Quantum computing

Quantum information

The bit is the basic concept of classical information theory. A bit is in one of two physical states, typically denoted 0 and 1.

The qubit is the unit of quantum information. A qubit is an abstract mathematical model that applies to any physical system that is represented by that model. When measured, a qubit takes one of two states ${\displaystyle |0\rangle }$ or ${\displaystyle |1\rangle }$. However, the quantum states ${\displaystyle |0\rangle }$ and ${\displaystyle |1\rangle }$ belong to a vector space, meaning that they can be multiplied by constants and added together, returning a valid quantum state. Such a combination is known as a superposition of ${\displaystyle |0\rangle }$ and ${\displaystyle |1\rangle }$.^[37][38]

A two-dimensional vector mathematically represents a qubit state. Physicists typically use bra–ket notation for quantum mechanical linear algebra, writing ${\displaystyle |\psi \rangle }$ 'ket psi' for a vector labeled ${\displaystyle \psi }$. Because a qubit is a two-state system, any qubit state takes the form ${\displaystyle \alpha |0\rangle +\beta |1\rangle }$, where ${\displaystyle |0\rangle }$ and ${\displaystyle |1\rangle }$ are the standard basis states,^[a] and ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are the probability amplitudes, which are in general complex numbers.^[38] If either ${\displaystyle \alpha }$ or ${\displaystyle \beta }$ 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 ${\displaystyle \alpha |0\rangle +\beta |1\rangle }$, the state collapses to ${\displaystyle |0\rangle }$ with probability ${\displaystyle |\alpha |^{2}}$, or to ${\displaystyle |1\rangle }$ with probability ${\displaystyle |\beta |^{2}}$. Any valid qubit state has coefficients ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ such that ${\displaystyle |\alpha |^{2}+|\beta |^{2}=1}$. As an example, measuring the qubit ${\displaystyle 1/{\sqrt {2}}|0\rangle +1/{\sqrt {2}}|1\rangle }$ would produce either ${\displaystyle |0\rangle }$ or ${\displaystyle |1\rangle }$ with equal probability.

Two particularly important superposition states are the plus state ${\displaystyle |+\rangle =1/{\sqrt {2}}|0\rangle +1/{\sqrt {2}}|1\rangle }$ and the minus state ${\displaystyle |-\rangle =1/{\sqrt {2}}|0\rangle -1/{\sqrt {2}}|1\rangle }$. 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 ${\displaystyle |0\rangle \leftrightarrow |+\rangle }$ and ${\displaystyle |1\rangle \leftrightarrow |-\rangle }$—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$${\displaystyle X:={\begin{pmatrix}0&1\\1&0\end{pmatrix}}.}$$Mathematically, the application of such a logic gate to a quantum state vector is modeled with matrix multiplication. Thus

${\displaystyle X|0\rangle =|1\rangle }$ and ${\displaystyle X|1\rangle =|0\rangle }$.

The mathematics of single-qubit gates can be extended to operate on multi-qubit quantum memories in two important ways. One way is 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:$${\displaystyle |00\rangle$$ :={\begin{pmatrix}1\\0\\0\\0\end{pmatrix}};\quad |01\rangle :={\begin{pmatrix}0\\1\\0\\0\end{pmatrix}};\quad |10\rangle :={\begin{pmatrix}0\\0\\1\\0\end{pmatrix}};\quad |11\rangle :={\begin{pmatrix}0\\0\\0\\1\end{pmatrix}}.} The controlled NOT (CNOT) gate can then be represented using the following matrix:$${\displaystyle \operatorname {CNOT}$$ :={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{pmatrix}}.} As a mathematical consequence of this definition, ${\textstyle \operatorname {CNOT} |00\rangle =|00\rangle }$, ${\textstyle \operatorname {CNOT} |01\rangle =|01\rangle }$, ${\textstyle \operatorname {CNOT} |10\rangle =|11\rangle }$, and ${\textstyle \operatorname {CNOT} |11\rangle =|10\rangle }$. In other words, the CNOT applies a NOT gate (${\textstyle X}$ from before) to the second qubit if and only if the first qubit is in the state ${\textstyle |1\rangle }$. If the first qubit is ${\textstyle |0\rangle }$, nothing is done to either qubit.

