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Quantum Algorithm for Linear Systems of Equations
Open accessAbstract
Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix $A$ and a vector $\stackrel{\ensuremath{\rightarrow}}{b}$, find a vector $\stackrel{\ensuremath{\rightarrow}}{x}$ such that $A\stackrel{\ensuremath{\rightarrow}}{x}=\stackrel{\ensuremath{\rightarrow}}{b}$. We consider the case where one does not need to know the solution $\stackrel{\ensuremath{\rightarrow}}{x}$ itself, but rather an approximation of the expectation value of some operator associated with $\stackrel{\ensuremath{\rightarrow}}{x}$, e.g., ${\stackrel{\ensuremath{\rightarrow}}{x}}^{\ifmmode\dagger\else\textdagger\fi{}}M\stackrel{\ensuremath{\rightarrow}}{x}$ for some matrix $M$. In this case, when $A$ is sparse, $N\ifmmode\times\else\texttimes\fi{}N$ and has condition number $\ensuremath{\kappa}$, the fastest known classical algorithms can find $\stackrel{\ensuremath{\rightarrow}}{x}$ and estimate ${\stackrel{\ensuremath{\rightarrow}}{x}}^{\ifmmode\dagger\else\textdagger\fi{}}M\stackrel{\ensuremath{\rightarrow}}{x}$ in time scaling roughly as $N\sqrt{\ensuremath{\kappa}}$. Here, we exhibit a quantum algorithm for estimating ${\stackrel{\ensuremath{\rightarrow}}{x}}^{\ifmmode\dagger\else\textdagger\fi{}}M\stackrel{\ensuremath{\rightarrow}}{x}$ whose runtime is a polynomial of $\mathrm{log}(N)$ and $\ensuremath{\kappa}$. Indeed, for small values of $\ensuremath{\kappa}$ [i.e., $\mathrm{poly}\mathrm{log}(N)$], we prove (using some common complexity-theoretic assumptions) that any classical algorithm for this problem generically requires exponentially more time than our quantum algorithm.
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“On the specific applications of this algorithm [75], its use is possible in computational biology, where numeric models can be solved, and also in the domain of machine learning techniques.”
From Quantum Computing in the Next-Generation Computational Biology Landscape: From Protein Folding to Molecular Dynamics (2024) · cites this paper as [75]
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“As proposed by Harrow etal. [75], the quantum algorithm is capable enough to give solutions to definite linear systems at exponentially rapid speed compared to any existing algorithm.”
From Quantum Computing in the Next-Generation Computational Biology Landscape: From Protein Folding to Molecular Dynamics (2024) · cites this paper as [75]
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“The classical algorithmic complexity for this task scales polynomially in the dimension N , whereas the now-famous Harrow-Hassidim-Lloyd (HHL) quantum algorithm [3] has a complexity that scales logarithmically in N , with some scaling improvements having been proposed [134-137].”
From Variational quantum algorithms (2021) · cites this paper as [3]
Experimental — accuracy not yet measuredThe cited source says what the claim attributes to it.
In the citing paper
“In particular, the task of matrix inversion admits an exponential quantum speedup, which could have many applications.”
From Quantum Computing in the NISQ era and beyond (2018) · cites this paper as [40]
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In the citing paper
“This quantum algorithm runs in time O (log N ) , an exponential speedup relative to classical matrix inversion (for fixed error and for fixed sparsity and condition number of A ).”
From Quantum Computing in the NISQ era and beyond (2018) · cites this paper as [40]
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“Available pieces of literature reveal the application of quantum algorithms to solve linear [75] and differential equations [76-78], where these algorithms play the role of 'subroutines' in algorithms on data processing.”
From Quantum Computing in the Next-Generation Computational Biology Landscape: From Protein Folding to Molecular Dynamics (2024) · cites this paper as [75]
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“They proved that even though these traditional sampling ones are quick, any traditional algorithm, in general, needs much more time than their [75] algorithm to complete the identical matrix inversion job.”
From Quantum Computing in the Next-Generation Computational Biology Landscape: From Protein Folding to Molecular Dynamics (2024) · cites this paper as [75]
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“This group [75] reported that their algorithm was related to the traditional Monte Carlo ones, which are pretty quick and operate with samples of a statistical distribution involving N objects instead of using all the N constituents of distribution.”
From Quantum Computing in the Next-Generation Computational Biology Landscape: From Protein Folding to Molecular Dynamics (2024) · cites this paper as [75]
The cited source says what the claim attributes to it.
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“Recent research trends indicate 'data sets' which are used to define these equations, which can be enormous depending on the specific application [75], where data of the order petabytes could be required to process to get a solution.”
From Quantum Computing in the Next-Generation Computational Biology Landscape: From Protein Folding to Molecular Dynamics (2024) · cites this paper as [75]
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“However, aside from the caveat that the matrix A must be sparse and well conditioned, note that both the input vector | b 〉 and the output vector | A -1 b 〉 are quantum states.”
