Quantum computing

Short description: Computer hardware technology that uses quantum mechanics

IBM quantum computer demo at ITU WTSA 2024 in Delhi Bloch sphere representation of a qubit. The state |ψ⟩=α|0⟩+β|1⟩ is a point on the surface of the sphere, partway between the poles, |0⟩ and |1⟩.

A quantum computer is a real or theoretical computer that exploits quantum phenomena like superposition and entanglement in an essential way. It is widely believed that a quantum computer could perform some calculations exponentially faster than any classical computer. For example, a large-scale quantum computer could break some widely used encryption schemes and aid physicists in performing physical simulations. However, current hardware implementations of quantum computation are largely experimental and only suitable for specialized tasks.

The basic unit of information in quantum computing, the qubit (or "quantum bit"), serves the same function as the bit in ordinary or "classical" computing.[1] However, unlike a classical bit, which can be in one of two states (a binary), a qubit can exist in a linear combination of two states known as a quantum superposition. The result of measuring a qubit is one of the two states given by a probabilistic rule. If a quantum computer manipulates the qubit in a particular way, wave interference effects amplify the probability of the desired measurement result. The design of quantum algorithms involves creating procedures that allow a quantum computer to perform this amplification.

Quantum computers are not yet practical for real-world applications. Physically engineering high-quality qubits has proven to be 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 aimed at developing scalable qubits with longer coherence times and lower error rates. Example implementations include superconductors (which isolate an electrical current by eliminating electrical resistance) and ion traps (which confine a single atomic particle using electromagnetic fields). Researchers have claimed, and are widely believed to be correct, that certain quantum devices can outperform classical computers on narrowly defined tasks, a milestone referred to as quantum advantage or quantum supremacy. These tasks are not necessarily useful for real-world applications. As a result, current demonstrations are best understood as scientific milestones rather than evidence of broad near-term deployment. In December 2024, Google's Willow chip achieved below-threshold error correction, a milestone 30 years in the making, while global government investment in quantum computing reached $10 billion by April 2025.[2]

## Contents

* 1 History
* 2 Quantum information processing
* 2.1 Quantum information
* 2.2 Unitary operators
* 2.3 Quantum parallelism
* 2.4 Quantum programming
* 2.4.1 Gate array
* 2.4.2 Measurement-based quantum computing
* 2.4.3 Adiabatic quantum computing
* 2.4.4 Neuromorphic quantum computing
* 2.4.5 Topological quantum computing
* 2.4.6 Quantum Turing machine
* 2.4.7 Noisy intermediate-scale quantum computing
* 2.4.8 Quantum cryptography and cybersecurity
* 3 Communication
* 3.1 Quantum communication protocols
* 4 Algorithms
* 4.1 Simulation of quantum systems
* 4.2 Post-quantum cryptography
* 4.3 Search problems
* 4.4 Quantum annealing
* 4.5 Machine learning
* 4.6 AI-assisted algorithm discovery
* 5 Engineering
* 5.1 Challenges
* 5.1.1 Coolant
* 5.1.2 Decoherence
* 5.2 Quantum supremacy
* 5.3 Skepticism
* 5.4 Physical realizations
* 6 Theory
* 6.1 Computability
* 6.2 Complexity
* 7 List of quantum computers
* 7.1 Types of quantum computers
* 8 See also
* 9 Notes
* 10 References
* 10.1 Sources
* 11 Further reading
* 11.1 Textbooks
* 11.2 Academic papers
* 12 External links

## History

For many years, the fields of quantum mechanics and computer science formed distinct academic communities.[3] Modern quantum theory was developed in the 1920s to explain perplexing physical phenomena observed at atomic scales,[4][5] and digital computers emerged in the following decades to replace human computers for tedious calculations.[6] Both disciplines had practical applications during World War II; computers played a major role in wartime cryptography,[7] and quantum physics was essential for nuclear physics used in the Manhattan Project.[8]

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.[9] When digital computers became faster, physicists faced an exponential increase in overhead when simulating quantum dynamics,[10] prompting Yuri Manin and Richard Feynman to independently suggest that hardware based on quantum phenomena might be more efficient for computer simulation.[11][12][13] 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.[14][15]

Quantum algorithms then emerged for solving oracle problems, such as Deutsch's algorithm in 1985,[16] the Bernstein–Vazirani algorithm in 1993,[17] and Simon's algorithm in 1994.[18] These algorithms did not solve practical problems, but demonstrated mathematically that one could obtain more information by querying a black box with a quantum state in superposition, sometimes referred to as quantum parallelism.[19]

Peter Shor (pictured here in 2017) showed in 1994 that a scalable quantum computer would be able to break RSA encryption.

Peter Shor built on these results with his 1994 algorithm for breaking the widely used RSA and Diffie–Hellman encryption protocols,[20] which drew significant attention to the field of quantum computing. In 1996, Grover's algorithm established a quantum speedup for the widely applicable unstructured search problem.[21][22] The same year, Seth Lloyd proved that quantum computers could simulate quantum systems without the exponential overhead present in classical simulations,[23] validating Feynman's 1982 conjecture.[24]

Over the years, experimentalists have constructed small-scale quantum computers using trapped ions and superconductors.[25] In 1998, a two-qubit quantum computer demonstrated the feasibility of the technology,[26][27] and subsequent experiments have increased the number of qubits and reduced error rates.[25]

In 2019, Google AI and NASA announced that they had achieved quantum supremacy with a 54-qubit machine, performing a computation that classical supercomputers would take an estimated 10,000 years to complete—a claim subsequently disputed by IBM, which argued the calculation could be done in approximately 2.5 days on its Summit supercomputer with optimized algorithms, sparking a debate over the precise threshold for this milestone.[28][29][30][31][32]

Recent milestones in quantum computing have increasingly focused on controlling decoherence through quantum error correction. In 2024, researchers demonstrated theoretical and practical approaches for high threshold, low-overhead fault-tolerant quantum memory. These developments represent a critical step toward scaling systems beyond the noisy intermediate-scale quantum (NISQ) era into reliable, fault-tolerant computing architectures, though large-scale physical implementation remains an ongoing engineering challenge.[33]

## Quantum information processing

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

The "classical" in classical computation thus refers to the computational model, not to whether the microscopic physics of the hardware is ultimately quantum-mechanical. A conventional digital computer can be described by classical states and transition rules: memory stores bits, while logic elements transform one configuration of bits into another. This computational behavior is not tied to electronics, and can be abstracted through the idea of a Turing machine, a mechanical device that performs deterministic transformations on a finite state. In principle, the same classical transition rules can be implemented by some entirely classical mechanical device, possibly with a fixed slow-down in physical time.[34] If a classical computation uses randomness, this can be modeled as access to random classical bits rather than as coherent quantum information.[35] A quantum computer, by contrast, uses coherent quantum states, so that superposition, relative phase, and interference are part of the computation itself, and has no classical counterpart.

Quantum programs instead 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.

Physicist Charlie Bennett noted that since classical computers are composed of quantum atoms, one might study them from the opposite direction:[36]

> 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 |0⟩ and |1⟩, serve as the quantum counterparts of the classical states 0 and 1. However, the quantum states |0⟩ and |1⟩ 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 |0⟩ and |1⟩.[37][38]

A two-dimensional vector mathematically represents a qubit state. Physicists typically use bra–ket notation for quantum mechanical linear algebra, writing |ψ⟩ 'ket psi' for a vector labeled ψ. Because a qubit is a two-state system, any qubit state takes the form α|0⟩+β|1⟩, where |0⟩ and |1⟩ are the standard basis states,[lower-alpha 1] and α and β are the probability amplitudes, which are in general complex numbers.[38] If either α or β is zero, the qubit is effectively a classical bit; when both are nonzero, the qubit is in superposition. Such a quantum state vector behaves similarly to a (classical) probability vector, with one key difference: unlike probabilities, probability amplitudes are not necessarily positive numbers.[40] 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 α|0⟩+β|1⟩, the state collapses to |0⟩ with probability |α|2, or to |1⟩ with probability |β|2. Any valid qubit state has coefficients α and β such that |α|2+|β|2=1. As an example, measuring the qubit 1/2|0⟩+1/2|1⟩ would produce either |0⟩ or |1⟩ with equal probability.

Two particularly important superposition states are the plus state |+⟩=1/2|0⟩+1/2|1⟩ and the minus state |−⟩=1/2|0⟩−1/2|1⟩. While both yield outcomes 0 and 1 with equal probability upon standard basis measurement, they behave differently under operations such as the Hadamard gate—which maps |0⟩↔|+⟩ and |1⟩↔|−⟩—demonstrating that relative phase differences carry meaningful quantum information.

Each additional qubit doubles the dimension of the state space.[39] 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⟩.

In general, the vector space for an n-qubit system is 2n-dimensional, and this makes it challenging for a classical computer to simulate a quantum one: representing a 100-qubit system requires storing 2100 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 X:=(0110). Mathematically, the application of such a logic gate to a quantum state vector is modeled with matrix multiplication. Thus

X|0⟩=|1⟩ and X|1⟩=|0⟩.

