A quantum computer is a computer that represents and processes information using quantum states. Quantum computations exploit phenomena such as superposition, interference, and entanglement. Quantum computers have the potential to complete some calculations exponentially faster than classical computers. For example, a large-scale quantum computer could break widely used encryption schemes and aid physicists in performing physical simulations. However, current hardware implementations of quantum computation are largely experimental and suitable for only certain specialized tasks. The basic unit of information in quantum computing, the qubit (quantum bit), serves a similar function as the bit in ordinary or "classical" computing. Unlike a classical bit, which can be in one of two states (a binary), a qubit can exist in a linear combination of 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. Quantum algorithm design involves creating procedures that allow a quantum computer to perform this amplification. Quantum computers are not yet practical for real-world applications. If a physical qubit is not sufficiently isolated from its environment, it suffers from quantum decoherence, introducing noise (error) into calculations. Governments have invested in research aimed at developing 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 that quantum devices can outperform classical computers on specific tasks, a metric referred to as quantum advantage or quantum supremacy. Such tasks are not necessarily useful for real-world applications. As a result, as of 2026 demonstrations are best understood as scientific milestones rather than evidence for near-term deployment. Global government investment in quantum computing reached $10 billion by April 2025.
History
Quantum mechanics and computer science formed distinct academic communities until the advent of quantum computing. Quantum theory was developed in the 1920s to explain perplexing physical phenomena. Computers emerged decades later. Both disciplines had practical applications during World War II; computers played a major role in wartime cryptography, while quantum physics was essential for nuclear physics, e.g., in the Manhattan Project. As physicists applied quantum mechanical models to computational problems and swapped bits for qubits, quantum mechanics and computer science began to converge. In 1980, Paul Benioff introduced the quantum Turing machine, which used quantum theory to describe a simplified computer. As digital computers became faster, physicists faced an exponential increase in overhead when simulating quantum dynamics, prompting Yuri Manin and Richard Feynman to independently suggest that hardware based on quantum phenomena might be more efficient for computer simulation. 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. Quantum algorithms then emerged for solving oracle problems, such as Deutsch's algorithm in 1985, the Bernstein–Vazirani algorithm in 1993, and Simon's algorithm in 1994. These algorithms did not solve practical problems, but demonstrated mathematically that more information could be obtained by querying a black box with a quantum state in superposition, sometimes referred to as quantum parallelism.
Peter Shor built on these results in 1994 with polynomial-time quantum algorithms for integer factorization and the discrete logarithm problem. A sufficiently large quantum computer could therefore break widely used public-key cryptography systems: efficient factorization would compromise RSA, while an efficient discrete-logarithm algorithm would compromise Diffie–Hellman key exchange. The cryptographic implications of Shor's algorithm drew attention to quantum computing. In 1996, Grover's algorithm established a quantum speedup for the unstructured search problem. The same year, Seth Lloyd proved that quantum computers could simulate quantum systems without the exponential overhead required by classical simulations, validating Feynman's 1982 conjecture. Experimentalists constructed small-scale quantum computers using trapped ions and superconductors. In 1998, a two-qubit quantum computer demonstrated technical feasibility. Subsequent experiments increased the number of qubits and reduced error rates. 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 threshold for this milestone. Quantum computing increasingly focused on controlling decoherence through quantum error correction. In 2024, researchers demonstrated approaches for high threshold, low-overhead fault-tolerant quantum memory. These developments represented 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 engineering challenge.
