The DOI matches its Crossref record; 35 of the 35 references it cites were found in a registry or an index.
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.
Overview
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“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]
The cited source says what the claim attributes to it.
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]
The cited source says what the claim attributes to it.
In the citing paper
“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]
The cited source says what the claim attributes to it.
In the citing paper
“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]
The cited source says what the claim attributes to it.
In the citing paper
“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.
In the citing paper
“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]
The cited source says what the claim attributes to it.
In the citing paper
“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]
The cited source says what the claim attributes to it.
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]
The cited source says what the claim attributes to it.
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]
The cited source says what the claim attributes to it.
In the citing paper
“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]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Citations
Cites
Algorithms for quantum computation: discrete logarithms and factoring
Shor, P. W. (2002) · Proceedings 35th Annual Symposium on Foundations of Computer Science · doi:10.1109/sfcs.1994.365700 · Cited by 8,713
A scheme for efficient quantum computation with linear optics
Knill, E., Laflamme, R., & Milburn, G. J. (2001) · Nature · doi:10.1038/35051009 · Cited by 5,880
The Mathematical Theory of Finite Element Methods
Brenner, S. C., & Scott, L. R. (2007) · Texts in applied mathematics · doi:10.1007/978-0-387-75934-0 · Cited by 4,009
Lloyd, S. (1996) · Science · doi:10.1126/science.273.5278.1073 · Cited by 3,088 · cited 2×
“[2] S.”
BackgroundSelf-citationIntroduction to the Theory of Computation
Sipser, M. (1996) · ACM SIGACT News · doi:10.1145/230514.571645 · Cited by 2,886
An Introduction to the Conjugate Gradient Method Without the Agonizing Pain
Shewchuk, J. R. (1994) · Cited by 2,193
Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion
Hansen, P. C. (1987) · Medical Entomology and Zoology · Cited by 2,142
A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem
Farhi, E., Goldstone, J., Gutmann, S., Lapan, J., Lundgren, A., & Preda, D. (2001) · Science · doi:10.1126/science.1057726 · Cited by 2,056
Quantum amplitude amplification and estimation
Brassard, G., Høyer, P., Mosca, M., & Tapp, A. (2002) · Contemporary Mathematics · doi:10.1090/conm/305/05215 · Cited by 1,680
Cleve, R., EKERT, A. K., Macchiavello, C., & Mosca, M. (1998) · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rspa.1998.0164 · Cited by 1,399
Research works printed but not linked: 3
- P. W. Shor, 1994, Proc. of the 35th FOCS
- P. Valiant, 2008, Proc. of the 40th Annual STOC
- L. Sheridan, 2009, J. Math Phys. A, 42, 185302
Cited by
Sorted by citations
Items 901–1,000 of 1,006
Quantum Algorithm for Boolean Equation Solving and Quantum Algebraic Attack on Cryptosystems
Chen, Y.-A., & Gao, X.-S. (2021) · Journal of Systems Science and Complexity · doi:10.1007/s11424-020-0028-6 · Cited by 28
Implementing smooth functions of a hermitian matrix on a quantum computer
Subramanian, S., Brierley, S., Jozsa, R., et al. (2019) · Cambridge University Engineering Department Publications Database · Cited by 19
Quantum Speedup for Graph Sparsification, Cut Approximation and Laplacian Solving
