{ "name": "Quantum algorithms list: complexity, speedup, and evidence", "description": "A row-by-row complexity table. Evidence rates the quantum statement: proved, conditional, heuristic, or demonstrated. Speedup is the comparison with the classical column, and none-proved means that comparison is not a theorem.", "license": "https://creativecommons.org/licenses/by/4.0/", "creator": "QuantumAlgorithms.ai editorial", "url": "https://quantumalgorithms.ai/quantum-algorithms/table", "dateModified": "2026-10-28", "attribution": "QuantumAlgorithms.ai editorial. AI-assisted compilation sourced from the primary papers named in each row. CC BY 4.0.", "algorithms": [ { "id": "shor", "algorithm": "Shor's algorithm", "problem": "Integer factorization and discrete logarithms", "model": "Gate array", "quantum_complexity": "O((log n)^2 (log log n) (log log log n)) gate-array steps for an integer n, plus classical post-processing polynomial in log n (Shor 1997). Resource estimates for a 2048-bit RSA integer are separate: about 20 million noisy qubits and about 8 hours under the assumptions in Gidney and Ekerå 2021, and fewer than one million noisy qubits and less than one week under the assumptions in Gidney 2025.", "classical_comparison": "Shor 1997 does not prove a classical lower bound. The polynomial quantum runtime is not a theorem that factoring is classically hard.", "speedup": "exponential", "evidence": "proved", "year": "1994", "reference_slug": "shor-1997", "reference_url": "https://quantumalgorithms.ai/references/shor-1997", "page": "https://quantumalgorithms.ai/algorithms/shor" }, { "id": "grover", "algorithm": "Grover's algorithm", "problem": "Unstructured search", "model": "Oracle", "quantum_complexity": "Θ(√(N/t)) queries when t items are marked. Optimal in the black-box model up to a small constant (Grover 1996; Boyer et al. 1998; Bennett et al. 1997).", "classical_comparison": "Θ(N/t) queries to the same oracle in the worst case.", "speedup": "quadratic", "evidence": "proved", "year": "1996", "reference_slug": "grover-1996", "reference_url": "https://quantumalgorithms.ai/references/grover-1996", "page": "https://quantumalgorithms.ai/algorithms/grover" }, { "id": "phase-estimation", "algorithm": "Quantum phase estimation", "problem": "Eigenphase of a unitary", "model": "Gate array, given an eigenstate", "quantum_complexity": "Precision 2^(−l−2) with error probability at most ε, using the powered unitary O(l log(l/ε)) times plus a term polynomial in l (Kitaev 1995, Lemma 10).", "classical_comparison": "No single classical counterpart. The bound assumes an eigenstate, or a state whose overlap is not exponentially small, and controlled powers the machine can apply.", "speedup": "none-proved", "evidence": "conditional", "year": "1995", "reference_slug": "kitaev-1995", "reference_url": "https://quantumalgorithms.ai/references/kitaev-1995", "page": "https://quantumalgorithms.ai/algorithms/phase-estimation" }, { "id": "hhl", "algorithm": "HHL algorithm", "problem": "Sparse linear systems", "model": "Gate array with sparse row access", "quantum_complexity": "Õ(κ T_B + κ² s² log(N)/ε) to prepare a state within vector distance ε of the normalized solution, for an s-sparse matrix of norm at most 1 (Harrow, Hassidim, and Lloyd 2009, Appendix A.1). The output is a quantum state, not the list of solution entries.", "classical_comparison": "Conjugate gradient, as that paper compares it, uses on the order of N s √κ log(1/ε) operations for the classical vector when A is positive definite. The two algorithms do not return the same kind of answer. A low-rank matrix with sampling access has a different classical algorithm (Gilyén, Lloyd, and Tang 2018).", "speedup": "exponential", "evidence": "conditional", "year": "2009", "reference_slug": "harrow-2009", "reference_url": "https://quantumalgorithms.ai/references/harrow-2009", "page": "https://quantumalgorithms.ai/algorithms/hhl" }, { "id": "vqe", "algorithm": "Variational quantum eigensolver", "problem": "Approximate ground-state energy", "model": "Hybrid", "quantum_complexity": "No proved convergence time. Prepare an ansatz, estimate the energy by measurement, and update the parameters classically (Peruzzo et al. 2014). Reported runs include HeH+ on a photonic processor and LiH, BeH2, and a four-qubit Heisenberg model on a superconducting processor (Kandala et al. 2017).", "classical_comparison": "No theorem in the papers this library cites says the loop beats classical quantum-chemistry methods. The published demonstrations are on very small