量子计算加速的解法器算法及应用综述

doi:10.19596/j.cnki.1001-246x.8778

摘要/Abstract

摘要：

量子计算作为一种基于量子力学原理的全新计算模型, 具有强大的并行性和潜在的颠覆性影响力, 为解决复杂问题提供了新的思路。本文的主要目标是对量子计算在大规模科学与工程计算领域中数值计算问题的解法器算法和应用进行综述。重点介绍量子计算在线性方程组、特征值问题、微分方程、哈密顿量与图计算、量子机器学习、量子解法器平台以及实际数值模拟等领域的具体应用。针对不同的数值计算问题, 详细讨论当前主流的量子计算算法, 并总结近年来国内外相关算法的研究进展。最后, 对量子计算在数值计算求解相关研究方向的未来发展趋势进行展望。

关键词: 量子计算, 量子解法器, 量子并行性, 方程求解问题

Abstract:

Quantum computing is a new computing model based on the principles of quantum mechanics. Because of its powerful parallelism far superior to classical computing, quantum computing is considered as a computational method that may have a subversive impact on the future, providing a new way to solve some complex problems. The algorithms and applications of quantum solvers in numerical computation-related problems of large-scale science and engineering are reviewed. In particular, systems of linear equations, eigenvalue problems, differential equations, Hamiltonian and graph computation, quantum machine learning, quantum solver platform, and practical numerical simulation have been introduced. Aiming at different numerical computing problems, the current mainstream quantum computing algorithms are introduced in detail, and the research progress of relevant algorithms at home and abroad in recent years is comprehensively summarized. Finally, the future development trend of quantum computing in numerical algebra solving is prospected.

Key words: quantum computing, quantum solver, quantum parallelism, equation solving

中图分类号:

TP3-0

许康, 李泽阳, 郭竹丰, 沈颖童, 王威, 缑敏辉, 王子正, 王玉坤, 刘伟峰. 量子计算加速的解法器算法及应用综述[J]. 计算物理, 2024, 41(1): 131-150.

Kang XU, Zeyang LI, Zhufeng GUO, Yingtong SHEN, Wei WANG, Minhui GOU, Zizheng WANG, Yukun WANG, Weifeng LIU. A Review of Algorithms and Applications of Solvers with Quantum Computing Acceleration[J]. Chinese Journal of Computational Physics, 2024, 41(1): 131-150.

图/表 3

图1 HHL算法整体线路图

Fig.1 Overall circuit diagram of HHL algorithm

图2 变分量子算法示意图(使用经典优化器训练参数化量子电路，其中量子计算机采用含参数的酉门来处理整个算法计算复杂度高的部分，经典计算机负责处理参数的迭代更新(比如梯度下降算法)部分。)

Fig.2 Schematic diagram of the Variational Quantum Algorithm (The classical optimizer is used to train theparameterized quantum circuits, in which quantum computer uses the parameterized unitary gates to process thecomputationally difficult subroutines of the whole algorithm and the classical computer is responsible for theiterative updating of parameters, such as gradient descent algorithm.)

表1 部分量子云平台使用的语言和性能参数

Table 1 The language and performance parameters used by some quantum cloud platforms

量子云平台	语言	性能
百度量易伏	Python, C++	8个量子比特T1^①: 31.0 μs T2^②: 8.7 μs保真度^③：99.80%
华为HiQ	Python	11个量子比特
本源	Python, C++	12个量子比特T1: 20.82 μs T2: 4.14 μs保真度：99.78%
IBM	Python	433个量子比特T1: 91.12 μs T2: 59.94 μs保真度：99.47%