Apers, S., & de Wolf, R. (2020) · Annual Symposium on Foundations of Computer Science · doi:10.1109/focs46700.2020.00065 · Cited by 22
Shaik, R. U., & Periasamy, S. (2022) · International Journal of Remote Sensing · doi:10.1080/01431161.2022.2061877 · Cited by 20
A novel fault-tolerant quantum divider and its simulation
Yuan, S., Gao, S., Wen, C., Wang, Y., Qu, H., & Wang, Y. (2022) · Quantum Information Processing · doi:10.1007/s11128-022-03523-8 · Cited by 19
QMLP: An Error-Tolerant Nonlinear Quantum MLP Architecture using Parameterized Two-Qubit Gates
Chu, C., Chia, N.-H., Jiang, L., & Chen, F. (2022) · International Symposium on Low Power Electronics and Design (ISLPED) · doi:10.1145/3531437.3539719 · Cited by 20
Compact quantum kernel-based binary classifier
Blank, C., da Silva, A. J., de Albuquerque, L. P., Petruccione, F., & Park, D. K. (2022) · Quantum Science and Technology · doi:10.1088/2058-9565/ac7ba3 · Cited by 21
Improved circuit implementation of the HHL algorithm and its simulations on QISKIT
Zhang, M., Dong, L., Zeng, Y., & Cao, N. (2022) · Scientific Reports · doi:10.1038/s41598-022-17660-8 · Cited by 22
Variational quantum extreme learning machine
Wang, Y., Lin, K.-Y., Cheng, S., & Li, L. (2022) · Neurocomputing · doi:10.1016/j.neucom.2022.09.068 · Cited by 20
Quantum Fuzzy Support Vector Machine for Binary Classification
Huang, X., Zhang, S., Chen, L., & Xia, J. (2022) · Computer Systems Science and Engineering · doi:10.32604/csse.2023.032190 · Cited by 19
An Inexact Feasible Quantum Interior Point Method for Linearly Constrained Quadratic Optimization
Wu, Z., Mohammadisiahroudi, M., Augustino, B., Yang, X., & Tamás Terlaky, B. (2023) · Entropy · doi:10.3390/e25020330 · Cited by 20
Depth analysis of variational quantum algorithms for the heat equation
Guseynov, N., Zhukov, A. A., Pogosov, W. V., & Lebedev, A. V. (2023) · Physical Review A · doi:10.1103/physreva.107.052422 · Cited by 25
Wen, J., Wang, Z., Huang, Z., Cai, D., Jia, B., Cao, C., Ma, Y., Wei, H., Wen, K., & Qian, L. (2023) · Science China Physics Mechanics and Astronomy · doi:10.1007/s11433-023-2147-3 · Cited by 20
Trainability and Expressivity of Hamming-Weight Preserving Quantum Circuits for Machine Learning
Monbroussou, L., Mamon, E. Z., Landman, J., Grilo, A. B., Kukla, R., & Kashefi, E. (2025) · Quantum · doi:10.22331/q-2025-05-15-1745 · Cited by 27
What is Quantum Parallelism, Anyhow?
Markidis, S. (2024) · ISC High Performance (ISC) · doi:10.23919/isc.2024.10528926 · Cited by 26
Polylogarithmic-depth controlled-NOT gates without ancilla qubits
Claudon, B., Zylberman, J., Feniou, C., Debbasch, F., Peruzzo, A., & Piquemal, J. (2024) · Nature Communications · doi:10.1038/s41467-024-50065-x · Cited by 28
Olivares-Sánchez, J., Casanova, J., Solano, E., & Lamata, L. (2020) · Quantum Reports · doi:10.3390/quantum2020019 · Cited by 19
Algorithmic cooling in liquid-state nuclear magnetic resonance
Atia, Y., Elias, Y., Mor, T., & Weinstein, Y. (2016) · Physical Review A · doi:10.1103/physreva.93.012325 · Cited by 19
A Complete Characterization of Unitary Quantum SpacePreprint
Fefferman, B., & Lin, C. Y.-Y. (2016) · arXiv (Cornell University) · doi:10.4230/lipics.itcs.2018.4 · Cited by 37
Superconductive Electronics: A 25-Year Review [Feature]
Bairamkulov, R., & Micheli, G. D. (2024) · IEEE Circuits and Systems Magazine · doi:10.1109/mcas.2024.3376492 · Cited by 21
Blockchain Security Risk Assessment in Quantum Era, Migration Strategies, and Proactive Defense
Baseri, Y., Hafid, A. S., Shahsavari, Y., Makrakis, D., & Khodaiemehr, H. (2025) · IEEE Communications Surveys & Tutorials · doi:10.1109/comst.2025.3621113 · Cited by 25
Quantum lattice Boltzmann method for simulating nonlinear fluid dynamics
