems, but demonstrated
mathematically that one could gain more information by querying a black box with a quantum state in superposition, sometimes referred to as *quantum parallelism*.^{[17]}

Peter Shor built on these results with his 1994 algorithms for breaking the widely used RSA and Diffie–Hellman encryption protocols,^{[18]} which drew significant attention to the field of quantum computing.^{[19]}
In 1996, Grover's algorithm established a quantum speedup for the widely applicable unstructured search problem.^{[20]}^{[21]} The same year, Seth Lloyd proved that quantum computers could simulate quantum systems without the exponential overhead present in classical simulations,^{[22]} validating Feynman's 1982 conjecture.^{[23]}

Over the years, experimentalists have constructed small-scale quantum computers using trapped ions and superconductors.^{[24]}
In 1998, a two-qubit quantum computer demonstrated the feasibility of the technology,^{[25]}^{[26]} and subsequent experiments have increased the number of qubits and reduced error rates.^{[24]}

In 2019, Google AI and NASA announced that they had achieved quantum supremacy with a 54-qubit machine, performing a computation that is impossible for any classical computer.^{[27]}^{[28]}^{[29]} However, the validity of this claim is still being actively researched.^{[30]}^{[31]}

The threshold theorem shows how increasing the number of qubits can mitigate errors,^{[32]} yet fully fault-tolerant quantum computing remains "a rather distant dream".^{[33]} According to some researchers, *noisy intermediate-scale quantum* (NISQ) machines may have specialized uses in the near future, but noise in quantum gates limits their reliability.^{[33]}

Investment in quantum computing research has increased in the public and private sectors.^{[34]}^{[35]}
As one consulting firm summarized,^{[36]}

... investment dollars are pouring in, and quantum-computing start-ups are proliferating. ... While quantum computing promises to help businesses solve problems that are beyond the reach and speed of conventional high-performance computers, use cases are largely experimental and hypothetical at this early stage.

With focus on business management’s point of view, the potential
applications of quantum computing into four major categories are
cybersecurity, data analytics and artificial intelligence, optimization
and simulation, and data management and searching.^{[37]}

In December 2023, physicists, for the first time, report the
entanglement of individual molecules, which may have significant
applications in quantum computing.^{[38]} Also in December 2023, scientists at Harvard
University successfully created "quantum circuits" that correct errors
more efficiently than alternative methods, which may potentially remove a
major obstacle to practical quantum computers.^{[39]}^{[40]} The Harvard research team was supported by MIT, QuEra Computing, Caltech, and Princeton University and funded by DARPA's Optimization with Noisy Intermediate-Scale Quantum devices (ONISQ) program.^{[41]}^{[42]} Research efforts are ongoing to jumpstart quantum computing through topological and photonic approaches as well.^{[43]}

In July 2024, quantum computing company Quantinuum announced that
their new 56-qubit H2-1 computer has broken a world record in "quantum
supremacy," topping the performance of benchmarking set by Google's
Sycamore machine by 100-fold.^{[44]}

## Quantum information processing

Computer engineers typically describe a modern computer's operation in terms of classical electrodynamics. Within these "classical" computers, some components (such as semiconductors and random number generators) may rely on quantum behavior, but these components are not isolated from their environment, so any quantum information quickly decoheres. While programmers may depend on probability theory when designing a randomized algorithm, quantum mechanical notions like superposition and interference are largely irrelevant for program analysis.

Quantum programs, in contrast, rely on precise control of coherent quantum systems. Physicists describe these systems mathematically using linear algebra. Complex numbers model probability amplitudes, vectors model quantum states, and matrices model the operations that can be performed on these states. Programming a quantum computer is then a matter of composing operations in such a way that the resulting program computes a useful result in theory and is implementable in practice.

As physicist Charlie Bennett describes the relationship between quantum and classical computers,^{[45]}

A classical computer is a quantum computer ... so we shouldn't be asking about "where do quantum speedups come from?" We should say, "well, all computers are quantum. ... Where do classical slowdowns come from?"