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 |00⟩:=(1000);|01⟩:=(0100);|10⟩:=(0010);|11⟩:=(0001). The controlled NOT (CNOT) gate can then be represented using the following matrix: CNOT⁡:=(1000010000010010). As a mathematical consequence of this definition, CNOT⁡|00⟩=|00⟩, CNOT⁡|01⟩=|01⟩, CNOT⁡|10⟩=|11⟩, and CNOT⁡|11⟩=|10⟩. In other words, the CNOT applies a NOT gate (X from before) to the second qubit if and only if the first qubit is in the state |1⟩. If the first qubit is |0⟩, 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, enabling the simultaneous computation of multiple outputs. 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.[41][42]

### Quantum programming

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

#### Gate array

A quantum circuit diagram implementing a Toffoli gate from more primitive gates

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 2n×2n over 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. Implementation of Boolean functions using the few-qubit quantum gates is presented here.[43]

#### 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.[44]

#### Neuromorphic quantum computing

Neuromorphic quantum computing (abbreviated 'n.quantum computing') is an unconventional process 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 do not 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.[45]

#### Quantum Turing machine

A quantum Turing machine is the quantum analog of a Turing machine.[9] All of these models of computation—quantum circuits,[46] one-way quantum computation,[47] adiabatic quantum computation,[48] and topological quantum computation[49]—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.

#### Noisy intermediate-scale quantum computing

The threshold theorem shows how increasing the number of qubits can mitigate errors,[50] yet fully fault-tolerant quantum computing remains "a rather distant dream".[51] 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.[51] 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.[52] 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.[53][54]

#### Quantum cryptography and cybersecurity

Main article: Quantum cryptography

Digital cryptography enables communications to remain private, preventing unauthorized parties from accessing them. Conventional encryption, the obscuring of a message with a key through an algorithm, relies on the algorithm being difficult to reverse. Encryption is also the basis for digital signatures and authentication mechanisms. Quantum computing may be sufficiently more powerful that difficult reversals are feasible, allowing messages relying on conventional encryption to be read.[55]

Quantum cryptography replaces conventional algorithms with computations based on quantum computing. In principle, quantum encryption would be impossible to decode even with a quantum computer. This advantage comes at a significant cost in terms of elaborate infrastructure, while effectively preventing legitimate decoding of messages by governmental security officials.[55]

Ongoing research in quantum and post-quantum cryptography has led to new algorithms for quantum key distribution, initial work on quantum random number generation and to some early technology demonstrations.[56]: 1012–1036

## Communication

Quantum cryptography enables new ways to transmit data securely; for example, quantum key distribution uses entangled quantum states to establish secure cryptographic keys.[56]: 1017 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.[57] With appropriate cryptographic protocols, the sender and receiver can thus establish shared private information resistant to eavesdropping.[14][58]

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.[59][60]

### Quantum communication protocols

Quantum teleportation is a protocol by which Alice can transmit the quantum state of a qubit to Bob using one shared entangled pair (e-bit) and two classical bits of communication. The state of Alice's qubit is not physically transmitted—instead, it is reconstructed at Bob's end through classically communicated measurement outcomes and local unitary corrections. This demonstrates that quantum communication requires both entanglement and classical communication; neither alone is sufficient. Teleportation cannot be used to transmit information faster than light because the classical bits must travel through normal channels.

Superdense coding is the complementary protocol: using one shared e-bit and sending only one qubit, Alice can transmit two classical bits to Bob. This appears to violate Holevo's theorem—which states that a single qubit can carry at most one bit of classical information—but the shared entanglement circumvents this limit. Superdense coding thus demonstrates that entanglement can effectively double the classical information-carrying capacity of quantum communication.

## Algorithms

Progress in finding quantum algorithms typically focuses on the quantum circuit model,[46] though exceptions like the quantum adiabatic algorithm exist. Quantum algorithms can be roughly categorized by the type of speedup achieved over corresponding classical algorithms.[61]

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.[61] 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.[62] 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 don'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" provides a super-polynomial speedup, which is believed to be unlikely.[63]

In addition to these problems, quantum algorithms are being explored for applications in cryptography, optimization, and machine learning, although most of these remain at the research stage and require significant advances in error correction and hardware scalability before practical implementation.[64]

Some quantum algorithms, such as Grover's algorithm and amplitude amplification, give polynomial speedups over corresponding classical algorithms.[61] Though these algorithms give comparably modest quadratic speedup, they are widely applicable and thus give speedups for a wide range of problems.[22] These speed-ups are, however, over the theoretical worst-case of classical algorithms, and concrete real-world speed-ups over algorithms used in practice have not been demonstrated.

### 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.[65] Recent reviews identify quantum chemistry as one of the most promising application areas for quantum computing, particularly for problems in electronic structure, chemical dynamics, and spectroscopy, while noting that useful implementations remain limited by current hardware.[66] Quantum simulation could also be used to simulate the behavior of atoms and particles at unusual conditions such as the reactions inside a collider.[67] In June 2023, IBM computer scientists reported that a quantum computer produced better results for a physics problem than a conventional supercomputer.[68][69]

About 2% of the annual global energy output is used for nitrogen fixation to produce ammonia for the Haber process in the agricultural fertiliser 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.[70] It is expected that an early use of quantum computing will be modeling that improves the efficiency of the Haber–Bosch process[71] by the mid-2020s[72] although some have predicted it will take longer.[73]

### Post-quantum cryptography

Main article: Post-quantum cryptography

A notable application of quantum computing is in attacking cryptographic systems that are currently in use. Integer factorization, which underpins the security of public key cryptographic systems, is believed to be computationally infeasible on a classical computer for large integers if they are the product of a few prime numbers (e.g., the product of two 300-digit primes).[74] By contrast, a quantum computer could solve this problem exponentially faster using Shor's algorithm to factor the integer.[75] 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.[76][77] Some public-key algorithms are based on problems other than the integer factorization and discrete logarithm problems to which Shor's algorithm applies, such as the McEliece cryptosystem, which relies on a hard problem in coding theory.[76][78] 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.[79] It has been shown 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,[80] meaning that symmetric key lengths are effectively halved: AES-256 would have comparable security against an attack using Grover's algorithm to that AES-128 has against classical brute-force search (see Key size).

### Search problems

Main article: Grover's algorithm

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 n items in a database. This can be solved by Grover's algorithm using O(n) queries to the database, quadratically fewer than the Ω(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. 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,[81] and Farhi, Goldstone, and Gutmann's algorithm for evaluating NAND trees.[82]

Problems that can be efficiently addressed with Grover's algorithm have the following properties:[83][84]

1. There is no searchable structure in the collection of possible answers,
2. The number of possible answers to check is the same as the number of inputs to the algorithm, and
3. There exists a Boolean function 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[85] 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.[86]

### Quantum annealing

A wafer of adiabatic quantum computers

Quantum annealing uses the adiabatic theorem to perform 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. Quantum annealing can solve Ising models and the (computationally equivalent) QUBO problem, which in turn can be used to encode a wide range of combinatorial optimization problems.[87] Adiabatic optimization may be helpful for solving computational biology problems.[88]

### Machine learning

Main article: Quantum 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.[51][89] However, review literature notes that many proposed quantum machine-learning advantages rely on assumptions about efficient data encoding or continued access to quantum hardware, and have not yet translated into broad practical end-to-end advantage on current devices.[90][91] For example, the HHL Algorithm, named after its discoverers Harrow, Hassidim, and Lloyd, is believed to provide speedup over classical counterparts.[51][92] Some research groups have recently explored the use of quantum annealing hardware for training Boltzmann machines and deep neural networks.[93][94][95]

Deep generative chemistry models have been explored for potential applications in drug discovery. Early experimental work has explored the use of near-term quantum hardware in molecular generative modeling for drug discovery. In 2023, researchers at Gero reported a hybrid quantum–classical generative model based on a restricted Boltzmann machine, implemented on a commercially available quantum annealing device, to generate novel drug-like small molecules with physicochemical properties comparable to known medicinal compounds.[96][97] 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[23] and thus may be instrumental in applications involving quantum chemistry. Therefore, one can expect that quantum-enhanced generative models[98] including quantum GANs[99] may eventually be developed into ultimate generative chemistry algorithms.

### AI-assisted algorithm discovery

Artificial intelligence has also been explored as a tool for discovering and optimizing algorithms relevant to quantum computing. AlphaEvolve, a Google DeepMind system based on large language models and evolutionary algorithms, has been described as a coding agent for scientific and algorithmic discovery.[100] In quantum-computing research, AlphaEvolve-optimized quantum circuits have been used in work on quantum computation of molecular geometry through many-body nuclear spin echoes.[101]

## Engineering

As of 2023,[update] 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.[102][103] In practice, improvements in qubit counts alone are not enough, because error rates, connectivity, and data movement also affect whether an end-to-end application can outperform classical methods.

### Challenges

There are a number of technical challenges in building a large-scale quantum computer.[104] Physicist David DiVincenzo has listed these requirements for a practical quantum computer:[105]

* 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.

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.[106]

The theoretical potential for large-scale quantum computers to eventually break widely used public-key encryption schemes has prompted significant motivated changes in global cybersecurity strategies. In response to this future challenge, organizations, including the National Institute of Standards and Technology (NIST), have initiated detailed standardization processes for post-quantum cryptography. These global efforts are designed to develop, evaluate, and deploy cryptographic algorithms that remain safe against both quantum and classical computer attacks, well before fully fault-tolerant quantum systems become available.[107]

#### Coolant

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.[108] On 27 January 2026, the US DARPA agency made a call for proposals for a quantum computing coolant below 1 Kelvin, which does not use helium-3. In February 2026, the Chinese Academy of Sciences announced the testing of a rare-earth alloy, EuCo2Al9, which could fill a similar role.[109]

#### Decoherence

One of the greatest challenges involved in 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, the lattice vibrations, and the 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 temperatures.[110] Currently, some quantum computers require their qubits to be cooled to 20 millikelvin (usually using a dilution refrigerator[111]) in order to prevent significant decoherence.[112] A 2020 study argues that ionizing radiation such as cosmic rays can nevertheless cause certain systems to decohere within milliseconds.[113]

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.[114]

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 L2, where L is the number of binary 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 104 bits without error correction.[115] With error correction, the figure would rise to about 107 bits. Computation time is about L2 or about 107 steps and at 1 MHz, about 10 seconds. However, the encoding and error-correction overheads increase the size of a real fault-tolerant quantum computer by several orders of magnitude. Careful estimates[116][117] 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[118] for practically useful integer factorization problem sizing 1,024-bit or larger.