Wang, B., Meng, Z., Zhao, Y., & Yang, Y. (2025) · npj Quantum Information · doi:10.1038/s41534-025-01142-6 · Cited by 20
An Introduction To Quantum Machine Learning Techniques
Pushpak, S. N., & Jain, S. (2021) · International Conference on Reliability, Infocom Technologies and Optimization (Trends and Future Directions) (ICRITO) · doi:10.1109/icrito51393.2021.9596240 · Cited by 18
Magic state injection on the rotated surface code
Lao, L., & Criger, B. (2022) · ACM International Conference on Computing Frontiers (CF) · doi:10.1145/3528416.3530237 · Cited by 25
Implications of Quantum Science on Industry 4.0: Challenges and Opportunities
Ahuja, N. J., & Dutt, S. (2022) · Lecture notes on data engineering and communications technologies · doi:10.1007/978-3-031-04613-1_6 · Cited by 19
Investigating hardware acceleration for simulation of CFD quantum circuits
Moawad, Y., Vanderbauwhede, W. A., & Steijl, R. (2022) · Frontiers in Mechanical Engineering · doi:10.3389/fmech.2022.925637 · Cited by 18
A quantum algorithm for heat conduction with symmetrization
Wei, S.-J., Wei, C., Lv, P., Shao, C., Gao, P., Zhou, Z., Li, K., Xin, T., & Long, G. (2023) · 中国科学通报：英文版 · doi:10.1016/j.scib.2023.02.016 · Cited by 18
Universal compilation for quantum state tomography
Hai, V. T., & Ho, L. B. (2023) · Scientific Reports · doi:10.1038/s41598-023-30983-4 · Cited by 16
Synthesizing efficient circuits for Hamiltonian simulation
Mukhopadhyay, P., Wiebe, N., & Zhang, H. T. (2023) · npj Quantum Information · doi:10.1038/s41534-023-00697-6 · Cited by 20
Applications of the vortex-surface field to flow visualization, modelling and simulation
Yang, Y., Xiong, S., & Zhen Hua, L. U. (2023) · Flow · doi:10.1017/flo.2023.27 · Cited by 17
Benefits of Open Quantum Systems for Quantum Machine Learning
Olivera‐Atencio, M. L., Lamata, L., & Casado‐Pascual, J. (2023) · Advanced Quantum Technologies · doi:10.1002/qute.202300247 · Cited by 18
Spacetime-Efficient Low-Depth Quantum State Preparation with Applications
Gui, K., Dalzell, A. M., Achille, A., Suchara, M., & Chong, F. T. (2024) · Quantum · doi:10.22331/q-2024-02-15-1257 · Cited by 33
On the expressivity of embedding quantum kernels
Gil-Fuster, E., Eisert, J., & Dunjko, V. (2024) · Machine Learning Science and Technology · doi:10.1088/2632-2153/ad2f51 · Cited by 19
Exact distributed quantum algorithm for generalized Simon’s problem
Li, H., Qiu, D., Luo, L., & Mateus, P. (2024) · Acta Informatica · doi:10.1007/s00236-024-00455-x · Cited by 18
Synthetiq: Fast and Versatile Quantum Circuit Synthesis
Paradis, A., Dekoninck, J., Bichsel, B., & Vechev, M. (2024) · Proceedings of the ACM on Programming Languages · doi:10.1145/3649813 · Cited by 26
Driven multiphoton qubit-resonator interactions
Ayyash, M., Xu, X., Ashhab, S., & Mariantoni, M. (2024) · Physical Review A · doi:10.1103/physreva.110.053711 · Cited by 23
Nonlinear transformation of complex amplitudes via quantum singular value transformation
Guo, N., Mitarai, K., & Fujii, K. (2024) · Physical Review Research · doi:10.1103/physrevresearch.6.043227 · Cited by 18
Goto, H., & Uchikawa, H. (2013) · Scientific Reports · doi:10.1038/srep02044 · Cited by 17
A quantum heuristic algorithm for the traveling salesman problem
Bang, J., Ryu, J., Lee, C., Yoo, S., Lim, J., & Lee, J. (2012) · Journal of the Korean Physical Society · doi:10.3938/jkps.61.1944 · Cited by 18
Ito, T., & Jeffery, S. (2018) · Algorithmica · doi:10.1007/s00453-018-0527-1 · Cited by 27
Optimal usage of quantum random access memory in quantum machine learning
Bang, J., Dutta, A., Lee, S.-W., & Kim, J. (2019) · Physical Review A · doi:10.1103/physreva.99.012326 · Cited by 17
Fast Solution of Linear Systems with Analog Resistive Switching Memory (RRAM)