### Quantum information

Just as the bit is the basic concept of classical information theory, the *qubit* is the fundamental unit of quantum information. The same term *qubit*
is used to refer to an abstract mathematical model and to any physical
system that is represented by that model. A classical bit, by
definition, exists in either of two physical states, which can be
denoted 0 and 1. A qubit is also described by a state, and two states
often written and serve as the quantum counterparts of the classical states 0 and 1. However, the quantum states and belong to a vector space,
meaning that they can be multiplied by constants and added together,
and the result is again a valid quantum state. Such a combination is
known as a *superposition* of and .^{[46]}^{[47]}

A two-dimensional vector mathematically represents a qubit state. Physicists typically use Dirac notation for quantum mechanical linear algebra, writing 'ket psi' for a vector labeled . Because a qubit is a two-state system, any qubit state takes the form , where and are the standard *basis states*,^{[a]} and and are the *probability amplitudes,* which are in general complex numbers.^{[47]} If either or is zero, the qubit is effectively a classical bit; when both are nonzero, the qubit is in superposition. Such a quantum state vector acts similarly to a (classical) probability vector, with one key difference: unlike probabilities, probability *amplitudes* are not necessarily positive numbers.^{[49]} Negative amplitudes allow for destructive wave interference.

When a qubit is measured in the standard basis, the result is a classical bit. The Born rule describes the norm-squared correspondence between amplitudes and probabilities—when measuring a qubit , the state collapses to with probability , or to with probability . Any valid qubit state has coefficients and such that . As an example, measuring the qubit would produce either or with equal probability.

Each additional qubit doubles the dimension of the state space.^{[48]}
As an example, the vector 1/√2|00⟩ + 1/√2|01⟩ represents a two-qubit state, a tensor product of the qubit |0⟩ with the qubit 1/√2|0⟩ + 1/√2|1⟩.
This vector inhabits a four-dimensional vector space spanned by the basis vectors |00⟩, |01⟩, |10⟩, and |11⟩.
The Bell state 1/√2|00⟩ + 1/√2|11⟩ is impossible to decompose into the tensor product of two individual qubits—the two qubits are *entangled* because their probability amplitudes are correlated.
In general, the vector space for an *n*-qubit system is 2^{n}-dimensional,
and this makes it challenging for a classical computer to simulate a
quantum one: representing a 100-qubit system requires storing 2^{100} classical values.

### Unitary operators

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 while 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 controlled NOT (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, quantum computation can be described as a network of quantum logic gates and measurements. However, any measurement can be deferred to the end of 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.

### Quantum parallelism

*Quantum parallelism*
is the heuristic that quantum computers can be thought of as evaluating
a function for multiple input values simultaneously. This can be
achieved by preparing a quantum system in a superposition of input
states, and applying a unitary transformation that encodes the function
to be evaluated. The resulting state encodes the function's output
values for all input values in the superposition, allowing for the
computation of multiple outputs simultaneously. This property is key to
the speedup of many quantum algorithms. However, "parallelism" in this
sense is insufficient to speed up a computation, because the measurement
at the end of the computation gives only one value. To be useful, a
quantum algorithm must also incorporate some other conceptual
ingredient.^{[50]}^{[51]}

### Quantum programming

There are a number of models of computation for quantum computing, distinguished by the basic elements in which the computation is decomposed.

#### Gate array

A quantum gate array decomposes computation into a sequence of few-qubit quantum gates. 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 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 of size 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.

#### Measurement-based quantum computing

A measurement-based quantum computer decomposes computation into a sequence of Bell state measurements and single-qubit quantum gates applied to a highly entangled initial state (a cluster state), using a technique called quantum gate teleportation.

#### Adiabatic quantum computing

An adiabatic quantum computer, based on quantum annealing, decomposes computation into a slow continuous transformation of an initial Hamiltonian into a final Hamiltonian, whose ground states contain the solution.^{[52]}

#### Neuromorphic quantum computing

Neuromorphic quantum computing (abbreviated as ‘n.quantum computing’) is an unconventional computing type of computing that uses neuromorphic computing to perform quantum operations. It was suggested that quantum algorithms, which are algorithms that run on a realistic model of quantum computation, can be computed equally efficiently with neuromorphic quantum computing. Both, traditional quantum computing and neuromorphic quantum computing are physics-based unconventional computing approaches to computations and don’t follow the von Neumann architecture. They both construct a system (a circuit) that represents the physical problem at hand, and then leverage their respective physics properties of the system to seek the “minimum”. Neuromorphic quantum computing and quantum computing share similar physical properties during computation.