One approach to overcoming errors combines low-density parity-check code with cat qubits that have intrinsic bit-flip error suppression. Implementing 100 logical qubits with 768 cat qubits could reduce the error rate to one part in 108 per cycle per bit.[119]

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.[120][121] Non-Abelian anyons can, in effect, remember how they have been manipulated, making them potentially useful in quantum computing.[122] As of 2025, Microsoft and other organizations are investing in quasi-particle research.[122]

### 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.[123][51][124] The problem need not be useful, so some view the quantum supremacy test only as a potential future benchmark.[125]

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.[29][126][127] This claim has been subsequently challenged: IBM has stated that Summit can perform samples much faster than claimed,[128][129] 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[130][131][132] and even beating it.[133][134][135]

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.[136][137][138] 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.[139]

Claims of quantum supremacy have generated hype around quantum computing,[140] but they are based on contrived benchmark tasks that do not directly imply useful real-world applications.[102][141] Accordingly, benchmark-level quantum advantage should not be interpreted as proof that quantum computers are already broadly useful across practical computing workloads.

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.[142]

### Skepticism

Despite high hopes for quantum computing, significant progress in hardware, and optimism about future applications, a 2023 Nature spotlight article summarized current quantum computers as being "For now, [good for] absolutely nothing".[102] 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[103] 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.[143][144] In particular, processing large amounts of non-quantum data is a challenge for quantum computers.[103]
* 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 the speedup offered by many quantum algorithms.[103]
* 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 a sufficiently high degree of entanglement for a 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 asymptotically improve upon those bounds.

Bill Unruh doubted the practicality of quantum computers in a paper published in 1994.[145] Paul Davies argued that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the holographic principle.[146] Skeptics like Gil Kalai doubt that quantum supremacy will ever be achieved.[147][148][149] 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 10300... Could we ever learn to control the more than 10300 continuously variable parameters defining the quantum state of such a system? My answer is simple. No, never."[150]

### Physical realizations

Quantum System One, a quantum computer by IBM from 2019 with 20 superconducting qubits[151]

A practical quantum computer must use a physical system as a programmable quantum register.[152] Researchers are exploring several technologies as candidates for reliable qubit implementations.[153] Superconductors and trapped ions are some of the most developed proposals, but experimentalists are considering other hardware possibilities as well.[154] For example, topological quantum computer approaches are being explored for more fault-tolerance computing systems.[155]

The first quantum logic gates were implemented with trapped ions and prototype general-purpose machines with up to 20 qubits have been realized. However, the technology behind these devices combines complex vacuum equipment, lasers, and microwave and radio frequency equipment, making full-scale processors difficult to integrate with standard computing equipment. Moreover, the trapped ion system itself has engineering challenges to overcome.[156]

The largest commercial systems are based on superconductor devices and have scaled to 2000 qubits. However, the error rates for larger machines have been on the order of 5%. Technologically, these devices are all cryogenic and scaling to large numbers of qubits requires wafer-scale integration, a serious engineering challenge by itself.[157]

In addition to cryogenic platforms, room-temperature approaches to spin–photon interfaces have been experimentally demonstrated. In 2025, researchers at Stanford University realized a nanoscale device in which a thin layer of molybdenum diselenide is integrated on a nanostructured silicon substrate, enabling a spin–photon interface that operates at ambient conditions using structured "twisted" light to couple electronic and photonic degrees of freedom.[158][159] Such room-temperature, chip-integrated spin–photon interfaces are being investigated as potential building blocks for heterogeneous quantum networks that combine different qubit modalities and reduce reliance on large cryogenic infrastructures.[158][160]

## Theory

### Computability

Any computational problem solvable by a classical computer is also solvable by a quantum computer.[161] 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.[162]

### Complexity

Main article: Quantum complexity theory

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.[163] It is known that BPP⊆BQP but there is no proof BQP≠BPP, which intuitively would mean that quantum computers are more powerful than classical computers in terms of time complexity.[164]

The suspected relationship of BQP to several classical complexity classes[63]

The exact relationship of BQP to P, NP, and PSPACE is not known. However, it is known that P⊆BQP⊆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 that there exist 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⊈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).[165]

## List of quantum computers

* Hanyuan-1 — 100-qubit neutral atom quantum computer from the Chinese Academy of Sciences in China.[166]
* IBM Quantum System One — IBM superconducting quantum-computing system introduced in 2019.[167]
* IBM Quantum System Two — modular superconducting system using IBM Heron processors.
* Jiuzhang — photonic quantum-computing prototype for Gaussian boson sampling.[168]
* QpiAI-Indus — 25-qubit superconducting quantum computer from QpiAI in India.[169]

### Types of quantum computers

* Cat qubit quantum computer — proposed approach based on cat-state qubits.
* Kane quantum computer — proposed silicon-based nuclear spin quantum-computer architecture.
* Linear optical quantum computing — photonic model using photons and linear optical elements.
* Neutral atom quantum computer — approach using neutral atoms trapped and controlled with optical techniques.
* Nuclear magnetic resonance quantum computer — approach using nuclear magnetic resonance and molecular nuclear-spin states.
* Spin qubit quantum computer — semiconductor architecture using spin states as qubits.
* Superconducting quantum computing — approach using superconducting electronic circuits.
* Topological quantum computer — proposed approach using topological states such as anyons.
* Trapped-ion quantum computer — approach using trapped charged atoms as qubits.

## See also

* Company:D-Wave Systems
* Engineering:Electronic quantum holography - Information storage technology
* Glossary of quantum computing - None
* Intelligence Advanced Research Projects Activity - American government agency
* QpiAI-Indus
* Company:IonQ
* Social:List of emerging technologies - New technologies actively in development
* List of quantum computing journals
* List of quantum computing books
* List of quantum processors
* Magic state distillation - Quantum computing algorithm
* Metacomputing - Computing for the purpose of computing
* Natural computing - Academic field
* Non-local quantum computation - Method of quantum computing via entanglement
* Optical computing - Computer that uses photons or light waves
* Quantum bus - Device to store or transfer information in quantum computing
* Quantum cognition
* Quantum sensor - Device measuring quantum mechanical effects
* Quantum volume - Metric for a quantum computer's capabilities
* Physics:Quantum weirdness - Unintuitive aspects of quantum mechanics
* Company:Rigetti Computing
* Supercomputer
* Theoretical computer science - Subfield of computer science and mathematics
* Unconventional computing - Computing by new or unusual methods
* Physics:Valleytronics - Experimental area in semiconductors

## Notes

1. ↑ The standard basis is also the computational basis.[39]