Sun, Z., Pedretti, G., & Ielmini, D. (2019) · IEEE International Conference on Rebooting Computing (ICRC) · doi:10.1109/icrc.2019.8914709 · Cited by 18
Doronin, S. I., Fel’dman, E. B., & Zenchuk, A. I. (2020) · Quantum Information Processing · doi:10.1007/s11128-019-2570-5 · Cited by 18
Implementation of a quantum transfer learning approach to image splicing detection
Gokhale, A., Pande, M. B., & Pramod, D. (2020) · International Journal of Quantum Information · doi:10.1142/s0219749920500240 · Cited by 18
Quantum advantage for differential equation analysis
Kiani, B. T., Palma, G. D., Englund, D. R., Kaminsky, W. M., Marvian, M., & Lloyd, S. (2022) · Physical Review A · doi:10.1103/physreva.105.022415 · Cited by 20
Self-citationQuantum Algorithms for Nonlinear Equations in Fluid Mechanics
Steijl, R. (2020) · IntechOpen eBooks · doi:10.5772/intechopen.95023 · Cited by 19
QKD-Enhanced Cybersecurity Protocols
Djordjević, I. B. (2021) · IEEE photonics journal · doi:10.1109/jphot.2021.3069510 · Cited by 24
A Quantum Algorithm for RF-based Fingerprinting Localization Systems
Shokry, A., & Youssef, M. A. (2022) · Conference on Local Computer Networks · doi:10.1109/lcn53696.2022.9843246 · Cited by 18
Schuld, M., & Petruccione, F. (2016) · Springer eBooks · doi:10.1007/978-1-4899-7502-7_913-1 · Cited by 19
Quantum Expectation-Maximization for Gaussian Mixture ModelsPreprint
Kerenidis, I., Luongo, A., & Prakash, A. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1908.06657 · Cited by 18
Heuristics for Quantum Compiling with a Continuous Gate SetPreprint
Davis, M., Smith, E., Tudor, A., Sen, K., Siddiqi, I. A., & Iancu, C. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1912.02727 · Cited by 29
A Hybrid Quantum-Classical Generative Adversarial Network for Near-Term Quantum Processors
Boyle, A. O., & Nikandish, G. (2024) · IEEE Access · doi:10.1109/access.2024.3433383 · Cited by 19
Quantum resource estimation for large scale quantum algorithms
Gheorghiu, V., & Mosca, M. (2024) · Future Generation Computer Systems · doi:10.1016/j.future.2024.107480 · Cited by 19
Quantum gradient descent and Newton’s method for constrained polynomial optimization
Rebentrost, P., Schuld, M., Wossnig, L., Petruccione, F., & Lloyd, S. (2019) · New Journal of Physics · doi:10.1088/1367-2630/ab2a9e · Cited by 152
Self-citationQuantum algorithm for total least squares data fitting
Wang, H., & Xiang, H. (2019) · Physics Letters A · doi:10.1016/j.physleta.2019.04.037 · Cited by 16
Orsucci, D., & Dunjko, V. (2021) · Quantum · doi:10.22331/q-2021-11-08-573 · Cited by 18
Gaussian Elimination versus Greedy Methods for the Synthesis of Linear Reversible Circuits
Brugière, T. G. D., Baboulin, M., Valiron, B., Martiel, S., & Allouche, C. (2021) · ACM Transactions on Quantum Computing · doi:10.1145/3474226 · Cited by 31
Quantum Attacks on 1K-AES and PRINCE
Cai, B., Wu, Y., Dong, J., Qin, S., Gao, F., & Wen, Q. (2021) · The Computer Journal · doi:10.1093/comjnl/bxab216 · Cited by 17
Intelligent recommender system based on quantum clustering and matrix completion
Ouedrhiri, O., Banouar, O., Hadaj, S. E., & Raghay, S. (2022) · Concurrency and Computation Practice and Experience · doi:10.1002/cpe.6943 · Cited by 18
Qurzon: A Prototype for a Divide and Conquer-Based Quantum Compiler for Distributed Quantum Systems
Chatterjee, T., Das, A., Mohtashim, S. I., Saha, A., & Chakrabarti, A. (2022) · SN Computer Science · doi:10.1007/s42979-022-01207-9 · Cited by 18
Hybrid classical-quantum machine learning based on dissipative two-qubit channels
Ghasemian, E., & Tavassoly, M. K. (2022) · Scientific Reports · doi:10.1038/s41598-022-24346-8 · Cited by 18
Carleman linearization approach for chemical kinetics integration toward quantum computation