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 a 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 acceleration 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

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

Gate array

A quantum gate array decomposes computation into a sequence of few-qubit quantum gates. A quantum computation can be described as a network of quantum logic gates and measurements. 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 (in the above formalism, any unitary matrix of size ${\displaystyle 2^{n}\times 2^{n}}$ over ${\displaystyle n}$ qubits) can be represented as a network of quantum logic gates from a fairly small family of gates. A choice of gate family that enables this construction is known as a universal gate set, since a computer that can run such circuits is a universal quantum computer. One common such set includes all single-qubit gates as well as the CNOT gate. 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]

Quantum-computing paradigms

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.

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.^[48]

A topological quantum computer decomposes computation into the braiding of anyons in a 2D lattice.^[49]

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 out of reach as of 2026.^[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 created "quantum circuits" that correct errors more efficiently than alternative methods, which may remove a major obstacle to practical quantum computers.^[52] The Harvard research team was supported by MIT, QuEra Computing, Caltech, and Princeton and funded by DARPA's Optimization with Noisy Intermediate-Scale Quantum devices (ONISQ) program.^[53][54]

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.

Simulation of quantum systems

Since chemistry and nanotechnology rely on understanding quantum systems, and such systems are impossible to efficiently simulate classically, quantum simulation may be an important application.^[64] Quantum computational chemistry is promising for quantum computing, particularly for problems in electronic structure, chemical dynamics, and spectroscopy; useful implementations remain hardware-limited.^[65] Quantum simulation could be used to simulate the behavior of atoms and particles under unusual conditions such as the reactions inside a collider.^[66] In June 2023, IBM computer scientists reported that a quantum computer produced better results for a physics problem than a conventional supercomputer.^[67][68]

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. Quantum simulations might be used to understand this process and increase energy efficiency.^{[69][70][71][72]}

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 underlies 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.^[73]

Thus quantum computing can in theory be used to attack currently-used cryptographic systems. Integer factorization, which underpins the security of public key cryptographic systems, is believed to be computationally infeasible on a classical computer for large integers that 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 would allow a quantum computer to break many widely-used cryptographic systems, in the sense that a polynomial time (in the number of digits of the integer) algorithm could do so. In particular, most popular public key ciphers rely on the difficulty of factoring integers or the discrete logarithm. In particular, RSA, Diffie–Hellman, and elliptic curve Diffie–Hellman algorithms could be broken. These are used to secure Web pages, encrypted emails, and many other data. Breaking these would have significant ramifications for electronic privacy and security.

Identifying cryptographic systems that are 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 that Shor's algorithm cannot solve, such as the McEliece cryptosystem, which relies on a hard problem in coding theory.^[76][78] Lattice-based cryptosystems are not known to be susceptible to 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] Applying Grover's algorithm to break a symmetric (secret-key) algorithm by brute force requires roughly 2^n/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 such an attack to that of AES-128 against classical brute-force search.

Post-quantum algorithms are designed to run but be difficult to break on a classical computer. Quantum cryptography replaces conventional encryption algorithms with techniques based on quantum mechanics such as entanglement. In principle, quantum encryption cannot be decoded even by a quantum computer. This advantage comes at a significant infrastructure cost, while effectively preventing legitimate decoding of messages.^[73]

Search problems

The most well-known example of a problem that allows for a polynomial quantum speedup is unstructured search, which involves finding a marked item out of a list of ${\displaystyle n}$ items in a database. This can be solved by Grover's algorithm using ${\displaystyle O({\sqrt {n}})}$ queries to the database, quadratically fewer than the ${\displaystyle \Omega (n)}$ queries required for classical algorithms. In this case, the advantage is not only provable but also optimal: it has been shown that Grover's algorithm gives the maximal possible probability of finding the desired element for any number of oracle lookups. Many examples of provable 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]

• The collection of possible answers has no searchable structure
• The number of possible answers to check is the same as the number of inputs to the algorithm,
• A Boolean function exists 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, in which the algorithm iterates through 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