#### Topological quantum computing

A topological quantum computer decomposes computation into the braiding of anyons in a 2D lattice.^{[53]}

#### Quantum Turing machine

A quantum Turing machine is the quantum analog of a Turing machine.^{[7]} All of these models of computation—quantum circuits,^{[54]} one-way quantum computation,^{[55]} adiabatic quantum computation,^{[56]} and topological quantum computation^{[57]}—have
been shown to be equivalent to the quantum Turing machine; given a
perfect implementation of one such quantum computer, it can simulate all
the others with no more than polynomial overhead. This equivalence need
not hold for practical quantum computers, since the overhead of
simulation may be too large to be practical.

### Quantum cryptography and cybersecurity

Quantum computing has significant potential applications in the fields of cryptography and cybersecurity. Quantum cryptography, which relies on the principles of quantum mechanics, offers the possibility of secure communication channels that are resistant to eavesdropping. Quantum key distribution (QKD) protocols, such as BB84, enable the secure exchange of cryptographic keys between parties, ensuring the confidentiality and integrity of communication. Moreover, quantum random number generators (QRNGs) can produce high-quality random numbers, which are essential for secure encryption.

However, quantum computing also poses challenges to traditional cryptographic systems. Shor's algorithm, a quantum algorithm for integer factorization, could potentially break widely used public-key cryptography schemes like RSA, which rely on the difficulty of factoring large numbers. Post-quantum cryptography, which involves the development of cryptographic algorithms that are resistant to attacks by both classical and quantum computers, is an active area of research aimed at addressing this concern.

Ongoing research in quantum cryptography and post-quantum
cryptography is crucial for ensuring the security of communication and
data in the face of evolving quantum computing capabilities. Advances in
these fields, such as the development of new QKD protocols, the
improvement of QRNGs, and the standardization of post-quantum
cryptographic algorithms, will play a key role in maintaining the
integrity and confidentiality of information in the quantum era.^{[58]}

## Communication

Quantum cryptography enables new ways to transmit data securely; for example, quantum key distribution uses entangled quantum states to establish secure cryptographic keys.^{[59]} When a sender and receiver exchange quantum states, they can guarantee that an adversary
does not intercept the message, as any unauthorized eavesdropper would
disturb the delicate quantum system and introduce a detectable change.^{[60]} With appropriate cryptographic protocols, the sender and receiver can thus establish shared private information resistant to eavesdropping.^{[12]}^{[61]}

Modern fiber-optic cables
can transmit quantum information over relatively short distances.
Ongoing experimental research aims to develop more reliable hardware
(such as quantum repeaters), hoping to scale this technology to
long-distance quantum networks
with end-to-end entanglement. Theoretically, this could enable novel
technological applications, such as distributed quantum computing and
enhanced quantum sensing.^{[62]}^{[63]}

## Algorithms

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.^{[64]}

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.^{[64]} 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, but evidence suggests that
this is unlikely.^{[65]} 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.^{[66]}

Some quantum algorithms, like Grover's algorithm and amplitude amplification, give polynomial speedups over corresponding classical algorithms.^{[64]}
Though these algorithms give comparably modest quadratic speedup, they
are widely applicable and thus give speedups for a wide range of
problems.^{[21]}

### Simulation of quantum systems

Since chemistry and nanotechnology rely on understanding quantum
systems, and such systems are impossible to simulate in an efficient
manner classically, quantum simulation may be an important application of quantum computing.^{[67]}
Quantum simulation could also be used to simulate the behavior of atoms
and particles at unusual conditions such as the reactions inside a collider.^{[68]}
In June 2023, IBM computer scientists reported that a quantum computer
produced better results for a physics problem than a conventional
supercomputer.^{[69]}^{[70]}

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 (even though naturally
occurring organisms also produce ammonia). Quantum simulations might be
used to understand this process and increase the energy efficiency of
production.^{[71]} It is expected that an early use of quantum computing will be modeling that improves the efficiency of the Haber–Bosch process^{[72]} by the mid 2020s^{[73]} although some have predicted it will take longer.^{[74]}

### Post-quantum cryptography

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).^{[75]} By comparison, a quantum computer could solve this problem exponentially faster using Shor's algorithm to find its factors.^{[76]} 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*.^{[77]}^{[78]}
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.^{[77]}^{[79]} 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.^{[80]} 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,^{[81]}
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*).