Akiba, T., Morii, Y., & Maruta, K. (2023) · Scientific Reports · doi:10.1038/s41598-023-31009-9 · Cited by 17
Quantum neural network cost function concentration dependency on the parametrization expressivity
Friedrich, L., & Maziero, J. (2023) · Scientific Reports · doi:10.1038/s41598-023-37003-5 · Cited by 16
An exponentially-growing family of universal quantum circuits
Kordzanganeh, M., Sekatski, P., Fedichkin, L. E., & Melnikov, A. A. (2023) · Machine Learning Science and Technology · doi:10.1088/2632-2153/ace757 · Cited by 17
Yu, G., Wu, C., Liu, Y., Du, Z., Zhao, N., Jiao, Y., Xu, Z., & Shi, W. (2023) · Science China Earth Sciences · doi:10.1007/s11430-022-1144-7 · Cited by 19
Li, H., Wang, Z., & Fei, S.-M. (2023) · Physical Review A · doi:10.1103/physreva.108.032418 · Cited by 17
End-To-End Resource Analysis for Quantum Interior-Point Methods and Portfolio Optimization
Dalzell, A. M., Clader, B. D., Salton, G., Berta, M., Lin, C. Y.-Y., Bader, D. A., Stamatopoulos, N., Schuetz, M. J. A., Brandão, F. G. S. L., Katzgraber, H. G., et al. (2023) · PRX Quantum · doi:10.1103/prxquantum.4.040325 · Cited by 30
Complexity-Theoretic Limitations on Quantum Algorithms for Topological Data Analysis
Schmidhuber, A., & Lloyd, S. (2023) · PRX Quantum · doi:10.1103/prxquantum.4.040349 · Cited by 40
Self-citationKocherla, S., Song, Z., Chrit, F. E., Gard, B., Dumitrescu, E., Alexeev, A., & Bryngelson, S. H. (2024) · AVS Quantum Science · doi:10.1116/5.0217675 · Cited by 26
Quantum neural networks under depolarization noise: exploring white-box attacks and defenses
Winderl, D., Franco, N., & Lorenz, J. M. (2024) · Quantum Machine Intelligence · doi:10.1007/s42484-024-00208-6 · Cited by 23
Sato, Y., Tezuka, H., Kondo, R., & Yamamoto, N. (2025) · Physical Review Applied · doi:10.1103/physrevapplied.23.014063 · Cited by 18
Quantum Algorithm for Linear Non-unitary Dynamics with Near-Optimal Dependence on All Parameters
An, D., Childs, A. M., & Lin, L. (2025) · Communications in Mathematical Physics · doi:10.1007/s00220-025-05509-w · Cited by 99
Row and column iteration methods to solve linear systems on a quantum computer
Shao, C., & Xiang, H. (2020) · Physical Review A · doi:10.1103/physreva.101.022322 · Cited by 17
Training and Meta-Training Binary Neural Networks with Quantum Computing
Fawaz, A., Klein, P., Piat, S., Simone Severini, B., & Mountney, P. (2019) · ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD) · doi:10.1145/3292500.3330953 · Cited by 17
A Short Path Quantum Algorithm for Exact Optimization
Hastings, M. B. (2018) · Quantum · doi:10.22331/q-2018-07-26-78 · Cited by 17
Protocol for Implementing Quantum Nonparametric Learning with Trapped Ions
Zhang, D.-B., Zhu, S.-L., & Wang, Z. D. (2020) · Physical Review Letters · doi:10.1103/physrevlett.124.010506 · Cited by 16
Hybrid classical-quantum approach to solve the heat equation using quantum annealers
Pollachini, G. G., Salazar, J. P. L. C., Góes, C. B. D., Maciel, T. O., & Duzzioni, E. I. (2021) · Physical Review A · doi:10.1103/physreva.104.032426 · Cited by 18
$$Q|SI\rangle $$Q|SI⟩ : A Quantum Programming Environment
Liu, S., Wang, X., Zhou, L., Guan, J., Li, Y., He, Y., Duan, R., & Ying, M. (2018) · Lecture notes in computer science · doi:10.1007/978-3-030-01461-2_8 · Cited by 17
Schuld, M., & Petruccione, F. (2018) · Quantum science and technology · doi:10.1007/978-3-319-96424-9_5 · Cited by 17
Obtaining a linear combination of the principal components of a matrix on quantum computers
Daskin, A. (2016) · Quantum Information Processing · doi:10.1007/s11128-016-1388-7 · Cited by 17
Quantum speedup in solving the maximal-clique problem
Chang, W.-L., Yu, Q., Li, Z., Chen, J., Peng, X., & Feng, M. (2018) · Physical Review A · doi:10.1103/physreva.97.032344 · Cited by 17
