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 specific applications of Harrow et al.'s algorithm, its use is possible in computational biology, where numeric models can be solved, and also in the domain of machine learning techniques.
“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]
The Introduction and Discussion (page not specified in our copy) describe using the algorithm to estimate features of solutions to linear equations, including equations arising from numerical models. They do not identify computational biology or machine-learning techniques as applications, so the claimed domains are not supported by this paper's text. (LLM, conf=0.83)
Experimental — accuracy not yet measuredAs proposed by Harrow et al., the quantum algorithm is capable enough to give solutions to definite linear systems at exponentially rapid speed compared to any existing algorithm.
“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]
The Abstract describes a quantum algorithm that can be exponentially faster than classical algorithms for certain linear-system tasks. But it estimates a feature of the solution rather than producing the full solution, and the speedup depends on conditions including sparsity and a sufficiently small condition number; the claim omits those limits. (LLM, conf=0.89)
Experimental — accuracy not yet measuredFor an N x N linear system, classical algorithmic complexity scales polynomially in N, whereas the HHL quantum algorithm has complexity scaling logarithmically in N, with some scaling improvements proposed.
“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]
The Abstract says classical time scales polynomially with N and that the HHL algorithm can scale logarithmically with N. But it limits that comparison to cases where the goal is to estimate a feature of the solution rather than output the full vector, and its runtime also depends on sparsity, condition number, and precision. The later scaling improvements are cited to other references in the citing sentence, not to HHL. (LLM, conf=0.86)
Experimental — accuracy not yet measuredThe task of matrix inversion admits an exponential quantum speedup.
“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]
The HHL 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.
“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]
Available literature reveals the application of quantum algorithms to solve linear and differential equations, where these algorithms play the role of subroutines in algorithms on data processing.
“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]
Harrow et al. proved that even though traditional sampling algorithms are quick, any traditional algorithm in general needs much more time than their algorithm to complete the identical matrix inversion job.
“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]
Harrow et al. reported that their algorithm was related to traditional Monte Carlo ones, which are quick and operate with samples of a statistical distribution involving N objects instead of using all N constituents of the distribution.
“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]
Recent research trends indicate that data sets used to define linear and differential equations can be enormous depending on the specific application, with data of the order of petabytes possibly required to process to get a solution.
“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]
Both the input vector |b> and the output vector |A^-1 b> of the HHL algorithm are quantum states.
“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]
Assuming A is sparse, accessible via a row-index function, and has small condition number κ, the HHL algorithm can approximately produce |x> in time polynomial in log N, d, and κ.
“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]
Achieving a runtime similar to the HHL algorithm classically would imply that classical computers could efficiently simulate any polynomial-time quantum computation.
“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]
The HHL algorithm solves systems of linear equations in a quantum sense: given the ability to create the quantum state |b> and access to A, it outputs a state approximately proportional to |x>.
“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]
By considering more favorable schemes for Hamiltonian decomposition, most modern-day simulation algorithms scale in terms of τ_max = t||H||_max instead of τ.
“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]
Childs 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.
“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]
Feeding the conditional operation into quantum phase estimation gives eigenvector and eigenvalue estimates to precision ε, with copy count scaling as O(1/ε^3).
“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]
Density matrix exponentiation works best when ρ has some large eigenvalues; when every eigenvalue is O(1/d), resolving them needs time t=O(d).
“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]
The HHL quantum algorithm can solve certain linear systems exponentially faster than any known classical algorithm.
“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]
The HHL algorithm allows the inversion of a matrix in O(log N) within some conditions.
“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]
Reading out function values from the quantum state in HHL-based algorithms generally requires exponential sampling, representing a data output problem.
“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]
HHL-based algorithms rely on amplitude encoding, which provides exponential memory advantage by compressing large grids into a small qubit register.
“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]
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“6.6 Quantummatrixinversion QRAMhasfurtherimplicationsforquantumalgorithms. Inparticular,thetaskofmatrix inversionadmitsanexponentialquantumspeedup,whichcouldhavemanyapplications. ThealgorithmwecallHHL[40](afterthethreeauthorswhofirstdiscusseditnineyears ago) takes as input a succinct representation of a very large N×N…”
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Quantum Simulation for High-Energy Physics
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Option Pricing using Quantum Computers
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Quantum Algorithm Implementations for Beginners
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Quantum Linear System Algorithm for Dense Matrices
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Quantum optical neural networks
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- 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 › Computer Science › Artificial Intelligence › 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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