### Search problems

The most well-known example of a problem that allows for a polynomial quantum speedup is *unstructured search*, which involves 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. Many examples of provable
quantum speedups for query problems are based on Grover's algorithm,
including Brassard, Høyer, and Tapp's algorithm for finding collisions in two-to-one functions,^{[82]} and Farhi, Goldstone, and Gutmann's algorithm for evaluating NAND trees.^{[83]}

Problems that can be efficiently addressed with Grover's algorithm have the following properties:^{[84]}^{[85]}

- 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 that 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^{[86]} is a 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. Breaking symmetric ciphers with this algorithm is of interest to government agencies.^{[87]}

### Quantum annealing

Quantum annealing
relies on the adiabatic theorem to undertake calculations. A system is
placed in the ground state for a simple Hamiltonian, which slowly
evolves 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. Adiabatic optimization may be helpful for solving computational biology problems.^{[88]}

### Machine learning

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]}^{[89]}

For example, the HHL Algorithm, named after its discoverers Harrow, Hassidim, and Lloyd, is believed to provide speedup over classical counterparts.^{[33]}^{[90]} Some research groups have recently explored the use of quantum annealing hardware for training Boltzmann machines and deep neural networks.^{[91]}^{[92]}^{[93]}

Deep generative chemistry models emerge as powerful tools to expedite drug discovery.
However, the immense size and complexity of the structural space of all
possible drug-like molecules pose significant obstacles, which could be
overcome in the future by quantum computers. Quantum computers are
naturally good for solving complex quantum many-body problems^{[22]}
and thus may be instrumental in applications involving quantum
chemistry. Therefore, one can expect that quantum-enhanced generative
models^{[94]} including quantum GANs^{[95]} may eventually be developed into ultimate generative chemistry algorithms.

## Engineering

As of 2023, classical computers outperform quantum computers for all
real-world applications. While current quantum computers may speed up
solutions to particular mathematical problems, they give no
computational advantage for practical tasks. Scientists and engineers
are exploring multiple technologies for quantum computing hardware and
hope to develop scalable quantum architectures, but serious obstacles
remain.^{[96]}^{[97]}

### Challenges

There are a number of technical challenges in building a large-scale quantum computer.^{[98]} Physicist David DiVincenzo has listed these requirements for a practical quantum computer:^{[99]}

- 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. Superconducting 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.^{[100]}

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
that enable interfacing with the qubits. Scaling these systems to
support a growing number of qubits is an additional challenge.^{[101]}

#### Decoherence

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.^{[102]} Currently, some quantum computers require their qubits to be cooled to 20 millikelvin (usually using a dilution refrigerator^{[103]}) in order to prevent significant decoherence.^{[104]} A 2020 study argues that ionizing radiation such as cosmic rays can nevertheless cause certain systems to decohere within milliseconds.^{[105]}

As a result, time-consuming tasks may render some quantum
algorithms inoperable, as attempting to maintain the state of qubits for
a long enough duration will eventually corrupt the superpositions.^{[106]}

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 by the 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.^{[107]} 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. However, the encoding and error-correction overheads
increase the size of a real fault-tolerant quantum computer by several
orders of magnitude. Careful estimates^{[108]}^{[109]} show that at least 3 million
physical qubits would factor 2,048-bit integer in 5 months on a fully
error-corrected trapped-ion quantum computer. In terms of the number of
physical qubits, to date, this remains the lowest estimate^{[110]} for practically useful integer factorization problem sizing 1,024-bit or larger.