Quantum annealing uses the adiabatic theorem to perform calculations. A system is placed in the ground state for a simple Hamiltonian, which 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 stays in its ground state throughout the process. Quantum annealing can solve Ising models and the (computationally equivalent) quadratic unconstrained binary optimisation (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

Since quantum computers can produce outputs that classical computers cannot produce efficiently, and since quantum computation is fundamentally linear algebra, so quantum algorithms that can speed up machine learning tasks may be possible.^[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 translated into practical advantage as of 2024.^[90][91] For example, the HHL Algorithm is believed to provide speedup over classical counterparts.^[51][92] Research groups have explored 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. Near-term quantum hardware has been explored for molecular generative modeling for drug discovery. In 2023, researchers reported a hybrid quantum–classical generative model based on a restricted Boltzmann machine, implemented on a commercially available quantum annealing device, to generate novel small molecules with physicochemical properties comparable to medicinal compounds.^[96][97] However, the immense size and complexity of the structural space of all possible relevant 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 apply to applications involving quantum chemistry. Quantum-enhanced generative models^[98] including quantum generative adversarial networks (GANs)^[99] may be developed into generative chemistry algorithms.

AI-assisted algorithm discovery

Artificial intelligence has been explored as a tool for discovering and optimizing quantum algorithms. 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]

Challenges

Many technical challenges confront the 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 are easily read.

The control of multi-qubit systems requires the rapid generation and coordination of a large number of electrical signals with deterministic timing. This has led to the development of quantum controllers that enable interfacing with the qubits. Scaling these systems to support many qubits is an additional challenge.^[106]

The potential to break public-key encryption has motivated changes in global cybersecurity strategies. The National Institute of Standards and Technology (NIST) 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 attacks.^[107]

Modular and distributed architectures

One approach to the scalability problem is to distribute a computation across multiple smaller quantum processing modules instead of increasing the number of qubits in a single device. In such modular architectures — also referred to as distributed quantum computing (DQC) — each module contains a limited number of qubits, and the modules are interconnected through quantum channels (for example, optical fibres) and classical communication links, forming a single logical computing system.^[123]

In one strategy, the quantum logic between qubits in different modules is applied using quantum gate teleportation, using remote entanglement between the modules, but local operations and measurements within each module, and classical communication of measurement outcomes.^[124] Quantum algorithms distributed across a photonic network link between trapped-ion modules, as well as teleported two-qubit gates between remote solid-state qubit registers based on nitrogen-vacancy centers in diamond, were demonstrated in 2025.^[123][125]

High rate and high fidelity remote entanglement generation across the network is the core challenge in distributed protocols. Quantum sensing may became integral to a distributed quantum computer.^[126]

Quantum supremacy

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 classical computers.^{[127][51][128]} The problem need not be useful, so quantum supremacy test may be just a future benchmark.^[129]

In October 2019, Google Quantum AI, 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, then generally considered the world's fastest computer.^{[29][130][131]} This claim was challenged: IBM stated that Summit can perform samples much faster than claimed.^[132][133] Researchers later developed better quantum algorithms for the sampling problem,^{[134][135][136]} possibly beating Summit.^{[137][138][139]}

In December 2020, a group at USTC implemented a type of boson sampling on 76 photons with a photonic quantum computer, Jiuzhang, seeking quantum supremacy.^{[140][141][142]} The authors claimed that a classical computer would require 600 million years to generate the number of samples their quantum processor generated in 20 seconds.^[143]

Hyped claims of quantum supremacy,^[144] were based on tasks that do not directly imply real-world applications.^[102][145]

A January 2024 study reported verification of quantum supremacy experiments by computing exact amplitudes for experimentally generated bitstrings using a Sunway supercomputer, demonstrating a significant leap in simulation capability built on a multiple-amplitude tensor network contraction algorithm.^[146]