## References

1. ↑ Mermin 2007, p. 1.
2. ↑ "Quantum Computing Just Hit a Milestone That Experts Said Was a Decade Away — and the Race Is Only Getting Faster" (in en). 2026-05-20. https://thefirmo.com/quantum-computing-just-hit-a-milestone-that-experts-said-was-a-decade-away-and-the-race-is-only-getting-faster/.
3. ↑ Aaronson 2013, p. 132.
4. ↑ Zwiebach, Barton (2022). Mastering Quantum Mechanics: Essentials, Theory, and Applications. MIT Press. §1. ISBN 978-0-262-04613-8. "Quantum physics has replaced classical physics as the correct fundamental description of our physical universe. It is used routinely to describe most phenomena that occur at short distances. [...] The era of quantum physics began in earnest in 1925 with the discoveries of Erwin Schrödinger and Werner Heisenberg. The seeds for these discoveries were planted by Max Planck, Albert Einstein, Niels Bohr, Louis de Broglie, and others."
5. ↑ Weinberg, Steven (2015). "Historical Introduction". Lectures on Quantum Mechanics (2nd ed.). Cambridge University Press. pp. 1–30. ISBN 978-1-107-11166-0.
6. ↑ Ceruzzi, Paul E. (2012) (in en-US). Computing: A Concise History. Cambridge, Massachusetts: MIT Press. pp. 3, 46. ISBN 978-0-262-31038-3. OCLC 796812982.
7. ↑ Hodges, Andrew (2014) (in en-US). Alan Turing: The Enigma. Princeton, New Jersey: Princeton University Press. p. xviii. ISBN 978-0-691-16472-4.
8. ↑ Mårtensson-Pendrill, Ann-Marie (2006-11-01). "The Manhattan project—a part of physics history" (in en-US). Physics Education 41 (6): 493–501. doi:10.1088/0031-9120/41/6/001. ISSN 0031-9120. Bibcode: 2006PhyEd..41..493M.
9. ↑ 9.0 9.1 Benioff, Paul (1980). "The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines". Journal of Statistical Physics 22 (5): 563–591. doi:10.1007/bf01011339. Bibcode: 1980JSP....22..563B.
10. ↑ Buluta, Iulia; Nori, Franco (2009-10-02). "Quantum Simulators" (in en). Science 326 (5949): 108–111. doi:10.1126/science.1177838. ISSN 0036-8075. PMID 19797653. Bibcode: 2009Sci...326..108B.
11. ↑ Manin, Yu. I. (1980) (in ru). Vychislimoe i nevychislimoe. Soviet Radio. pp. 13–15. http://publ.lib.ru/ARCHIVES/M/MANIN_Yuriy_Ivanovich/Manin_Yu.I._Vychislimoe_i_nevychislimoe.(1980).[djv-fax].zip. Retrieved 4 March 2013.
12. ↑ Feynman, Richard (June 1982). "Simulating Physics with Computers". International Journal of Theoretical Physics 21 (6/7): 467–488. doi:10.1007/BF02650179. Bibcode: 1982IJTP...21..467F. https://people.eecs.berkeley.edu/~christos/classics/Feynman.pdf. Retrieved 28 February 2019.
13. ↑ Nielsen & Chuang 2010, p. 214.
14. ↑ 14.0 14.1 Bennett, C. H.; Brassard, G. (1984). "Quantum cryptography: Public key distribution and coin tossing". Proceedings of the International Conference on Computers, Systems & Signal Processing, Bangalore, India. 1. New York: IEEE. pp. 175–179. Reprinted as Bennett, C. H.; Brassard, G. (4 December 2014). "Quantum cryptography: Public key distribution and coin tossing". Theoretical Computer Science. Theoretical Aspects of Quantum Cryptography – celebrating 30 years of BB84 560 (1): 7–11. doi:10.1016/j.tcs.2014.05.025. Bibcode: 2014TComS.560....7B.
15. ↑ Brassard, G. (2005). "Brief history of quantum cryptography: A personal perspective". IEEE Information Theory Workshop on Theory and Practice in Information-Theoretic Security, 2005. Awaji Island, Japan: IEEE. pp. 19–23. doi:10.1109/ITWTPI.2005.1543949. ISBN 978-0-7803-9491-9.
16. ↑ Deutsch, D. (1985-07-08). "Quantum theory, the Church–Turing principle and the universal quantum computer" (in en). Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 400 (1818): 97–117. doi:10.1098/rspa.1985.0070. ISSN 0080-4630. Bibcode: 1985RSPSA.400...97D.
17. ↑ Bernstein, Ethan; Vazirani, Umesh (1993). "Quantum complexity theory" (in en). Proceedings of the twenty-fifth annual ACM symposium on Theory of computing – STOC '93. San Diego, California, United States: ACM Press. pp. 11–20. doi:10.1145/167088.167097. ISBN 978-0-89791-591-5. http://portal.acm.org/citation.cfm?doid=167088.167097.
18. ↑ Simon, D. R. (1994). "On the power of quantum computation". Proceedings 35th Annual Symposium on Foundations of Computer Science. Santa Fe, New Mexico, USA: IEEE Comput. Soc. Press. pp. 116–123. doi:10.1109/SFCS.1994.365701. ISBN 978-0-8186-6580-6.
19. ↑ Nielsen & Chuang 2010, p. 30-32.
20. ↑ Shor, Peter W. (1994). "Algorithms for Quantum Computation: Discrete Logarithms and Factoring". Symposium on Foundations of Computer Science. Santa Fe, New Mexico: IEEE. pp. 124–134. doi:10.1109/SFCS.1994.365700. ISBN 978-0-8186-6580-6.
21. ↑ Grover, Lov K. (1996). "A fast quantum mechanical algorithm for database search" (in en). ACM symposium on Theory of computing. Philadelphia: ACM Press. pp. 212–219. doi:10.1145/237814.237866. ISBN 978-0-89791-785-8.
22. ↑ 22.0 22.1 Nielsen & Chuang 2010, p. 7.
23. ↑ 23.0 23.1 Lloyd, Seth (1996-08-23). "Universal Quantum Simulators". Science 273 (5278): 1073–1078. doi:10.1126/science.273.5278.1073. ISSN 0036-8075. PMID 8688088. Bibcode: 1996Sci...273.1073L.
24. ↑ Cao, Yudong; Romero, Jonathan; Olson, Jonathan P.; Degroote, Matthias; Johnson, Peter D. et al. (2019-10-09). "Quantum Chemistry in the Age of Quantum Computing" (in en-US). Chemical Reviews 119 (19): 10856–10915. doi:10.1021/acs.chemrev.8b00803. ISSN 0009-2665. PMID 31469277. Bibcode: 2019ChRv..11910856C.
25. ↑ 25.0 25.1 Grumbling & Horowitz 2019, pp. 164–169.
26. ↑ Chuang, Isaac L.; Gershenfeld, Neil; Kubinec, Markdoi (April 1998). "Experimental Implementation of Fast Quantum Searching". Physical Review Letters (American Physical Society) 80 (15): 3408–3411. doi:10.1103/PhysRevLett.80.3408. Bibcode: 1998PhRvL..80.3408C.
27. ↑ Holton, William Coffeen. "quantum computer". Encyclopedia Britannica (Encyclopædia Britannica). https://www.britannica.com/technology/quantum-computer.
28. ↑ Gibney, Elizabeth (2019-10-23). "Hello quantum world! Google publishes landmark quantum supremacy claim" (in en). Nature 574 (7779): 461–462. doi:10.1038/d41586-019-03213-z. PMID 31645740. Bibcode: 2019Natur.574..461G.
29. ↑ 29.0 29.1 Lay summary: Martinis, John; Boixo, Sergio (October 23, 2019). "Quantum Supremacy Using a Programmable Superconducting Processor". Nature (Google AI) 574 (7779): 505–510. doi:10.1038/s41586-019-1666-5. PMID 31645734. Bibcode: 2019Natur.574..505A. https://ai.googleblog.com/2019/10/quantum-supremacy-using-programmable.html. Retrieved 2022-04-27.