From Quantum Computing in the NISQ era and beyond (2018) · cites this paper as [40]
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In the citing paper
“Assuming these constraints, x j i can be approximately produced in time polynomial in log N , d and κ . 58,59”
From Quantum algorithms: an overview (2016) · cites this paper as [58]
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In the citing paper
“Indeed, one can even show that achieving a similar runtime classically would imply that classical computers could ef fi ciently simulate any polynomial-time quantum computation. 58”
From Quantum algorithms: an overview (2016) · cites this paper as [58]
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“The quantum algorithm of Harrow, Hassidim and Lloyd 58 (HHL) for solving systems of linear equations sidesteps this issue by ' solving ' the equations in a peculiarly quantum sense: given the ability to create the quantum state b j i ¼ P N i ¼ 1 bi i j i , and access to A , the algorithm outputs a state approximately proportional to x j i ¼ P N i ¼ 1 xi i j i .”
From Quantum algorithms: an overview (2016) · cites this paper as [58]
In the citing paper
“Second, by considering more favorable schemes for Hamiltonian decomposition 256,257 , most modern-day simulation algorithms scale in terms of τ max = t ‖ H ‖ max instead of τ , where ‖ H ‖ max is the absolute value of the largest entry of H as a matrix in the computational basis.”
From Quantum Chemistry in the Age of Quantum Computing (2019) · cites this paper as [257]
In the citing paper
“Childs 257 showed that given oracle access to a Hamiltonian with eigenvalues λ , one can construct a unitary quantum walk operator with eigenvalues e ± i arcsin λ .”
From Quantum Chemistry in the Age of Quantum Computing (2019) · cites this paper as [257]
In the citing paper
“Inserting this conditional operation in the quantum phase algorithm and using the improved phaseestimation techniques of ref. 15 yields the eigenvectors and eigenvalues to accuracy by applying the quantum phase algorithm for time t = O( -1 ), and so requires n = O(1/ 3 ) copies of the state ρ.”
From Quantum principal component analysis (2014) · cites this paper as [15]
In the citing paper
“Note that density matrix exponentiation is most effective when some of the eigenvalues of ρ are large.If all of the eigenvalues are of size O(1/d) then we require time t = O(d) to resolve the eigenvalues 15 .”
From Quantum principal component analysis (2014) · cites this paper as [15]
In the citing paper
“The Harrow-Hassidim-Lloyd (HHL) quantum algorithm [16] can solve certain linear systems exponentially faster than any known classical algorithm”
From The prospects of quantum computing in computational molecular biology (2021) · cites this paper as [16]
In the citing paper
“However, the algorithm proposed by Harrow, Hassidim and Lloyd [16] allows the inversion of a matrix in O log N ð Þ within some conditions.”
From The prospects of quantum computing in computational molecular biology (2021) · cites this paper as [16]
In the citing paper
“Reading out the function values from the quantum state |u in general requires exponential sampling, representing a so-called data”
From Solving nonlinear differential equations with differentiable quantum circuits (2021) · cites this paper as [9]
In the citing paper
“The power of HHL-based algorithms relies on amplitude encoding, where a function is represented as a quantum state |u = 2 N k=1 u k |k encoded in complex u k amplitudes of computational basis states |k for N qubits.This allows for compressing large grids into a small qubit register, providing exponential memory advantage.”
From Solving nonlinear differential equations with differentiable quantum circuits (2021) · cites this paper as [9]
In the citing paper
“When two particles for example are sent into a multiport, e.g.multimode waveguide [153,154], multimode fiber [155] or integrated waveguide structures [156], the transmission can be described as a complex bi-partite quantum walk”
From Two-photon interference: the Hong–Ou–Mandel effect (2021) · cites this paper as [154]
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Quantum Algorithms for Solving Ordinary Differential Equations via Classical Integration Methods
Zanger, B., Mendl, C. B., Schulz, M., & Schreiber, M. (2021) · Quantum · doi:10.22331/q-2021-07-13-502 · Cited by 46
New frontiers of quantum computing in chemical engineering
Ajagekar, A., & You, F. (2022) · Korean Journal of Chemical Engineering · doi:10.1007/s11814-021-1027-6 · Cited by 40
A universal set of qubit quantum channels
Braun, D., Giraud, O., Nechita, I., Pellegrini, C., & Žnidarič, M. (2014) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/47/13/135302 · Cited by 42
Quantum Computing: Lecture NotesPreprint
de Wolf, R. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1907.09415 · Cited by 197