Quantum information processing Computer engineers typically describe a modern computer's operation in terms of classical electrodynamics. In these computers, 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. If a classical computation uses randomness, this can be modeled as access to random classical bits rather than as coherent quantum information. A quantum computer, by contrast, uses coherent quantum states, so that superposition, relative phase, and interference are part of the computation itself, and have 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:
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 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 | 0 ⟩ {\displaystyle |0\rangle } or | 1 ⟩ {\displaystyle |1\rangle } . However, the quantum states | 0 ⟩ {\displaystyle |0\rangle } and | 1 ⟩ {\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 | 0 ⟩ {\displaystyle |0\rangle } and | 1 ⟩ {\displaystyle |1\rangle } . A two-dimensional vector mathematically represents a qubit state. Physicists typically use bra–ket notation for quantum mechanical linear algebra, writing | ψ ⟩ {\displaystyle |\psi \rangle } 'ket psi' for a vector labeled ψ {\displaystyle \psi } . Because a qubit is a two-state system, any qubit state takes the form α | 0 ⟩ + β | 1 ⟩ {\displaystyle \alpha |0\rangle +\beta |1\rangle } , where | 0 ⟩ {\displaystyle |0\rangle } and | 1 ⟩ {\displaystyle |1\rangle } are the standard basis states, and α {\displaystyle \alpha } and β {\displaystyle \beta } are the probability amplitudes, which are in general complex numbers. If either α {\displaystyle \alpha } or β {\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. 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 ⟩ {\displaystyle \alpha |0\rangle +\beta |1\rangle } , the state collapses to | 0 ⟩ {\displaystyle |0\rangle } with probability | α | 2 {\displaystyle |\alpha |^{2}} , or to | 1 ⟩ {\displaystyle |1\rangle } with probability | β | 2 {\displaystyle |\beta |^{2}} . Any valid qubit state has coefficients α {\displaystyle \alpha } and β {\displaystyle \beta } such that | α | 2 + | β | 2 = 1 {\displaystyle |\alpha |^{2}+|\beta |^{2}=1} . As an example, measuring the qubit 1 / 2 | 0 ⟩ + 1 / 2 | 1 ⟩ {\displaystyle 1/{\sqrt {2}}|0\rangle +1/{\sqrt {2}}|1\rangle } would produce either | 0 ⟩ {\displaystyle |0\rangle } or | 1 ⟩ {\displaystyle |1\rangle } with equal probability. Two particularly important superposition states are the plus state | + ⟩ = 1 / 2 | 0 ⟩ + 1 / 2 | 1 ⟩ {\displaystyle |+\rangle =1/{\sqrt {2}}|0\rangle +1/{\sqrt {2}}|1\rangle } and the minus state | − ⟩ = 1 / 2 | 0 ⟩ − 1 / 2 | 1 ⟩ {\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 | 0 ⟩ ↔ | + ⟩ {\displaystyle |0\rangle \leftrightarrow |+\rangle } and | 1 ⟩ ↔ | − ⟩ {\displaystyle |1\rangle \leftrightarrow |-\rangle } —demonstrating that relative phase differences carry meaningful quantum information. Each additional qubit doubles the dimension of the state space. 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 := ( 0 1 1 0 ) . {\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
X | 0 ⟩ = | 1 ⟩ {\displaystyle X|0\rangle =|1\rangle } and X | 1 ⟩ = | 0 ⟩ {\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: | 00 ⟩ := ( 1 0 0 0 ) ; | 01 ⟩ := ( 0 1 0 0 ) ; | 10 ⟩ := ( 0 0 1 0 ) ; | 11 ⟩ := ( 0 0 0 1 ) . {\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: CNOT := ( 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 ) . {\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, CNOT | 00 ⟩ = | 00 ⟩ {\textstyle \operatorname {CNOT} |00\rangle =|00\rangle } , CNOT | 01 ⟩ = | 01 ⟩ {\textstyle \operatorname {CNOT} |01\rangle =|01\rangle } , CNOT | 10 ⟩ = | 11 ⟩ {\textstyle \operatorname {CNOT} |10\rangle =|11\rangle } , and CNOT | 11 ⟩ = | 10 ⟩ {\textstyle \operatorname {CNOT} |11\rangle =|10\rangle } . In other words, the CNOT applies a NOT gate ( X {\textstyle X} from before) to the second qubit if and only if the first qubit is in the state | 1 ⟩ {\textstyle |1\rangle } . If the first qubit is | 0 ⟩ {\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.
