This is a list of quantum algorithms, including algorithms, algorithmic techniques, computational models, and problem frameworks used in quantum computing. A quantum algorithm is an algorithm designed to run on a quantum computer or on another formal model of quantum computation. Quantum algorithms are often grouped by the mathematical technique used, such as the quantum Fourier transform, amplitude amplification, quantum walks, phase estimation, or hybrid quantum-classical optimization.
Foundational and oracle algorithms
Algorithm or problem Description Deutsch–Jozsa algorithm Determines whether a black-box Boolean function is constant or balanced Bernstein–Vazirani algorithm Determines a hidden bit string encoded in a black-box function Simon's problem Finds a hidden period in a function and helped inspire later period-finding algorithms Hidden subgroup problem General framework for many quantum algorithms based on hidden algebraic structure Hidden shift problem Problem of finding a shift relating two functions Hidden linear function problem Oracle problem involving a hidden linear function
Fourier transform, phase estimation, and algebraic algorithms Algorithm or technique Description Quantum Fourier transform Quantum analogue of the discrete Fourier transform, used in several quantum algorithms Hadamard transform Transform used in many quantum circuits and query algorithms Quantum phase estimation algorithm Estimates the phase of an eigenvalue of a unitary operator Hadamard test Circuit technique for estimating real or imaginary parts of expectation values Swap test Estimates the overlap or similarity between quantum states Shor's algorithm Factoring and discrete-logarithm algorithm with major implications for public-key cryptography Aharonov–Jones–Landau algorithm Quantum algorithm related to approximating the Jones polynomial
Search and amplitude-amplification algorithms Algorithm or technique Description Grover's algorithm Search algorithm giving a quadratic speedup for unstructured search Amplitude amplification Generalization of Grover's algorithm used to amplify the probability of desired outcomes Quantum counting algorithm Estimates the number of marked items in a search space Quantum sort Quantum algorithmic approach to sorting BHT algorithm Quantum algorithm for the Collision problem
Quantum walk algorithms
Algorithm or model Description Quantum walk Quantum analogue of a classical random walk Continuous-time quantum walk Quantum-walk model in which the system evolves continuously in time Quantum walk search Search algorithms based on quantum walks
Simulation and linear-algebra algorithms
Algorithm or technique Description Hamiltonian simulation Problem and family of quantum algorithms for simulating the time evolution of quantum systems governed by a Hamiltonian Feynman's algorithm Early quantum-computing approach associated with simulating quantum systems Path integral Monte Carlo Monte Carlo method related to path-integral formulations of quantum systems HHL algorithm Quantum algorithm for solving certain systems of linear equations Quantum singular value transformation Framework for transforming singular values of encoded matrices
Optimization and variational algorithms Algorithm or family Description Quantum annealing Optimization approach based on adiabatic evolution or annealing-like quantum dynamics Quantum optimization algorithms Family of quantum algorithms for optimization problems Quantum approximate optimization algorithm Hybrid quantum-classical algorithm for approximate combinatorial optimization, based on alternating cost and mixer Hamiltonians Variational quantum eigensolver Hybrid quantum-classical algorithm for estimating ground-state energies
Sampling and non-local computation Algorithm or model Description Boson sampling Sampling problem using noninteracting bosons, often discussed in relation to quantum advantage Non-local quantum computation Quantum-information task involving separated parties and nonlocal operations Quantum artificial life Quantum-computing model inspired by artificial life and biological systems
See also Adiabatic quantum computation BQP Glossary of quantum computing List of algorithms List of artificial intelligence algorithms List of quantum and post-quantum cryptography List of quantum computing journals List of quantum processors List of quantum software Quantum circuit Quantum complexity theory Quantum information science Quantum programming Quantum supremacy
References