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Lao pronunciation: [{{{1}}}]Tibetan pronunciation: [{{{1}}}]Khmer pronunciation: [{{{1}}}]data-sort-value="" style="vertical-align:middle; text-align:center" class="table-na" | —`{{{1}}}`Welsh pronunciation: [{{{1}}}] Partially doneAthabaskan pronunciation: [{{{1}}}]Māori pronunciation: [{{{1}}}]IPA: [{{{1}}}]Bulgarian pronunciation: [{{{1}}}]Korean pronunciation: [{{{1}}}]Icelandic pronunciation: ​[{{{1}}}]Sanskrit pronunciation: [{{{1}}}]Bengali pronunciation: [{{{1}}}]Indonesian pronunciation: [{{{1}}}]Serbo-Croatian pronunciation: [{{{1}}}]CROSBI {{{1}}}Journal article: Arute, Frank; Arya, Kunal; Babbush, Ryan; Bacon, Dave; Bardin, Joseph C. et al. (October 23, 2019). "Quantum supremacy using a programmable superconducting processor". Nature 574 (7779): 505–510. doi:10.1038/s41586-019-1666-5. PMID 31645734. Bibcode: 2019Natur.574..505A.
30. ↑ Aaronson, Scott (2019-10-30). "Opinion | Why Google's Quantum Supremacy Milestone Matters". The New York Times. ISSN 0362-4331. https://www.nytimes.com/2019/10/30/opinion/google-quantum-computer-sycamore.html.
31. ↑ Pan, Feng; Zhang, Pan (2021-03-04). "Simulating the Sycamore quantum supremacy circuits". arXiv:2103.03074 [quant-ph].
32. ↑ Sample, Ian (2019-10-23). "Google claims it has achieved 'quantum supremacy' – but IBM disagrees" (in en-GB). The Guardian. ISSN 0261-3077. https://www.theguardian.com/technology/2019/oct/23/google-claims-it-has-achieved-quantum-supremacy-but-ibm-disagrees.
33. ↑ Bravyi (2024). "High-threshold and low-overhead fault-tolerant quantum memory". Nature 627 (8005): 778–782. doi:10.1038/s41586-024-07107-7. PMID 38538939. Bibcode: 2024Natur.627..778B.
34. ↑ Fredkin, Edward; Toffoli, Tommaso (1982). "Conservative logic". International Journal of Theoretical Physics 21 (3–4): 219–253.
35. ↑ Arora, Sanjeev; Barak, Boaz (2009). Computational Complexity: A Modern Approach. Cambridge University Press. pp. 123–125.
36. ↑ Bennett, Charlie (2020-07-31). Information Is Quantum: How Physics Helped Explain the Nature of Information and What Can Be Done With It (Videotape). Event occurs at 1:08:22 - via YouTube.
37. ↑ Nielsen & Chuang 2010, p. 13.
38. ↑ 38.0 38.1 Mermin 2007, p. 17.
39. ↑ 39.0 39.1 Mermin 2007, p. 18.
40. ↑ Aaronson 2013, p. 110.
41. ↑ Nielsen & Chuang 2010, p. 30–32.
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43. ↑ Kurgalin, Sergei; Borzunov, Sergei (2021). Concise guide to quantum computing: algorithms, exercises, and implementations. Texts in computer science. Cham: Springer. ISBN 978-3-030-65054-4.
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45. ↑ Nayak, Chetan; Simon, Steven; Stern, Ady; Das Sarma, Sankar (2008). "Nonabelian Anyons and Quantum Computation". Reviews of Modern Physics 80 (3): 1083–1159. doi:10.1103/RevModPhys.80.1083. Bibcode: 2008RvMP...80.1083N.
46. ↑ 46.0 46.1 Chi-Chih Yao, A. (1993). "Quantum circuit complexity". Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science. pp. 352–361. doi:10.1109/SFCS.1993.366852. ISBN 0-8186-4370-6.
47. ↑ Raussendorf, Robert; Browne, Daniel E.; Briegel, Hans J. (2003-08-25). "Measurement-based quantum computation on cluster states". Physical Review A 68 (2). doi:10.1103/PhysRevA.68.022312. Bibcode: 2003PhRvA..68b2312R.
48. ↑ Aharonov, Dorit; van Dam, Wim; Kempe, Julia; Landau, Zeph; Lloyd, Seth; Regev, Oded (2008-01-01). "Adiabatic Quantum Computation Is Equivalent to Standard Quantum Computation". SIAM Review 50 (4): 755–787. doi:10.1137/080734479. ISSN 0036-1445. Bibcode: 2008SIAMR..50..755A.
49. ↑ Freedman, Michael H.; Larsen, Michael; Wang, Zhenghan (2002-06-01). "A Modular Functor Which is Universal for Quantum Computation". Communications in Mathematical Physics 227 (3): 605–622. doi:10.1007/s002200200645. ISSN 0010-3616. Bibcode: 2002CMaPh.227..605F.
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54. ↑ Choudhury, Rizwan (2023-12-30). "Top 7 innovation stories of 2023 – Interesting Engineering" (in en-US). https://interestingengineering.com/lists/top-7-innovation-stories-of-2023-interesting-engineering.
55. ↑ 55.0 55.1 Gisin, Nicolas; Ribordy, Grégoire; Tittel, Wolfgang; Zbinden, Hugo (2002-03-08). "Quantum cryptography" (in en). Reviews of Modern Physics 74 (1): 145–195. doi:10.1103/RevModPhys.74.145. ISSN 0034-6861. Bibcode: 2002RvMP...74..145G. https://link.aps.org/doi/10.1103/RevModPhys.74.145.
56. ↑ 56.0 56.1 Pirandola, S.; Andersen, U. L.; Banchi, L.; Berta, M.; Bunandar, D.; Colbeck, R.; Englund, D.; Gehring, T. et al. (2020). "Advances in quantum cryptography". Advances in Optics and Photonics 12 (4): 1012. doi:10.1364/AOP.361502. Bibcode: 2020AdOP...12.1012P.
57. ↑ Xu, Feihu; Ma, Xiongfeng; Zhang, Qiang; Lo, Hoi-Kwong; Pan, Jian-Wei (2020-05-26). "Secure quantum key distribution with realistic devices". Reviews of Modern Physics 92 (2): 025002-3. doi:10.1103/RevModPhys.92.025002. Bibcode: 2020RvMP...92b5002X.
58. ↑ Xu, Guobin; Mao, Jianzhou; Sakk, Eric; Wang, Shuangbao Paul (2023-03-22). "2023 57th Annual Conference on Information Sciences and Systems (CISS)". IEEE. p. 3. doi:10.1109/CISS56502.2023.10089619. ISBN 978-1-6654-5181-9.
59. ↑ Kozlowski, Wojciech; Wehner, Stephanie (2019-09-25). "Proceedings of the Sixth Annual ACM International Conference on Nanoscale Computing and Communication" (in en). ACM. pp. 1–7. doi:10.1145/3345312.3345497. ISBN 978-1-4503-6897-1.
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62. ↑ Aaronson, Scott; Arkhipov, Alex (2011-06-06). "The computational complexity of linear optics" (in en). San Jose, California: Association for Computing Machinery. pp. 333–342. doi:10.1145/1993636.1993682. ISBN 978-1-4503-0691-1.
63. ↑ 63.0 63.1 Nielsen & Chuang 2010, p. 42.
64. ↑ Preskill 2018.
65. ↑ Norton, Quinn (15 February 2007). "The Father of Quantum Computing". Wired. http://archive.wired.com/science/discoveries/news/2007/02/72734.
66. ↑ Weidman, Jared D.; Sajjan, Manas; Mikolas, Camille; Stewart, Zachary J.; Pollanen, Johannes; Kais, Sabre; Wilson, Angela K. (2024-09-18). "Quantum computing and chemistry". Cell Reports Physical Science 5 (9). doi:10.1016/j.xcrp.2024.102105. Bibcode: 2024CRPS....502105W.
67. ↑ Ambainis, Andris (Spring 2014). "What Can We Do with a Quantum Computer?". Institute for Advanced Study. http://www.ias.edu/ias-letter/ambainis-quantum-computing.
68. ↑ Chang, Kenneth (14 June 2023). "Quantum Computing Advance Begins New Era, IBM Says – A quantum computer came up with better answers to a physics problem than a conventional supercomputer.". The New York Times. https://www.nytimes.com/2023/06/14/science/ibm-quantum-computing.html.
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71. ↑ Ruane, Jonathan; McAfee, Andrew; Oliver, William D. (2022-01-01). "Quantum Computing for Business Leaders". Harvard Business Review. ISSN 0017-8012. https://hbr.org/2022/01/quantum-computing-for-business-leaders.
72. ↑ Budde, Florian; Volz, Daniel (July 12, 2019). "Quantum computing and the chemical industry | McKinsey". McKinsey and Company. https://www.mckinsey.com/industries/chemicals/our-insights/the-next-big-thing-quantum-computings-potential-impact-on-chemicals.
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75. ↑ Nielsen & Chuang 2010, p. 216.
76. ↑ 76.0 76.1 Bernstein, Daniel J. (2009). "Introduction to post-quantum cryptography". Post-Quantum Cryptography. Berlin, Heidelberg: Springer. pp. 1–14. doi:10.1007/978-3-540-88702-7_1. ISBN 978-3-540-88701-0.
77. ↑ See also pqcrypto.org, a bibliography maintained by Daniel J. Bernstein and Tanja Lange on cryptography not known to be broken by quantum computing.
78. ↑ McEliece, R. J. (January 1978). "A Public-Key Cryptosystem Based On Algebraic Coding Theory". DSNPR 44: 114–116. Bibcode: 1978DSNPR..44..114M. http://ipnpr.jpl.nasa.gov/progress_report2/42-44/44N.PDF.
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124. ↑ Boixo, Sergio; Isakov, Sergei V.; Smelyanskiy, Vadim N.; Babbush, Ryan; Ding, Nan et al. (2018). "Characterizing Quantum Supremacy in Near-Term Devices". Nature Physics 14 (6): 595–600. doi:10.1038/s41567-018-0124-x. Bibcode: 2018NatPh..14..595B.
125. ↑ Savage, Neil (5 July 2017). "Quantum Computers Compete for "Supremacy"". Scientific American. https://www.scientificamerican.com/article/quantum-computers-compete-for-supremacy/.
126. ↑ Giles, Martin (September 20, 2019). "Google researchers have reportedly achieved 'quantum supremacy'" (in en). https://www.technologyreview.com/f/614416/google-researchers-have-reportedly-achieved-quantum-supremacy/.
127. ↑ Tavares, Frank (2019-10-23). "Google and NASA Achieve Quantum Supremacy" (in en-US). http://www.nasa.gov/feature/ames/quantum-supremacy.
128. ↑ Pednault, Edwin; Gunnels, John A.; Nannicini, Giacomo; Horesh, Lior; Wisnieff, Robert (2019-10-22). "Leveraging Secondary Storage to Simulate Deep 54-qubit Sycamore Circuits". arXiv:1910.09534 [quant-ph].
129. ↑ Cho, Adrian (2019-10-23). "IBM casts doubt on Google's claims of quantum supremacy". Science. doi:10.1126/science.aaz6080. ISSN 0036-8075. https://www.science.org/content/article/ibm-casts-doubt-googles-claims-quantum-supremacy.
130. ↑ Liu, Yong (Alexander); Liu, Xin (Lucy); Li, Fang (Nancy); Fu, Haohuan; Yang, Yuling et al. (2021-11-14). "Closing the "quantum supremacy" gap". Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis. SC '21. New York, New York: Association for Computing Machinery. pp. 1–12. doi:10.1145/3458817.3487399. ISBN 978-1-4503-8442-1.
131. ↑ Bulmer, Jacob F. F.; Bell, Bryn A.; Chadwick, Rachel S.; Jones, Alex E.; Moise, Diana et al. (2022-01-28). "The boundary for quantum advantage in Gaussian boson sampling" (in en). Science Advances 8 (4). doi:10.1126/sciadv.abl9236. ISSN 2375-2548. PMID 35080972. Bibcode: 2022SciA....8.9236B.
132. ↑ McCormick, Katie (2022-02-10). "Race Not Over Between Classical and Quantum Computers" (in en). Physics 15. doi:10.1103/Physics.15.19. Bibcode: 2022PhyOJ..15...19M. https://physics.aps.org/articles/v15/19.
133. ↑ Pan, Feng; Chen, Keyang; Zhang, Pan (2022). "Solving the Sampling Problem of the Sycamore Quantum Circuits". Physical Review Letters 129 (9). doi:10.1103/PhysRevLett.129.090502. PMID 36083655. Bibcode: 2022PhRvL.129i0502P.
134. ↑ Cho, Adrian (2022-08-02). "Ordinary computers can beat Google's quantum computer after all". Science 377. doi:10.1126/science.ade2364. https://www.science.org/content/article/ordinary-computers-can-beat-google-s-quantum-computer-after-all.
135. ↑ "Google's 'quantum supremacy' usurped by researchers using ordinary supercomputer" (in en-US). 5 August 2022. https://techcrunch.com/2022/08/05/googles-quantum-supremacy-usurped-by-researchers-using-ordinary-supercomputer/.
136. ↑ Ball, Philip (2020-12-03). "Physicists in China challenge Google's 'quantum advantage'" (in en). Nature 588 (7838): 380. doi:10.1038/d41586-020-03434-7. PMID 33273711. Bibcode: 2020Natur.588..380B.
137. ↑ Garisto, Daniel. "Light-based Quantum Computer Exceeds Fastest Classical Supercomputers" (in en). https://www.scientificamerican.com/article/light-based-quantum-computer-exceeds-fastest-classical-supercomputers/.
138. ↑ Conover, Emily (2020-12-03). "The new light-based quantum computer Jiuzhang has achieved quantum supremacy" (in en-US). https://www.sciencenews.org/article/new-light-based-quantum-computer-jiuzhang-supremacy.
139. ↑ Zhong, Han-Sen; Wang, Hui; Deng, Yu-Hao; Chen, Ming-Cheng; Peng, Li-Chao et al. (2020-12-03). "Quantum computational advantage using photons" (in en). Science 370 (6523): 1460–1463. doi:10.1126/science.abe8770. ISSN 0036-8075. PMID 33273064. Bibcode: 2020Sci...370.1460Z.
140. ↑ Roberson, Tara M. (2020-05-21). "Can Hype Be a Force for Good?" (in en). Public Understanding of Science 29 (5): 544–552. doi:10.1177/0963662520923109. ISSN 0963-6625. PMID 32438851.
141. ↑ Cavaliere, Fabio; Mattsson, John; Smeets, Ben (September 2020). "The security implications of quantum cryptography and quantum computing" (in en). Network Security 2020 (9): 9–15. doi:10.1016/S1353-4858(20)30105-7. ISSN 1353-4858. http://www.magonlinelibrary.com/doi/10.1016/S1353-4858(20)30105-7.
142. ↑ Liu, Yong; Chen, Yaojian; Guo, Chu; Song, Jiawei; Shi, Xinmin; Gan, Lin; Wu, Wenzhao; Wu, Wei et al. (2024-01-16). "Verifying Quantum Advantage Experiments with Multiple Amplitude Tensor Network Contraction" (in en). Physical Review Letters 132 (3). doi:10.1103/PhysRevLett.132.030601. ISSN 0031-9007. PMID 38307065. Bibcode: 2024PhRvL.132c0601L. https://link.aps.org/doi/10.1103/PhysRevLett.132.030601.
143. ↑ Monroe, Don (December 2022). "Quantum Computers and the Universe". Communications of the ACM. https://m-cacm.acm.org/magazines/2022/12/266916-quantum-computers-and-the-universe/fulltext.
144. ↑ Swayne, Matt (June 20, 2023). "PsiQuantum Sees 700x Reduction in Computational Resource Requirements to Break Elliptic Curve Cryptography With a Fault Tolerant Quantum Computer". https://thequantuminsider.com/2023/06/20/psiquantum-sees-700x-reduction-in-computational-resource-requirements-to-break-elliptic-curve-cryptography-with-a-fault-tolerant-quantum-computer/.
145. ↑ Unruh, Bill (1995). "Maintaining coherence in Quantum Computers". Physical Review A 51 (2): 992–997. doi:10.1103/PhysRevA.51.992. PMID 9911677. Bibcode: 1995PhRvA..51..992U.
146. ↑ Davies, Paul (6 March 2007). "The implications of a holographic universe for quantum information science and the nature of physical law". arXiv:quant-ph/0703041.
147. ↑ Regan, K. W. (23 April 2016). "Quantum Supremacy and Complexity". https://rjlipton.wordpress.com/2016/04/22/quantum-supremacy-and-complexity/.
148. ↑ Kalai, Gil (May 2016). "The Quantum Computer Puzzle". Notices of the AMS 63 (5): 508–516. https://www.ams.org/journals/notices/201605/rnoti-p508.pdf.
149. ↑ Rinott, Yosef; Shoham, Tomer; Kalai, Gil (2021-07-13). "Statistical Aspects of the Quantum Supremacy Demonstration". arXiv:2008.05177 [quant-ph].
150. ↑ Dyakonov, Mikhail (15 November 2018). "The Case Against Quantum Computing". https://spectrum.ieee.org/the-case-against-quantum-computing.
151. ↑ Russell, John (January 10, 2019). "IBM Quantum Update: Q System One Launch, New Collaborators, and QC Center Plans" (in en-US). https://www.hpcwire.com/2019/01/10/ibm-quantum-update-q-system-one-launch-new-collaborators-and-qc-center-plans/.
152. ↑ Tacchino, Francesco; Chiesa, Alessandro; Carretta, Stefano; Gerace, Dario (2019-12-19). "Quantum Computers as Universal Quantum Simulators: State-of-the-Art and Perspectives" (in en). Advanced Quantum Technologies 3 (3). doi:10.1002/qute.201900052. ISSN 2511-9044. https://onlinelibrary.wiley.com/doi/10.1002/qute.201900052.
153. ↑ Grumbling & Horowitz 2019, p. 127.
154. ↑ Grumbling & Horowitz 2019, p. 114.
155. ↑ Nayak, Chetan; Simon, Steven H.; Stern, Ady; Freedman, Michael; Das Sarma, Sankar (2008-09-12). "Non-Abelian anyons and topological quantum computation". Reviews of Modern Physics 80 (3): 1083–1159. doi:10.1103/RevModPhys.80.1083. Bibcode: 2008RvMP...80.1083N. https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.80.1083.
156. ↑ Grumbling & Horowitz 2019, p. 119.
157. ↑ Grumbling & Horowitz 2019, p. 126.
158. ↑ 158.0 158.1 "Scientists achieve breakthrough on quantum signaling". Stanford University. 1 December 2025. https://news.stanford.edu/stories/2025/12/quantum-communication-room-temperature-breakthrough-research.
159. ↑ Pan, F.; Liu, F.; Heinz, T. F.; Dionne, J. A. (2025). "Room-temperature spin–photon interface in a molybdenum diselenide–silicon nanostructured device". Nature Communications.
160. ↑ "Room-Temperature Device Advances Quantum Communication". 2 December 2025. https://quantumzeitgeist.com/quantum-communication-molybdenum-diselenide-quantum-device/.
161. ↑ Nielsen & Chuang 2010, p. 29.
162. ↑ Nielsen & Chuang 2010, p. 126.
163. ↑ Nielsen & Chuang 2010, p. 41.
164. ↑ Nielsen & Chuang 2010, p. 201.
165. ↑ Bernstein, Ethan; Vazirani, Umesh (1997). "Quantum Complexity Theory". SIAM Journal on Computing 26 (5): 1411–1473. doi:10.1137/S0097539796300921. http://www.cs.berkeley.edu/~vazirani/bv.ps.
166. ↑ "Hanyuan No. 1 Becomes China's First Commercial Quantum Computer". The Quantum Insider. November 2, 2025. https://thequantuminsider.com/2025/11/02/chinese-report-neutral-atom-quantum-computer-enters-commercial-use/.
167. ↑ "IBM Unveils World's First Integrated Quantum Computing System for Commercial Use" (Press release). IBM. January 8, 2019. Retrieved May 21, 2026.
168. ↑ Zhong, Han-Sen (2020). "Quantum computational advantage using photons". Science 370 (6523): 1460–1463. doi:10.1126/science.abe8770.
169. ↑ "QpiAI Launches 25-Qubit Superconducting Quantum System in India". HPCwire. April 16, 2025. https://www.hpcwire.com/off-the-wire/qpiai-launches-25-qubit-superconducting-quantum-system-in-india/.