Another 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.^{[111]}^{[112]}

### Quantum supremacy

Physicist John Preskill coined the term *quantum supremacy*
to describe the engineering feat of demonstrating that a programmable
quantum device can solve a problem beyond the capabilities of
state-of-the-art classical computers.^{[113]}^{[114]}^{[115]} The problem need not be useful, so some view the quantum supremacy test only as a potential future benchmark.^{[116]}

In October 2019, Google AI Quantum, with the help of NASA, became
the first to claim to have achieved quantum supremacy by performing
calculations on the Sycamore quantum computer more than 3,000,000 times faster than they could be done on Summit, generally considered the world's fastest computer.^{[28]}^{[117]}^{[118]} This claim has been subsequently challenged: IBM has stated that Summit can perform samples much faster than claimed,^{[119]}^{[120]}
and researchers have since developed better algorithms for the sampling
problem used to claim quantum supremacy, giving substantial reductions
to the gap between Sycamore and classical supercomputers^{[121]}^{[122]}^{[123]} and even beating it.^{[124]}^{[125]}^{[126]}

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.^{[127]}^{[128]}^{[129]}
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.^{[130]}

Claims of quantum supremacy have generated hype around quantum computing,^{[131]} but they are based on contrived benchmark tasks that do not directly imply useful real-world applications.^{[96]}^{[132]}

In January 2024, a study published in *Physical Review Letters*
provided direct verification of quantum supremacy experiments by
computing exact amplitudes for experimentally generated bitstrings using
a new-generation Sunway supercomputer, demonstrating a significant leap
in simulation capability built on a multiple-amplitude tensor network
contraction algorithm. This development underscores the evolving
landscape of quantum computing, highlighting both the progress and the
complexities involved in validating quantum supremacy claims.^{[133]}

### Skepticism

Despite high hopes for quantum computing, significant progress in hardware, and optimism about future applications, a 2023 Nature spotlight article summarised current quantum computers as being "For now, [good for] absolutely nothing".^{[96]}
The article elaborated that quantum computers are yet to be more useful
or efficient than conventional computers in any case, though it also
argued that in the long term such computers are likely to be useful. A
2023 Communications of the ACM article^{[97]}
found that current quantum computing algorithms are "insufficient for
practical quantum advantage without significant improvements across the
software/hardware stack". It argues that the most promising candidates
for achieving speedup with quantum computers are "small-data problems",
for example in chemistry and materials science. However, the article
also concludes that a large range of the potential applications it
considered, such as machine learning, "will not achieve quantum
advantage with current quantum algorithms in the foreseeable future",
and it identified I/O constraints that make speedup unlikely for "big
data problems, unstructured linear systems, and database search based on
Grover's algorithm".

This state of affairs can be traced to several current and long-term considerations.

- Conventional computer hardware and algorithms are not only optimized for practical tasks, but are still improving rapidly, particularly GPU accelerators.
- Current quantum computing hardware generates only a limited amount of entanglement before getting overwhelmed by noise.
- Quantum algorithms
provide speedup over conventional algorithms only for some tasks, and
matching these tasks with practical applications proved challenging.
Some promising tasks and applications require resources far beyond those
available today.
^{[134]}^{[135]}In particular, processing large amounts of non-quantum data is a challenge for quantum computers.^{[97]} - Some promising algorithms have been "dequantized", i.e., their non-quantum analogues with similar complexity have been found.
- If quantum error correction
is used to scale quantum computers to practical applications, its
overhead may undermine speedup offered by many quantum algorithms.
^{[97]} - Complexity analysis of algorithms sometimes makes abstract assumptions that do not hold in applications. For example, input data may not already be available encoded in quantum states, and "oracle functions" used in Grover's algorithm often have internal structure that can be exploited for faster algorithms.

In particular, building computers with large numbers of qubits may be futile if those qubits are not connected well enough and cannot maintain sufficiently high degree of entanglement for long time. When trying to outperform conventional computers, quantum computing researchers often look for new tasks that can be solved on quantum computers, but this leaves the possibility that efficient non-quantum techniques will be developed in response, as seen for Quantum supremacy demonstrations. Therefore, it is desirable to prove lower bounds on the complexity of best possible non-quantum algorithms (which may be unknown) and show that some quantum algorithms asymptomatically improve upon those bounds.

Some researchers have expressed skepticism that scalable quantum computers could ever be built, typically because of the issue of maintaining coherence at large scales, but also for other reasons.