On physics-informed neural networks for quantum computers
Markidis, S. (2022) · Frontiers in Applied Mathematics and Statistics · doi:10.3389/fams.2022.1036711 · Cited by 45
Sukulthanasorn, N., Xiao, J., Wagatsuma, K., Nomura, R., Moriguchi, S., & Terada, K. (2025) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/j.cma.2025.117746 · Cited by 35
Hamiltonian simulation with nearly optimal dependence on spectral norm
Low, G. H. (2019) · ACM Symposium on Theory of Computing (STOC) · doi:10.1145/3313276.3316386 · Cited by 35
Efficient simulation of sparse Markovian quantum dynamics
Childs, A. M., & Li, T. (2017) · Quantum Information and Computation · doi:10.26421/qic17.11-12-1 · Cited by 65
Quantum algorithms for Second-Order Cone Programming and Support Vector Machines
Kerenidis, I., Prakash, A., & Szilágyi, D. (2021) · Quantum · doi:10.22331/q-2021-04-08-427 · Cited by 52
Quantum k-means algorithm based on Manhattan distance
Wu, Z., Song, T., & Zhang, Y. (2021) · Quantum Information Processing · doi:10.1007/s11128-021-03384-7 · Cited by 36
Quantum Approach to Accelerate Finite Volume Method on Steady Computational Fluid Dynamics Problems
Chen, Z., Xue, C., Chen, S., Lu, B.-H., Wu, Y.-C., Ding, J.-C., Huang, S., & Guo, G. (2022) · Quantum Information Processing · doi:10.1007/s11128-022-03478-w · Cited by 63
Fast quantum algorithm for protein structure prediction in hydrophobic-hydrophilic model
Wong, R., & Chang, W.-L. (2022) · Journal of Parallel and Distributed Computing · doi:10.1016/j.jpdc.2022.03.011 · Cited by 37
Quantum readout error mitigation via deep learning
Kim, J., Oh, B., Chong, Y., Hwang, E. H., & Park, D. K. (2022) · New Journal of Physics · doi:10.1088/1367-2630/ac7b3d · Cited by 39
CoqQ: Foundational Verification of Quantum Programs
Zhou, L., Barthe, G., Strub, P.-Y., Liu, J., & Ying, M. (2023) · Proceedings of the ACM on Programming Languages · doi:10.1145/3571222 · Cited by 35
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Yuan, P., & Zhang, S. (2023) · Quantum · doi:10.22331/q-2023-03-20-956 · Cited by 55
Quantum Algorithms for Scientific Computing and Approximate OptimizationPreprint
Hadfield, S. (2018) · arXiv (Cornell University) · doi:10.7916/d8x650c9 · Cited by 76
Quantum-enhanced deep neural network architecture for image scene classification
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Ingelmann, J., Bharadwaj, S. S., Pfeffer, P., Sreenivasan, K. R., & Schumacher, J. (2024) · Computers & Fluids · doi:10.1016/j.compfluid.2024.106369 · Cited by 35
Characterizing Grover search algorithm on large-scale superconducting quantum computers
AbuGhanem, M. (2025) · Scientific Reports · doi:10.1038/s41598-024-80188-6 · Cited by 42
Certified randomness using a trapped-ion quantum processor
Liu, M., Shaydulin, R., Niroula, P., DeCross, M. P., Hung, S.-H., Kon, W. Y., Cervero-Martín, E., Chakraborty, K., Amer, O., Aaronson, S., et al. (2025) · Nature · doi:10.1038/s41586-025-08737-1 · Cited by 55
Job, J., & Lidar, D. A. (2018) · Quantum Science and Technology · doi:10.1088/2058-9565/aabd9b · Cited by 34
Quantum remote sensing with asymmetric information gain
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Box algorithm for the solution of differential equations on a quantum annealer
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Reachability analysis of quantum Markov decision processes
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Random circuit block-encoded matrix and a proposal of quantum LINPACK benchmark
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Metadata
- Authors
- Harrow, A. W., Hassidim, A., & Lloyd, S.
- Affiliations
- University of Bristol (GB)
- Massachusetts Institute of Technology (US)
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2009-10-07
- Volume
- 103
- Issue
- 15
- Pages
- 150502-150502
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W1492999010
- Semantic Scholar
- ed562f0c86c80f75a8b9ac7344567e8b44c8d643
- Cited by
- 3,325
- References
- 34
- Citations by year
- 2026: 355 · 2025: 553 · 2024: 478 · 2023: 429 · 2022: 332 · 2021: 337 · 2020: 231 · 2019: 181 · 2018: 98 · 2017: 102 · 2016: 53 · 2015: 35 · 2014: 39 · 2013: 33 · 2012: 40
- FWCI
- 23.75
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Physical Review Letters (publishedVersion)
- arXiv (Cornell University) (publishedVersion)
- PubMed (publishedVersion)
- DSpace@MIT (Massachusetts Institute of Technology) (submittedVersion, cc-by-nc)
- arXiv (Cornell University)
- PIRSA
- Bulletin of the American Physical Society
- Funders
- W. M. Keck Foundation
- Engineering and Physical Sciences Research Council (GR/S82169/01)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Keywords
- quantum algorithms, systems of linear equations, sparse matrices, condition number, expectation value estimation, quantum advantage
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