### Sources

* Aaronson, Scott (2013). Quantum Computing Since Democritus. Cambridge University Press. doi:10.1017/CBO9780511979309. ISBN 978-0-521-19956-8. OCLC 829706638.
* Quantum Computing: Progress and Prospects. Washington, DC: The National Academies Press. 2019. doi:10.17226/25196. ISBN 978-0-309-47970-7. OCLC 1091904777.
* Mermin, N. David (2007). Quantum Computer Science: An Introduction. doi:10.1017/CBO9780511813870. ISBN 978-0-511-34258-5. OCLC 422727925.
* Nielsen, Michael; Chuang, Isaac (2010). Quantum Computation and Quantum Information (10th anniversary ed.). doi:10.1017/CBO9780511976667. ISBN 978-0-511-99277-3. OCLC 700706156.

## Further reading

### Textbooks

* 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.
* Bernhardt, Chris (2019). Quantum Computing for Everyone. MIT Press. ISBN 978-0-262-35091-4. OCLC 1082867954.
* Quantum Software: Aspects of Theory and System Design. Springer Nature. 2024. doi:10.1007/978-3-031-64136-7. ISBN 978-3-031-64136-7. https://link.springer.com/book/10.1007/978-3-031-64136-7.
* 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.
* Quantum Computation and Information: From Theory to Experiment. Topics in Applied Physics. 102. 2006. 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. doi:10.1007/978-3-030-61601-4. ISBN 978-3-03-061601-4. OCLC 1244536372. https://link.springer.com/book/10.1007/978-3-030-61601-4.
* 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.
* Kurgalin, Sergei; Borzunov, Sergei (2021). Concise Guide to Quantum Computing: Algorithms, Exercises, and Implementations. Springer. doi:10.1007/978-3-030-65052-0. ISBN 978-3-030-65052-0.
* 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.
* Wong, Thomas (2022). Introduction to Classical and Quantum Computing. Rooted Grove. ISBN 979-8-9855931-0-5. OCLC 1308951401. https://www.thomaswong.net/#publications.
* Zeng, Bei; Chen, Xie; Zhou, Duan-Lu; Wen, Xiao-Gang (2019). Quantum Information Meets Quantum Matter. doi:10.1007/978-1-4939-9084-9. ISBN 978-1-4939-9084-9. OCLC 1091358969.

### 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. doi:10.1023/B:QINP.0000042203.24782.9a. Bibcode: 2003QuIP....2..449A.
* 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.
* DiVincenzo, David P. (2000). "The Physical Implementation of Quantum Computation". Fortschritte der Physik 48 (9–11): 771–783. doi:10.1002/1521-3978(200009)48:9/11<771::AID-PROP771>3.0.CO;2-E. Bibcode: 2000ForPh..48..771D.
* DiVincenzo, David P. (1995). "Quantum Computation". Science 270 (5234): 255–261. doi:10.1126/science.270.5234.255. Bibcode: 1995Sci...270..255D. Table 1 lists switching and dephasing times for various systems.
* 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. https://lup.lub.lu.se/record/e034e7b7-d17c-4863-9cee-7e654f97225b.
* Krantz, P.; Kjaergaard, M.; Yan, F.; Orlando, T. P.; Gustavsson, S.; Oliver, W. D. (2019-06-17). "A Quantum Engineer's Guide to Superconducting Qubits" (in en). Applied Physics Reviews 6 (2): 021318. doi:10.1063/1.5089550. ISSN 1931-9401. Bibcode: 2019ApPRv...6b1318K.
* Mitchell, Ian (1998). "Computing Power into the 21st Century: Moore's Law and Beyond". http://citeseer.ist.psu.edu/mitchell98computing.html.
* Simon, Daniel R. (1994). "On the Power of Quantum Computation". Institute of Electrical and Electronics Engineers Computer Society Press. http://citeseer.ist.psu.edu/simon94power.html.