State of affairs: 2020s

Despite high hopes for quantum computing, significant progress in hardware, and optimism about future applications, a 2023 article summarized current quantum computers as "For now, [good for] absolutely nothing". The article claimed that quantum computers are yet of no practical use although they are likely to be useful, someday.^[102] A 2023 article stated that quantum computing algorithms are "insufficient for practical quantum advantage without significant improvements across the software/hardware stack". It foretold that the most promising candidates for achieving speedup with quantum computers are "small-data problems", for example, in chemistry and materials science. It concluded that many application domains, 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".^[103]

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

• Conventional computer hardware and algorithms are optimized for practical tasks and are still improving rapidly.
• Quantum computing hardware generates is overwhelmed by noise before completing any useful tasks.
• Quantum algorithms provide speedup only for some tasks, and matching these tasks with practical applications is challenging. Some of these require resources far beyond those available.^[147][148] In particular, processing large amounts of data is a challenge.^[103]
• Some promising algorithms have been "dequantized", i.e., non-quantum analogues with similar complexity have been found.
• The overhead of quantum error correction may undermine the speedup offered by many quantum algorithms.^[103]
• Algorithm complexity analysis may make abstract assumptions that do not hold in applications. For example, input data may not 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 sustain sufficient entanglement for enough time. Researchers often choose novel tasks to differentiate quantum devices, and to prove lower bounds on the complexity of classical algorithms, but this is not always possible.

Bill Unruh doubted the practicality of quantum computers in a paper published in 1994.^[149] Paul Davies argued that a 400-qubit computer would conflict with the cosmological information bound implied by the holographic principle.^[150] Gil Kalai doubted that quantum supremacy would ever be achieved.^{[151][152][153]} Physicist Mikhail Dyakonov 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."^[154]

Physical realizations

A practical quantum computer must use a physical system as a programmable quantum register.^[156] Researchers are exploring several technologies as candidates for reliable qubit implementations.^[157] Superconductors and trapped ions are some of the most developed proposals, but experimentalists are considering other hardware possibilities as well.^[158] For example, topological quantum computer approaches are being explored for more fault-tolerance computing systems.^[159]

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.^[160]

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.^[161]

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.^[162][163] 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.^[162][164]

Computability

Any computational problem solvable by a classical computer is also solvable by a quantum computer.^[165] Intuitively, this is because 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. Formally, any quantum or classical computer can be simulated by a Turing machine. Quantum computers provide no computability power over classical computers. Thus, quantum computers cannot solve undecidable problems like the halting problem, and the existence of quantum computers does not disprove the Church–Turing thesis.^[166]

Complexity

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

The class of problems that can be efficiently solved by a quantum computer with bounded error is called BQP, for "bounded error, quantum, polynomial time". More formally, BQP is the class of problems that can be solved by a polynomial-time quantum Turing machine with an error probability of at most 1/3. As a class of probabilistic problems, BQP is the quantum counterpart to BPP ("bounded error, probabilistic, polynomial time"), the class of problems that can be solved by polynomial-time probabilistic Turing machines with bounded error.^[167] ${\displaystyle {\mathsf {BPP\subseteq BQP}}}$ but no proof demonstrates that ${\displaystyle {\mathsf {BQP\neq BPP}}}$, which intuitively would mean that quantum computers offer superior time complexity over classical computers.^[168]

The exact relationship of BQP to P, NP, and PSPACE is not known. However, it is known that ${\displaystyle {\mathsf {P\subseteq BQP\subseteq PSPACE}}}$; that is, all problems that can be efficiently solved by a classical computer can be efficiently solved by a quantum computer, and all problems that can be efficiently solved by a quantum computer can be solved by a classical computer with polynomial space resources.

It is suspected that BQP is a strict superset of P, meaning that problems exist that are efficiently solvable by quantum computers that are not efficiently solvable by classical computers. For instance, integer factorization and the discrete logarithm problem are in BQP and are suspected to be outside of P. On the relationship of BQP to NP, little is known except that NP problems that are not in P are in BQP (integer factorization and the discrete logarithm problem are both in NP, for example). It is suspected that ${\displaystyle {\mathsf {NP\nsubseteq BQP}}}$; that is, it is believed that some efficiently checkable problems are not efficiently solvable by a quantum computer. A direct consequence is 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).^[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.