Lin, J., Zhang, D.-B., Zhang, S., Li, T., Wang, X., & Bao, W. (2020) · Physics Letters A · doi:10.1016/j.physleta.2020.126590 · Cited by 17
Logic Synthesis for Quantum ComputingPreprint
Soeken, M., Roetteler, M., Wiebe, N., & Micheli, G. D. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1706.02721 · Cited by 17
Quantum Simulation of Klein Gordon Equation and Observation of Klein Paradox in IBM Quantum ComputerPreprint
Kapil, M., Behera, B. K., & Panigrahi, P. K. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1807.00521 · Cited by 19
Assessing the potential of quantum computing in agriculture
Pook, T., Vandenplas, J., Boschero, J. C., Aguilera, E., Leijnse, K., Chauhan, A., Bouzembrak, Y., Knapen, R., & Aldridge, M. N. (2025) · Computers and Electronics in Agriculture · doi:10.1016/j.compag.2025.110332 · Cited by 18
Guseynov, N., Huang, X., & Liu, N. (2025) · Physical Review Research · doi:10.1103/xlpd-fb1g · Cited by 18
Sun, J., & Wang, W. (2025) · Computer Science Review · doi:10.1016/j.cosrev.2025.100842 · Cited by 16
Quantum Version of the k‐NN Classifier Based on a Quantum Sorting Algorithm
Quezada, L. F., Sun, G.-H., & Dong, S. (2022) · Annalen der Physik · doi:10.1002/andp.202100449 · Cited by 16
Nguyen, Q. T., Kiani, B. T., & Lloyd, S. (2022) · Quantum · doi:10.22331/q-2022-12-13-876 · Cited by 16
Self-citationIterative quantum-phase-estimation protocol for shallow circuits
Smith, J. G., Barnes, C. H. W., & Arvidsson-Shukur, D. R. M. (2022) · Physical Review A · doi:10.1103/physreva.106.062615 · Cited by 16
Liu, B., Zhu, L., Yang, Z., & He, G. (2023) · Communications in Computational Physics · doi:10.4208/cicp.oa-2022-0081 · Cited by 14
Quantum Circuit Design for Integer Multiplication Based on Schönhage–Strassen Algorithm
Nie, J., Zhu, Q., Li, M., & Sun, X. (2023) · IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems · doi:10.1109/tcad.2023.3279300 · Cited by 15
On efficient quantum block encoding of pseudo-differential operators
Li, H., Ni, H., & Lexing Ying, B. (2023) · Quantum · doi:10.22331/q-2023-06-02-1031 · Cited by 15
Moreira, M. S., Guerreschi, G. G., Vlothuizen, W., Marques, J., van Straten, J., Premaratne, S., Zou, X., Ali, H. A., Muthusubramanian, N., Zachariadis, C., et al. (2023) · npj Quantum Information · doi:10.1038/s41534-023-00779-5 · Cited by 16
Complexity of life sciences in quantum and AI era
Pyrkov, A. N., Aliper, A., Bezrukov, D. S., Podolskiy, D. I., Ren, F., & Zhavoronkov, A. A. (2024) · Wiley Interdisciplinary Reviews Computational Molecular Science · doi:10.1002/wcms.1701 · Cited by 18
Quantum Random Access Memory Architectures Using 3D Superconducting Cavities
Weiss, D., Puri, S., & Girvin, S. M. (2024) · PRX Quantum · doi:10.1103/prxquantum.5.020312 · Cited by 23
Variational Quantum Algorithms for Semidefinite Programming
Patel, D., Coles, P. J., & Wilde, M. M. (2024) · Quantum · doi:10.22331/q-2024-06-17-1374 · Cited by 22
Hamiltonian dynamics on digital quantum computers without discretization error
Granet, E., & Dreyer, H. (2024) · npj Quantum Information · doi:10.1038/s41534-024-00877-y · Cited by 24
Potential of quantum scientific machine learning applied to weather modeling
Jaderberg, B., Gentile, A. A., Ghosh, A., Elfving, V. E., Jones, C. R., Vodola, D., Manobianco, J., & Weiß, H. (2024) · Physical Review A · doi:10.1103/physreva.110.052423 · Cited by 22
On proving the robustness of algorithms for early fault-tolerant quantum computers
Kshirsagar, R., Katabarwa, A., & Johnson, P. (2024) · Quantum · doi:10.22331/q-2024-11-20-1531 · Cited by 16
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
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