Bill Unruh doubted the practicality of quantum computers in a paper published in 1994.^{[136]} Paul Davies argued that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the holographic principle.^{[137]} Skeptics like Gil Kalai doubt that quantum supremacy will ever be achieved.^{[138]}^{[139]}^{[140]} 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.*"^{[141]}^{[142]}

### Candidates for physical realizations

A practical quantum computer must use a physical system as a programmable quantum register.^{[143]} Researchers are exploring several technologies as candidates for reliable qubit implementations.^{[144]} Superconductors and trapped ions are some of the most developed proposals, but experimentalists are considering other hardware possibilities as well.^{[145]}

## Theory

### Computability

Any computational problem solvable by a classical computer is also solvable by a quantum computer.^{[146]}
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. It is possible to simulate both
quantum and classical computers manually with just some paper and a pen,
if given enough time. 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.^{[147]}

### Complexity

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 an 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.^{[148]} It is known that and is widely suspected that , which intuitively would mean that quantum computers are more powerful than classical computers in terms of time complexity.^{[149]}

The exact relationship of BQP to P, NP, and PSPACE is not known. However, it is known that ;
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 ;
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).^{[150]}

## See also

- D-Wave Systems – Canadian quantum computing company
- Electronic quantum holography
- Glossary of quantum computing
- IARPA – American government agency
- IonQ – US information technology company
- List of emerging technologies – New technologies actively in development
- List of quantum processors – List of quantum computer components
- Magic state distillation – Quantum computing algorithm
- Natural computing – terminology introduced to encompass three classes of methods
- Optical computing – Computer that uses photons or light waves
- Quantum bus – device which can be used to store or transfer information between independent qubits in a quantum computer
- Quantum cognition – application of quantum mechanics to cognitive phenomena
- Quantum volume – Metric for a quantum computer's capabilities
- Quantum weirdness – Unintuitive aspects of quantum mechanics
- Rigetti Computing – American quantum computing company
- Supercomputer – Type of extremely powerful computer
- Theoretical computer science – Subfield of computer science and mathematics
- Unconventional computing – Computing by new or unusual methods
- Valleytronics – Experimental area in semiconductors

## Notes

- The standard basis is also the
*computational basis*.^{[48]}

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## Further reading

### Textbooks

- Akama, Seiki (2014).
*Elements of Quantum Computing: History, Theories and Engineering Applications*. Springer. doi:10.1007/978-3-319-08284-4. ISBN 978-3-319-08284-4. OCLC 884786739. - Benenti, Giuliano; Casati, Giulio; Rossini, Davide; Strini, Giuliano (2019).
*Principles of Quantum Computation and Information: A Comprehensive Textbook*(2nd ed.). doi:10.1142/10909. ISBN 978-981-3237-23-0. OCLC 1084428655. S2CID 62280636. - Bernhardt, Chris (2019).
*Quantum Computing for Everyone*. MIT Press. ISBN 978-0-262-35091-4. OCLC 1082867954. - Hidary, Jack D. (2021).
*Quantum Computing: An Applied Approach*(2nd ed.). doi:10.1007/978-3-030-83274-2. ISBN 978-3-03-083274-2. OCLC 1272953643. S2CID 238223274. - Hiroshi, Imai; Masahito, Hayashi, eds. (2006).
*Quantum Computation and Information: From Theory to Experiment*. Topics in Applied Physics. Vol. 102. doi:10.1007/3-540-33133-6. ISBN 978-3-540-33133-9. - Hughes, Ciaran; Isaacson, Joshua; Perry, Anastasia; Sun, Ranbel F.; Turner, Jessica (2021).
*Quantum Computing for the Quantum Curious*(PDF). doi:10.1007/978-3-030-61601-4. ISBN 978-3-03-061601-4. OCLC 1244536372. S2CID 242566636. - Jaeger, Gregg (2007).
*Quantum Information: An Overview*. doi:10.1007/978-0-387-36944-0. ISBN 978-0-387-36944-0. OCLC 186509710. - Johnston, Eric R.; Harrigan, Nic; Gimeno-Segovia, Mercedes (2019).
*Programming Quantum Computers: Essential Algorithms and Code Samples*. O'Reilly Media, Incorporated. ISBN 978-1-4920-3968-6. OCLC 1111634190. - Kaye, Phillip; Laflamme, Raymond; Mosca, Michele (2007).
*An Introduction to Quantum Computing*. OUP Oxford. ISBN 978-0-19-857000-4. OCLC 85896383. - Kitaev, Alexei Yu.; Shen, Alexander H.; Vyalyi, Mikhail N. (2002).
*Classical and Quantum Computation*. American Mathematical Soc. ISBN 978-0-8218-3229-5. OCLC 907358694. - Stolze, Joachim; Suter, Dieter (2004).
*Quantum Computing: A Short Course from Theory to Experiment*. doi:10.1002/9783527617760. ISBN 978-3-527-61776-0. OCLC 212140089. - Susskind, Leonard; Friedman, Art (2014).
*Quantum Mechanics: The Theoretical Minimum*. New York: Basic Books. ISBN 978-0-465-08061-8. - Wichert, Andreas (2020).
*Principles of Quantum Artificial Intelligence: Quantum Problem Solving and Machine Learning*(2nd ed.). doi:10.1142/11938. ISBN 978-981-12-2431-7. OCLC 1178715016. S2CID 225498497. - Wong, Thomas (2022).
*Introduction to Classical and Quantum Computing*(PDF). Rooted Grove. ISBN 979-8-9855931-0-5. OCLC 1308951401. Archived from the original (PDF) on 29 January 2022. Retrieved 6 February 2022. - Zeng, Bei; Chen, Xie; Zhou, Duan-Lu; Wen, Xiao-Gang (2019).
*Quantum Information Meets Quantum Matter*. arXiv:1508.02595. doi:10.1007/978-1-4939-9084-9. ISBN 978-1-4939-9084-9. OCLC 1091358969. S2CID 118528258.

