Variational Quantum Linear Solver

Carlos Bravo-Prieto1,2,3, Ryan LaRose4, M. Cerezo1,5, Yigit Subasi6, Lukasz Cincio1, and Patrick J. Coles1

1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA.
2Barcelona Supercomputing Center, Barcelona, Spain.
3Institut de Ciències del Cosmos, Universitat de Barcelona, Barcelona, Spain.
4Department of Computational Mathematics, Science, and Engineering & Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48823, USA.
5Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM, USA
6Computer, Computational and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

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Abstract

Previously proposed quantum algorithms for solving linear systems of equations cannot be implemented in the near term due to the required circuit depth. Here, we propose a hybrid quantum-classical algorithm, called Variational Quantum Linear Solver (VQLS), for solving linear systems on near-term quantum computers. VQLS seeks to variationally prepare $|x\rangle$ such that $A|x\rangle\propto|b\rangle$. We derive an operationally meaningful termination condition for VQLS that allows one to guarantee that a desired solution precision $\epsilon$ is achieved. Specifically, we prove that $C \geqslant \epsilon^2 / \kappa^2$, where $C$ is the VQLS cost function and $\kappa$ is the condition number of $A$. We present efficient quantum circuits to estimate $C$, while providing evidence for the classical hardness of its estimation. Using Rigetti's quantum computer, we successfully implement VQLS up to a problem size of $1024\times1024$. Finally, we numerically solve non-trivial problems of size up to $2^{50}\times2^{50}$. For the specific examples that we consider, we heuristically find that the time complexity of VQLS scales efficiently in $\epsilon$, $\kappa$, and the system size $N$.

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[176] Hyeong-Gyu Kim, Siheon Park, and June-Koo Kevin Rhee, "Variational Quantum Approximate Spectral Clustering for Binary Clustering Problems", arXiv:2309.04465, (2023).

[177] Yulun Wang and Predrag S. Krstić, "Multistate transition dynamics by strong time-dependent perturbation in NISQ era", Journal of Physics Communications 7 7, 075004 (2023).

[178] Payal Kaushik, Sayantan Pramanik, M Girish Chandra, and C V Sridhar, "One-Step Time Series Forecasting Using Variational Quantum Circuits", arXiv:2207.07982, (2022).

[179] Mina Doosti, "Unclonability and Quantum Cryptanalysis: From Foundations to Applications", arXiv:2210.17545, (2022).

[180] Aidan Pellow-Jarman, Ilya Sinayskiy, Anban Pillay, and Francesco Petruccione, "Near term algorithms for linear systems of equations", Quantum Information Processing 22 6, 258 (2023).

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[182] Junyu Liu, Han Zheng, Masanori Hanada, Kanav Setia, and Dan Wu, "Quantum Power Flows: From Theory to Practice", arXiv:2211.05728, (2022).

[183] Shao-Hen Chiew and Leong-Chuan Kwek, "Scalable Quantum Computation of Highly Excited Eigenstates with Spectral Transforms", arXiv:2302.06638, (2023).

[184] Yangyang Liu, Zhen Chen, Chang Shu, Patrick Rebentrost, Yaguang Liu, S. C. Chew, B. C. Khoo, and Y. D. Cui, "A variational quantum algorithm-based numerical method for solving potential and Stokes flows", arXiv:2303.01805, (2023).

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[187] Leonardo Zambrano, Andrés Damián Muñoz-Moller, Mario Muñoz, Luciano Pereira, and Aldo Delgado, "Avoiding barren plateaus in the variational determination of geometric entanglement", arXiv:2304.13388, (2023).

[188] Dylan Herman, Rudy Raymond, Muyuan Li, Nicolas Robles, Antonio Mezzacapo, and Marco Pistoia, "Expressivity of Variational Quantum Machine Learning on the Boolean Cube", arXiv:2204.05286, (2022).

[189] Stefano Mangini, Alessia Marruzzo, Marco Piantanida, Dario Gerace, Daniele Bajoni, and Chiara Macchiavello, "Quantum neural network autoencoder and classifier applied to an industrial case study", arXiv:2205.04127, (2022).

[190] Yunya Liu, Jiakun Liu, Jordan R. Raney, and Pai Wang, "Quantum Computing for Solid Mechanics and Structural Engineering -- a Demonstration with Variational Quantum Eigensolver", arXiv:2308.14745, (2023).

[191] Hansheng Jiang, Zuo-Jun Max Shen, and Junyu Liu, "Quantum Computing Methods for Supply Chain Management", arXiv:2209.08246, (2022).

[192] Pablo Bermejo, Borja Aizpurua, and Roman Orus, "Improving Gradient Methods via Coordinate Transformations: Applications to Quantum Machine Learning", arXiv:2304.06768, (2023).

[193] Jessie M. Henderson, Marianna Podzorova, M. Cerezo, John K. Golden, Leonard Gleyzer, Hari S. Viswanathan, and Daniel O'Malley, "Quantum algorithms for geologic fracture networks", Scientific Reports 13, 2906 (2023).

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[196] Oliver Knitter, James Stokes, and Shravan Veerapaneni, "Toward Neural Network Simulation of Variational Quantum Algorithms", arXiv:2211.02929, (2022).

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[201] Anton Simen Albino, Otto Menegasso Pires, Peterson Nogueira, Renato Ferreira de Souza, and Erick Giovani Sperandio Nascimento, "Quantum computational intelligence for traveltime seismic inversion", arXiv:2208.05794, (2022).

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[203] Ruimin Shang, Zhimin Wang, Shangshang Shi, Jiaxin Li, Yanan Li, and Yongjian Gu, "Algorithm for simulating ocean circulation on a quantum computer", Science China Earth Sciences 66 10, 2254 (2023).

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[206] Willie Aboumrad and Dominic Widdows, "Mod2VQLS: a Variational Quantum Algorithm for Solving Systems of Linear Equations Modulo 2", arXiv:2311.12771, (2023).

[207] Nicolas PD Sawaya and Joonsuk Huh, "Improved resource-tunable near-term quantum algorithms for transition probabilities, with applications in physics and variational quantum linear algebra", arXiv:2206.14213, (2022).

[208] Minati Rath and Hema Date, "Quantum-Assisted Simulation: A Framework for Designing Machine Learning Models in the Quantum Computing Domain", arXiv:2311.10363, (2023).

[209] Yoshiyuki Saito, Xinwei Lee, Dongsheng Cai, and Nobuyoshi Asai, "Quantum Multi-Resolution Measurement with application to Quantum Linear Solver", arXiv:2304.05960, (2023).

[210] Po-Wei Huang, Xiufan Li, Kelvin Koor, and Patrick Rebentrost, "Hybrid quantum-classical and quantum-inspired classical algorithms for solving banded circulant linear systems", arXiv:2309.11451, (2023).

[211] Dingjie Lu, Zhao Wang, Jun Liu, Yangfan Li, Wei-Bin Ewe, and Zhuangjian Liu, "From Ad-Hoc to Systematic: A Strategy for Imposing General Boundary Conditions in Discretized PDEs in variational quantum algorithm", arXiv:2310.11764, (2023).

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The above citations are from Crossref's cited-by service (last updated successfully 2024-02-27 21:24:13) and SAO/NASA ADS (last updated successfully 2024-02-27 21:24:15). The list may be incomplete as not all publishers provide suitable and complete citation data.