systems.", "speedup": "none-proved", "evidence": "heuristic", "year": "2014", "reference_slug": "peruzzo-2014", "reference_url": "https://quantumalgorithms.ai/references/peruzzo-2014", "page": "https://quantumalgorithms.ai/algorithms/vqe" }, { "id": "qaoa", "algorithm": "QAOA", "problem": "Approximate combinatorial optimization", "model": "Hybrid", "quantum_complexity": "For p = 1, a worst-case approximation ratio of 0.6924 on MaxCut for connected 3-regular graphs (Farhi, Goldstone, and Gutmann 2014).", "classical_comparison": "Goemans and Williamson (1995) give an expected approximation ratio of at least 0.878 for MaxCut, and that guarantee is not restricted to 3-regular graphs. The QAOA authors say the p = 1 result is weaker than known classical algorithms for the same case.", "speedup": "none-proved", "evidence": "proved", "year": "2014", "reference_slug": "farhi-2014", "reference_url": "https://quantumalgorithms.ai/references/farhi-2014", "page": "https://quantumalgorithms.ai/algorithms/qaoa" }, { "id": "hamiltonian-simulation", "algorithm": "Hamiltonian simulation", "problem": "Approximate e^{−iHt}", "model": "Gate array with Hamiltonian access", "quantum_complexity": "For a block encoding of norm at most 1, O(t + log(1/ε)) queries to simulate the exponential to spectral-norm error ε (Low and Chuang 2019, Theorem 1). Product formulas and the truncated Taylor series are different access models, with a polynomial and a logarithmic dependence on 1/ε respectively.", "classical_comparison": "The query bound is not a gate count until the oracle is a circuit. It is not one complexity that can be copied onto an arbitrary molecule.", "speedup": "exponential", "evidence": "proved", "year": "2019", "reference_slug": "low-chuang-2019", "reference_url": "https://quantumalgorithms.ai/references/low-chuang-2019", "page": "https://quantumalgorithms.ai/algorithms/hamiltonian-simulation" }, { "id": "quantum-fourier-transform", "algorithm": "Quantum Fourier transform", "problem": "Discrete Fourier transform on amplitudes", "model": "Gate array", "quantum_complexity": "On n qubits the transform uses on the order of n² gates (Shor 1997, attributing the gate array to Coppersmith and to Deutsch).", "classical_comparison": "The classical fast Fourier transform writes all N = 2^n coefficients. The quantum transform does not return that list. A measurement yields one basis state. In Shor's algorithm the transform is not the bottleneck; modular exponentiation is.", "speedup": "exponential", "evidence": "proved", "year": "1994", "reference_slug": "shor-1997", "reference_url": "https://quantumalgorithms.ai/references/shor-1997", "page": "https://quantumalgorithms.ai/glossary/quantum-fourier-transform" }, { "id": "deutsch-jozsa", "algorithm": "Deutsch–Jozsa algorithm", "problem": "Constant versus balanced Boolean function, under a promise", "model": "Oracle", "quantum_complexity": "One query, exactly (Deutsch and Jozsa 1992).", "classical_comparison": "Any deterministic classical algorithm needs 2^{n−1}+1 queries in the worst case on n-bit inputs. The exponential gap is that exact, deterministic comparison. It is not an exponential separation from bounded-error classical sampling.", "speedup": "exponential", "evidence": "proved", "year": "1992", "reference_slug": "deutsch-jozsa-1992", "reference_url": "https://quantumalgorithms.ai/references/deutsch-jozsa-1992", "page": "" }, { "id": "bernstein-vazirani", "algorithm": "Bernstein–Vazirani algorithm", "problem": "Hidden string s, given f(x) = s·x mod 2", "model": "Oracle", "quantum_complexity": "One query. The algorithm was introduced at STOC 1993; this row cites the 1997 journal version (Bernstein and Vazirani).", "classical_comparison": "n queries to learn an n-bit string. The separation is polynomial (linear versus constant), not exponential.", "speedup": "polynomial", "evidence": "proved", "year": "1997", "reference_slug": "bernstein-vazirani-1997", "reference_url": "https://quantumalgorithms.ai/references/bernstein-vazirani-1997", "page": "" }, { "id": "simon", "algorithm": "Simon's algorithm", "problem": "Hidden xor period, under a promise", "model": "Oracle", "quantum_complexity": "O(n) queries to find the hidden string s, for the promise that f(x) = f(y) exactly when x ⊕ y is 0 or s (Simon 1997; FOCS 1994).", "classical_comparison": "Ω(2^{n/2}) queries, including for bounded-error classical algorithms. The separation is an oracle promise problem. It is not, by itself, a factoring