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 2 n × 2 n {\displaystyle 2^{n}\times 2^{n}} over n {\displaystyle n} qubits) can be represented as a network of quantum logic gates from a fairly small family of gates. A choice of gate family that enables this construction is known as a universal gate set, since a computer that can run such circuits is a universal quantum computer. One common such set includes all single-qubit gates as well as the CNOT gate. 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.
Quantum Turing machine A quantum Turing machine is the quantum analog of a Turing machine. All of these models of computation—quantum circuits, one-way quantum computation, adiabatic quantum computation, and topological quantum computation—have been shown to be equivalent to the quantum Turing machine; given a perfect implementation of one such quantum computer, it can simulate all the others with no more than polynomial overhead. This equivalence need not hold for practical quantum computers, since the overhead of simulation may be too large to be practical.
Quantum-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. A topological quantum computer decomposes computation into the braiding of anyons in a 2D lattice.
Noisy intermediate-scale quantum computing The threshold theorem shows how increasing the number of qubits can mitigate errors, yet fully fault-tolerant quantum computing remains out of reach as of 2026. 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. Scientists at Harvard created "quantum circuits" that correct errors more efficiently than alternative methods, which may remove a major obstacle to practical quantum computers. 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.
Communication
Quantum cryptography enables methods for secure data transmission; for example, quantum key distribution uses entangled quantum states to establish secure cryptographic keys. When a sender and receiver exchange quantum states, they can guarantee that an adversary does not intercept the message, as any eavesdropper would disturb the delicate quantum system and introduce a detectable change. With appropriate cryptographic protocols, the sender and receiver can privately share information resistant to eavesdropping. 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.
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, though exceptions such as the quantum adiabatic algorithm exist. Quantum algorithms can be roughly categorized by the type of speedup achieved over corresponding classical algorithms. 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. 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. 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 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 unlikely. In addition to these problems, quantum algorithms are 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 for practical implementation. Some quantum algorithms, such as Grover's algorithm and amplitude amplification, give polynomial speedups over corresponding classical algorithms. Though these algorithms give comparably modest quadratic speedup, they are widely applicable and thus accelerate a wide range of problems. These improvements are, however, over the theoretical worst-case of classical algorithms, and real-world speed-ups over traditional algorithms have not been demonstrated.
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. Quantum computational chemistry is promising for quantum computing, particularly for problems in electronic structure, chemical dynamics, and spectroscopy; useful implementations remain hardware-limited. Quantum simulation could be used to simulate the behavior of atoms and particles under unusual conditions such as the reactions inside a collider. In June 2023, IBM computer scientists reported that a quantum computer produced better results for a physics problem than a conventional supercomputer. 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.
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. 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). By contrast, a quantum computer could solve this problem exponentially faster using Shor's algorithm to factor the integer. 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. 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. 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. Applying Grover's algorithm to break a symmetric (secret-key) algorithm by brute force requires roughly 2n/2 invocations of the underlying cryptographic algorithm, compared with roughly 2n in the classical case, 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.
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 n {\displaystyle n} items in a database. This can be solved by Grover's algorithm using O ( n ) {\displaystyle O({\sqrt {n}})} queries to the database, quadratically fewer than the Ω ( n ) {\displaystyle \Omega (n)} queries required for classical algorithms. In this case, the advantage is not only provable but also optimal: it has been shown that Grover's algorithm gives the maximal possible probability of finding the desired element for any number of oracle lookups. 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, and Farhi, Goldstone, and Gutmann's algorithm for evaluating NAND trees. Problems that can be efficiently addressed with Grover's algorithm have the following properties:
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 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.
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. Adiabatic optimization may be helpful for solving computational biology problems.
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. 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. For example, the HHL Algorithm is believed to provide speedup over classical counterparts. Research groups have explored quantum annealing hardware for training Boltzmann machines and deep neural networks. 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. 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 and thus may apply to applications involving quantum chemistry. Quantum-enhanced generative models including