## External links

* Stanford Encyclopedia of Philosophy: "Quantum Computing" by Amit Hagar and Michael E. Cuffaro
* Hazewinkel, Michiel, ed. (2001), "Quantum computation, theory of", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/q130020
* Introduction to Quantum Computing for Business by Koen Groenland
* Schneider, J., & Smalley, I. (2024, August 5). What Is Quantum Computing? | IBM. https://www.ibm.com/think/topics/quantum-computing

Lectures

* Quantum computing for the determined – 22 video lectures by Michael Nielsen
* Video Lectures by David Deutsch
* Lomonaco, Sam. Four Lectures on Quantum Computing given at Oxford University in July 2006

* v
* t
* e

Processor technologies

Models|

* Turing machine
* Universal
* Post–Turing
* Quantum
* Belt machine
* Stack machine
* Finite-state machine
* with datapath
* Hierarchical
* Queue automaton
* Register machines
* Counter
* Pointer
* Random-access
* Random-access stored program

Architecture|

* Microarchitecture
* Von Neumann
* Harvard
* modified
* Dataflow
* Transport-triggered
* Cellular
* Endianness
* Memory access
* NUMA
* HUMA
* Load/store
* Register/memory
* Cache hierarchy
* Memory hierarchy
* Virtual memory
* Secondary storage
* Heterogeneous
* Fabric
* Multiprocessing
* Cognitive
* Neuromorphic

Instruction set
architectures|

| Types|

* CISC
* RISC
* Application-specific
* EDGE
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* VLIW
* EPIC
* MISC
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* NISC
* ZISC
* comparison
* addressing modes

|

* x86
* ARM
* MIPS
* Power ISA
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* Itanium
* Unicore
* MicroBlaze
* RISC-V
* others

Execution|

| Instruction pipelining|

* Pipeline stall
* Operand forwarding
* Classic RISC pipeline

|

Hazards|

* Data dependency
* Structural
* Control
* False sharing

Out-of-order|

* Tomasulo algorithm
* Reservation station
* Re-order buffer
* Register renaming

Speculative|

* Branch prediction
* Memory dependence prediction

Parallelism|

| Level|

* Bit
* Bit-serial
* Word
* Instruction
* Pipelining
* Scalar
* Superscalar
* Task
* Thread
* Process
* Data
* Vector
* Memory
* Distributed

|

Multithreading|

* Temporal
* Simultaneous
* Hyperthreading
* Speculative
* Preemptive
* Cooperative

Flynn's taxonomy|

* SISD
* SIMD
* SWAR
* SIMT
* MISD
* MIMD
* SPMD

Processor
performance|

* Transistor count
* Instructions per cycle (IPC)
* Cycles per instruction (CPI)
* Instructions per second (IPS)
* Floating-point operations per second (FLOPS)
* Transactions per second (TPS)
* Synaptic updates per second (SUPS)
* Performance per watt (PPW)
* Cache performance metrics
* Computer performance by orders of magnitude

Types|

* Central processing unit (CPU)
* Graphics processing unit (GPU)
* GPGPU
* Vector
* Barrel
* Stream
* Coprocessor
* ASIC
* FPGA
* CPLD
* Multi-chip module (MCM)
* System in package (SiP)

| By application|

* Microprocessor
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* Mobile
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* Ultra-low-voltage
* ASIP

|

Systems
on chip|

* System on a chip (SoC)
* Multiprocessor (MPSoC)
* Programmable (PSoC)
* Network on a chip (NoC)

Hardware
accelerators|

* AI accelerator
* Vision processing unit (VPU)
* Physics processing unit (PPU)
* Digital signal processor (DSP)
* Tensor processing unit (TPU)
* Secure cryptoprocessor
* Network processor
* Baseband processor

Word size|

* 1-bit
* 2-bit
* 4-bit
* 8-bit
* 16-bit
* 32-bit
* 48-bit
* 64-bit
* 128-bit
* 256-bit
* 512-bit
* others
* variable

Core count|

* Single-core
* Multi-core
* Manycore
* Heterogeneous architecture

Components|

* Core
* Cache
* CPU cache
* replacement policies
* coherence
* Bus
* Clock rate
* Clock signal
* FIFO

| Functional units|

* Arithmetic logic unit (ALU)
* Address generation unit (AGU)
* Floating-point unit (FPU)
* Memory management unit (MMU)
* Load–store unit
* Translation lookaside buffer (TLB)
* Integrated memory controller (IMC)

|

Logic|

* Combinational
* Sequential
* Glue
* Logic gate
* Quantum
* Array

Registers|

* Processor register
* Status register
* Stack register
* Register file
* Memory buffer
* Program counter

Control unit|

* Instruction unit
* Data buffer
* Write buffer
* Microcode ROM
* Counter

Datapath|

* Multiplexer
* Demultiplexer
* Adder
* Multiplier
* CPU
* Binary decoder
* Address decoder
* Sum addressed decoder
* Barrel shifter

Circuitry|

* Integrated circuit
* 3D
* Mixed-signal
* Power management
* Boolean
* Digital
* Analog
* Quantum
* Switch

Power
management|

* PMU
* APM
* ACPI
* Dynamic frequency scaling
* Dynamic voltage scaling
* Clock gating
* Performance per watt (PPW)
* Race to sleep

Related|

* History of general-purpose CPUs
* Microprocessor chronology
* Processor design
* Digital electronics
* Hardware security module
* Semiconductor device fabrication
* Tick–tock model

* v
* t
* e

Quantum information science

General|

* Quantum computing
* Qubit
* physical vs. logical
* DiVincenzo's criteria
* Quantum information
* Quantum programming
* Quantum processors
* Cloud-based quantum computing
* Timeline of quantum computing

Theorems|

* Bell's theorem
* Gleason's theorem
* Gottesman–Knill theorem
* Holevo's theorem
* Margolus–Levitin theorem
* No-broadcast theorem
* No-cloning theorem
* No-communication theorem
* No-deleting theorem
* No-hiding theorem
* No-teleportation theorem
* PBR theorem
* Quantum threshold theorem

Quantum
communication|

* Quantum capacity
* Classical capacity
* Entanglement-assisted classical capacity
* Quantum channel
* Quantum network
* Quantum cryptography
* Quantum key distribution
* BB84
* SARG04
* Three-stage quantum cryptography protocol
* Quantum teleportation
* Superdense coding
* LOCC
* Entanglement distillation

Quantum algorithms|

* Universal quantum simulator
* Deutsch–Jozsa algorithm
* Grover's algorithm
* Quantum Fourier transform
* Shor's algorithm
* Simon's problem
* Quantum phase estimation algorithm
* Quantum counting algorithm
* Quantum annealing
* Quantum algorithm for linear systems of equations
* Amplitude amplification

Quantum
complexity theory|

* Quantum Turing machine
* EQP
* BQP
* QMA
* PostBQP
* QIP

Quantum
computing models|

* Quantum circuit
* Quantum logic gate
* One-way quantum computer
* cluster state
* Adiabatic quantum computation
* Topological quantum computer

Quantum
error correction|

* Stabilizer codes
* Entanglement-Assisted Quantum Error Correction
* Shor code
* Steane code
* CSS code
* Quantum convolutional codes
* Toric code

Physical
implementations|

| Quantum optics|

* Cavity QED
* Circuit QED
* Linear optical quantum computing
* KLM protocol
* Boson sampling

|

Ultracold atoms|

* Trapped ion quantum computer
* Optical lattice

Spin-based|

* Nuclear magnetic resonance QC
* Kane QC
* Loss–DiVincenzo QC
* Nitrogen-vacancy center

Superconducting
quantum computing|

* Charge qubit
* Flux qubit
* Phase qubit
* Transmon

Software|

* libquantum
* OpenQASM
* Q#
* Qiskit
* IBM Q Experience

* v
* t
* e

Emerging technologies

Fields|

| Agriculture|

* Agricultural robot
* Cellular agriculture
* Closed ecological systems
* Cultured meat
* Genetically modified food
* Precision agriculture
* Vertical farming

|

Architecture|

* Arcology
* Building printing
* Contour crafting
* Domed city

Biomedical|

* Artificial uterus
* Ampakine
* Brain transplant
* Cryonics
* Cryoprotectant
* Cryopreservation
* Vitrification
* Suspended animation
* De-extinction
* Genetic engineering
* Gene therapy
* Head transplant
* Isolated brain
* Life extension
* Strategies for Engineered Negligible Senescence
* Nanomedicine
* Nanosensors
* Organ printing
* Personalized medicine
* Regenerative medicine
* Stem-cell therapy
* Tissue engineering
* Robot-assisted surgery
* Synthetic biology
* Synthetic genomics
* Virotherapy
* Oncolytic virus
* Tricorder
* Whole genome sequencing

Displays|

| Next generation|

* FED
* FLCD
* iMoD
* Laser
* LPD
* OLED
* OLET
* QD-LED
* SED
* TPD
* TDEL
* TMOS
* MicroLED

|

Screenless|

* Bionic contact lens
* Head-mounted display
* Head-up display
* Optical head-mounted display
* Virtual retinal display