algorithm.", "speedup": "exponential", "evidence": "proved", "year": "1997", "reference_slug": "simon-1997", "reference_url": "https://quantumalgorithms.ai/references/simon-1997", "page": "" }, { "id": "amplitude-amplification", "algorithm": "Amplitude amplification", "problem": "Boost the success probability of a quantum procedure", "model": "Oracle (a procedure and its inverse)", "quantum_complexity": "On the order of 1/√a uses of a procedure that succeeds with probability a, to reach a constant success probability (Brassard, Høyer, Mosca, and Tapp 2002).", "classical_comparison": "On the order of 1/a repetitions of the same procedure.", "speedup": "quadratic", "evidence": "proved", "year": "2002", "reference_slug": "brassard-2002", "reference_url": "https://quantumalgorithms.ai/references/brassard-2002", "page": "https://quantumalgorithms.ai/glossary/amplitude-amplification" }, { "id": "amplitude-estimation", "algorithm": "Amplitude estimation", "problem": "Estimate an unknown success probability", "model": "Oracle", "quantum_complexity": "Phase estimation on the Grover iterate. After M iterations the additive error in the success probability is on the order of √(a(1−a))/M plus 1/M², with constant probability (Brassard, Høyer, Mosca, and Tapp 2002). This row does not restate the paper's leading constants.", "classical_comparison": "Sampling to additive error ε takes on the order of 1/ε² trials.", "speedup": "quadratic", "evidence": "proved", "year": "2002", "reference_slug": "brassard-2002", "reference_url": "https://quantumalgorithms.ai/references/brassard-2002", "page": "https://quantumalgorithms.ai/glossary/amplitude-amplification" }, { "id": "quantum-counting", "algorithm": "Quantum counting", "problem": "Estimate the number of marked items", "model": "Oracle", "quantum_complexity": "Approximate counting by phase estimation on the Grover iterate (Brassard, Høyer, and Tapp 1998). The error in the fraction of marked items is the amplitude-estimation bound: on the order of √(a(1−a))/M plus 1/M² after M iterations (Brassard, Høyer, Mosca, and Tapp 2002).", "classical_comparison": "Sampling to additive error ε in that fraction takes on the order of 1/ε² trials.", "speedup": "quadratic", "evidence": "proved", "year": "1998", "reference_slug": "brassard-hoyer-tapp-1998", "reference_url": "https://quantumalgorithms.ai/references/brassard-hoyer-tapp-1998", "page": "" }, { "id": "quantum-walk-search", "algorithm": "Quantum walk search", "problem": "Oracle search by a discrete-time quantum walk", "model": "Oracle", "quantum_complexity": "O(√N) oracle calls to find one marked item among N, for the hypercube walk in Shenvi, Kempe, and Whaley (2003). The abstract describes a speedup similar to other quantum search algorithms. It does not improve on Grover's query exponent.", "classical_comparison": "Θ(N) queries to the same unstructured oracle.", "speedup": "quadratic", "evidence": "proved", "year": "2003", "reference_slug": "shenvi-kempe-whaley-2003", "reference_url": "https://quantumalgorithms.ai/references/shenvi-kempe-whaley-2003", "page": "" }, { "id": "qsvt", "algorithm": "Quantum singular value transformation", "problem": "Polynomial transformation of singular values", "model": "Block encoding", "quantum_complexity": "Applies a polynomial to the singular values of a block of a unitary, generalizing optimal Hamiltonian simulation, usually with a constant number of ancilla qubits (Gilyén, Su, Low, and Wiebe 2019). The abstract does not state a single end-to-end classical speedup, and this row does not add a degree-to-query leading constant.", "classical_comparison": "No single classical baseline. The comparison depends on the polynomial and on how the matrix is given.", "speedup": "none-proved", "evidence": "proved", "year": "2019", "reference_slug": "gilyen-su-low-wiebe-2019", "reference_url": "https://quantumalgorithms.ai/references/gilyen-su-low-wiebe-2019", "page": "" }, { "id": "quantum-annealing", "algorithm": "Quantum annealing", "problem": "Heuristic optimization of an Ising cost", "model": "Analog annealing, not the gate-array model", "quantum_complexity": "No proved runtime that beats classical optimization on general instances. Kadowaki and Nishimori (1998) introduce quantum annealing as a numerical method for the transverse Ising model.", "classical_comparison": "This row does not claim a classical lower bound. Simulated annealing is a different heuristic, not a theorem being compared.", "speedup": "none-proved", "evidence": "heuristic", "year": "1998", "reference_slug": "kadowaki-nishimori-1998", "reference_url": "https://quantumalgorithms.ai/references/kadowaki-nishimori-1998", "page": "" } ] }