### Academic papers

- Abbot, Derek; Doering, Charles R.; Caves, Carlton M.; Lidar, Daniel M.; Brandt, Howard E.; et al. (2003). "Dreams versus Reality: Plenary Debate Session on Quantum Computing".
*Quantum Information Processing*.**2**(6): 449–472. arXiv:quant-ph/0310130. doi:10.1023/B:QINP.0000042203.24782.9a. hdl:2027.42/45526. S2CID 34885835. - Berthiaume, Andre (1 December 1998). "Quantum Computation".
*Solution Manual for Quantum Mechanics*. pp. 233–234. doi:10.1142/9789814541893_0016. ISBN 978-981-4541-88-6. S2CID 128255429 – via Semantic Scholar. - DiVincenzo, David P. (2000). "The Physical Implementation of Quantum Computation".
*Fortschritte der Physik*.**48**(9–11): 771–783. arXiv:quant-ph/0002077. Bibcode:2000ForPh..48..771D. doi:10.1002/1521-3978(200009)48:9/11<771::AID-PROP771>3.0.CO;2-E. S2CID 15439711. - DiVincenzo, David P. (1995). "Quantum Computation".
*Science*.**270**(5234): 255–261. Bibcode:1995Sci...270..255D. CiteSeerX 10.1.1.242.2165. doi:10.1126/science.270.5234.255. S2CID 220110562. Table 1 lists switching and dephasing times for various systems. - Feynman, Richard (1982). "Simulating physics with computers".
*International Journal of Theoretical Physics*.**21**(6–7): 467–488. Bibcode:1982IJTP...21..467F. CiteSeerX 10.1.1.45.9310. doi:10.1007/BF02650179. S2CID 124545445. - Jeutner, Valentin (2021). "The Quantum Imperative: Addressing the Legal Dimension of Quantum Computers".
*Morals & Machines*.**1**(1): 52–59. doi:10.5771/2747-5174-2021-1-52. S2CID 236664155. - Krantz,
P.; Kjaergaard, M.; Yan, F.; Orlando, T. P.; Gustavsson, S.; Oliver, W.
D. (17 June 2019). "A Quantum Engineer's Guide to Superconducting
Qubits".
*Applied Physics Reviews*.**6**(2): 021318. arXiv:1904.06560. Bibcode:2019ApPRv...6b1318K. doi:10.1063/1.5089550. ISSN 1931-9401. S2CID 119104251. - Mitchell, Ian (1998). "Computing Power into the 21st Century: Moore's Law and Beyond".
- Simon, Daniel R. (1994). "On the Power of Quantum Computation". Institute of Electrical and Electronics Engineers Computer Society Press.

## External links

- Media related to Quantum computer at Wikimedia Commons
- Learning materials related to Quantum computing at Wikiversity
- 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

- 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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