Other|

* Autostereoscopy
* Flexible display
* Holographic display
* Computer-generated holography
* Multi-primary color display
* Ultra HD
* Volumetric display

Electronics|

* Electronic nose
* E-textiles
* Flexible electronics
* Molecular electronics
* Nanoelectromechanical systems
* Memristor
* Spintronics
* Thermal copper pillar bump
* Twistronics

Energy|

| Production|

* Airborne wind turbine
* Artificial photosynthesis
* Biofuels
* Carbon-neutral fuel
* Concentrated solar power
* Fusion power
* Home fuel cell
* Hydrogen economy
* Methanol economy
* Molten salt reactor
* Nantenna
* Photovoltaic pavement
* Space-based solar power
* Vortex engine

|

Storage|

* Beltway battery
* Compressed air energy storage
* Flywheel energy storage
* Grid energy storage
* Lithium–air battery
* Molten-salt battery
* Nanowire battery
* Research in lithium-ion batteries
* Silicon–air battery
* Thermal energy storage
* Ultracapacitor

Other|

* Smart grid
* Wireless power

Information and
communications

* Ambient intelligence
* Internet of things
* Artificial intelligence
* Applications of artificial intelligence
* Progress in artificial intelligence
* Machine translation
* Mobile translation
* Machine vision
* Semantic Web
* Speech recognition
* Atomtronics
* Carbon nanotube field-effect transistor
* Cybermethodology
* Fourth-generation optical discs
* 3D optical data storage
* Holographic data storage
* GPGPU
* Memory
* CBRAM
* FRAM
* Millipede
* MRAM
* NRAM
* PRAM
* Racetrack memory
* RRAM
* SONOS
* Optical computing
* RFID
* Chipless RFID
* Software-defined radio
* Three-dimensional integrated circuit

Manufacturing|

* 3D printing
* Claytronics
* Molecular assembler
* Utility fog

Materials science|

* Aerogel
* Amorphous metal
* Artificial muscle
* Conductive polymer
* Femtotechnology
* Fullerene
* Graphene
* High-temperature superconductivity
* High-temperature superfluidity
* Linear acetylenic carbon
* Metamaterials
* Metamaterial cloaking
* Metal foam
* Multi-function structures
* Nanotechnology
* Carbon nanotubes
* Molecular nanotechnology
* Nanomaterials
* Picotechnology
* Programmable matter
* Quantum dots
* Silicene
* Superalloy
* Synthetic diamond

Military|

* Antimatter weapon
* Caseless ammunition
* Directed-energy weapon
* Laser
* Maser
* Particle-beam weapon
* Sonic weapon
* Coilgun
* Railgun
* Plasma weapon
* Pure fusion weapon
* Stealth technology
* Vortex ring gun

Neuroscience|

* Artificial brain
* Brain–computer interface
* Electroencephalography
* Mind uploading
* Brain-reading
* Neuroinformatics
* Neuroprosthetics
* Bionic eye
* Brain implant
* Exocortex
* Retinal implant

Quantum|

* Quantum algorithms
* Quantum amplifier
* Quantum bus
* Quantum channel
* Quantum circuit
* Quantum complexity theory
* Quantum computing
* Quantum cryptography
* Quantum dynamics
* Quantum electronics
* Quantum error correction
* Quantum imaging
* Quantum information
* Quantum key distribution
* Quantum logic
* Quantum logic gates
* Quantum machine
* Quantum machine learning
* Quantum metamaterial
* Quantum metrology
* Quantum network
* Quantum neural network
* Quantum optics
* Quantum programming
* Quantum sensing
* Quantum simulator
* Quantum teleportation

Robotics|

* Domotics
* Nanorobotics
* Powered exoskeleton
* Self-reconfiguring modular robot
* Swarm robotics
* Uncrewed vehicle

Space science|

| Launch|

* Fusion rocket
* Non-rocket spacelaunch
* Mass driver
* Orbital ring
* Skyhook
* Space elevator
* Space fountain
* Space tether
* Reusable launch system

|

Propulsion|

* Beam-powered propulsion
* Ion thruster
* Laser propulsion
* Plasma propulsion engine
* Helicon thruster
* VASIMR
* Nuclear pulse propulsion
* Solar sail

Other|

* Interstellar travel
* Propellant depot
* Laser communication in space

Transport|

| Aerial|

* Adaptive compliant wing
* Backpack helicopter
* Delivery drone
* Flying car
* High-altitude platform
* Jet pack
* Pulse detonation engine
* Scramjet
* Spaceplane
* Supersonic transport

|

Land|

* Airless tire
* Alternative fuel vehicle
* Hydrogen vehicle
* Driverless car
* Ground effect train
* Hyperloop
* Maglev train
* Personal rapid transit
* Transit Elevated Bus
* Vactrain
* Vehicular communication systems

Pipeline|

* Pneumatic transport
* Automated vacuum collection

Other|

* Anti-gravity
* Cloak of invisibility
* Digital scent technology
* Force field
* Plasma window
* Immersive virtual reality
* Magnetic refrigeration
* Phased-array optics

Topics|

* Collingridge dilemma
* Differential technological development
* Disruptive Innovation
* Ephemeralization
* Exploratory engineering
* Fictional technology
* Proactionary principle
* Technological change
* Technological unemployment
* Technological convergence
* Technological evolution
* Technological paradigm
* Technology forecasting
* Accelerating change
* Moore's law
* Technological singularity
* Technology scouting
* Technology readiness level
* Technology roadmap
* Transhumanism

* Category
* List

* v
* t
* e

Quantum mechanics

Background|

* Introduction
* History
* timeline
* Glossary
* Classical mechanics
* Old quantum theory

Fundamentals|

* Bra–ket notation
* Casimir effect
* Complementarity
* Density matrix
* Energy level
* ground state
* excited state
* degenerate levels
* Vacuum state
* Zero-point energy
* QED vacuum
* QCD vacuum
* Hamiltonian
* Operator
* Quantum coherence
* Quantum decoherence
* Measurement
* Quantum
* Quantum realm
* Quantum system
* Quantum state
* Quantum number
* Quantum entanglement
* Quantum superposition
* Quantum nonlocality
* Quantum tunnelling
* Quantum levitation
* Quantum fluctuation
* Quantum annealing
* Quantum foam
* Quantum noise
* Heisenberg uncertainty principle
* Spontaneous parametric down-conversion
* Von Neumann entropy
* Spin
* Scattering theory
* Symmetry in quantum mechanics
* Symmetry breaking
* Spontaneous symmetry breaking
* Wave propagation
* Quantum interference
* Wave function
* Wave function collapse
* Wave–particle duality
* Matter wave
* Qubit
* Qutrit
* Observable
* Probability distribution

Formulations|

* Formulations
* Heisenberg
* Interaction
* Matrix mechanics
* Schrödinger
* Path integral formulation
* Phase space

Equations|

* Dirac
* Klein–Gordon
* Pauli
* Rydberg
* Schrödinger

Interpretations|

* Interpretations
* Bayesian
* Consistent histories
* Copenhagen
* de Broglie–Bohm
* Ensemble
* Hidden variables
* Many worlds
* Objective collapse
* Quantum logic
* Relational
* Stochastic
* Transactional
* Cosmological

Experiments|

* Afshar
* Bell's inequality
* Cold Atom Laboratory
* Davisson–Germer
* Delayed-choice quantum eraser
* Double-slit
* Franck–Hertz experiment
* Leggett–Garg inequality
* Mach-Zehnder inter.
* Elitzur–Vaidman
* Popper
* Quantum eraser
* Schrödinger's cat
* Quantum suicide and immortality
* Stern–Gerlach
* Wheeler's delayed choice

Science|

* Q-analog
* List
* Quantum algebra
* Quantum calculus
* Quantum differential calculus
* Quantum geometry
* Quantum group
* Quantum Bayesianism
* Quantum biology
* Quantum chemistry
* Quantum chaos
* Quantum cognition
* Quantum cosmology
* Quantum dynamics
* Quantum economics
* Quantum evolution
* Quantum finance
* Quantum game theory
* Quantum measurement problem
* Quantum mind
* Quantum probability
* Quantum social science
* Quantum stochastic calculus
* Quantum spacetime

Technology|

* Quantum algorithms
* Quantum amplifier
* Quantum cellular automata
* Quantum finite automata
* Quantum electronics
* Quantum logic gates
* Quantum clock
* Quantum channel
* Quantum bus
* Quantum circuit
* Phase qubit
* Matrix isolation
* Quantum dot
* Quantum dot display
* Quantum dot solar cell
* Quantum dot cellular automaton
* Quantum dot single-photon source
* Quantum dot laser
* Quantum well
* Quantum computing
* Timeline
* Quantum cryptography
* Post-quantum cryptography
* Quantum error correction
* Quantum imaging
* Quantum image processing
* Quantum information
* Quantum key distribution
* Quantum machine
* Quantum machine learning
* Quantum metamaterial
* Quantum metrology
* Quantum network
* Quantum neural network
* Quantum optics
* Quantum programming
* Quantum sensors
* Quantum simulator
* Quantum teleportation
* Quantum levitation
* Time travel
* Quantum complexity theory

Extensions|

* Quantum statistical mechanics
* Relativistic quantum mechanics
* Quantum field theory
* Axiomatic quantum field theory
* Quantum field theory in curved spacetime
* Thermal quantum field theory
* Topological quantum field theory
* Local quantum field theory
* Conformal field theory
* Two-dimensional conformal field theory
* Liouville field theory
* History
* Quantum gravity

* Category
* Physics Portal
* Physics WikiProject
* Commons

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