参考文献 138

1	FEYNMAN R P .Simulating physics with computers[J].International Journal of Theoretical Physics,1982,21(6):467-488.
2	DEUTSCH D .Quantum theory, the Church-Turing principle and the universal quantum computer[J].Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences,1985,400(1818):97-117.
3	BERNSTEIN E , VAZIRANI U .Quantum complexity theory[J].SIAM Journal on Computing,1997,26(5):1411-1473. DOI
4	CHI-CHIH YAO A. Quantum circuit complexity[C]//Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science. Palo Alto, CA, USA: IEEE, 1993: 352-361.
5	DEUTSCH D , JOZSA R .Rapid solution of problems by quantum computation[J].Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences,1992,439(1907):553-558.
6	SIMON D R .On the power of quantum computation[J].SIAM Journal on Computing,1997,26(5):1474-1483. DOI
7	SHOR P W. Algorithms for quantum computation: discrete logarithms and factoring[C]//Proceedings 35th Annual Symposium on Foundations of Computer Science. Santa Fe, NM, USA: IEEE, 1994: 124-134.
8	GROVER L K. A fast quantum mechanical algorithm for database search[C]//Proceedings of the twenty-eighth annual ACM symposium on Theory of Computing. Philadelphia, Pennsylvania, USA: Association for Computing Machinery, 1996: 212-219.
9	FARHI E, GOLDSTONE J, GUTMANN S. A quantum approximate optimization algorithm[EB/OL]. (2014-11-14)[2023-05-10]. https://doi.org/10.48550/arXiv.1411.4028.
10	HARROW A W , HASSIDIM A , LLOYD S .Quantum algorithm for linear systems of equations[J].Physical Review Letters,2009,103(15):150502. DOI
11	REBENTROST P , SCHULD M , WOSSNIG L , et al.Quantum gradient descent and Newton's method for constrained polynomial optimization[J].New Journal of Physics,2019,21(7):073023. DOI
12	KERENIDIS I , PRAKASH A .Quantum gradient descent for linear systems and least squares[J].Physical Review A,2020,101(2):022316. DOI
13	SHAO Changpeng , XIANG Hua .Row and column iteration methods to solve linear systems on a quantum computer[J].Physical Review A,2020,101(2):022322. DOI
14	AMBAINIS A. Variable time amplitude amplification and quantum algorithms for linear algebra problems[C]//29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Paris, France: Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2012: 636-647.
15	CHILDS A M , KOTHARI R , SOMMA R D .Quantum algorithm for systems of linear equations with exponentially improved dependence on precision[J].SIAM Journal on Computing,2017,46(6):1920-1950. DOI
16	GILYÉN A, SU Yuan, LOW G H, et al. Quantum singular value transformation and beyond: Exponential improvements for quantum matrix arithmetics[C]//Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing. Phoenix, AZ, USA: Association for Computing Machinery, 2019: 193-204.
17	WOSSNIG L , ZHAO Zhikuan , PRAKASH A .Quantum linear system algorithm for dense matrices[J].Physical Review Letters,2018,120(5):050502. DOI
18	WANG Chunhao , WOSSNIG L .A quantum algorithm for simulating non-sparse Hamiltonians[J].Quantum Information and Computation,2020,20(7/8):597-615.
19	CLADER B D , JACOBS B C , SPROUSE C R .Preconditioned quantum linear system algorithm[J].Physical Review Letters,2013,110(25):250504. DOI
20	SHAO Changpeng , XIANG Hua .Quantum circulant preconditioner for a linear system of equations[J].Physical Review A,2018,98(6):062321. DOI
21	PERUZZO A , MCCLEAN J , SHADBOLT P , et al.A variational eigenvalue solver on a photonic quantum processor[J].Nature Communications,2014,5,4213. DOI
22	XU Xiaosi , SUN Jinzhao , ENDO S , et al.Variational algorithms for linear algebra[J].Science Bulletin,2021,66(21):2181-2188. DOI
23	BRAVO-PRIETO C, LAROSE R, CEREZO M, et al. Variational quantum linear solver[EB/OL]. (2020-06-02)[2023-05-10]. https://doi.org/10.48550/arXiv.1909.05820.
24	AN Dong , LIN Lin .Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm[J].ACM Transactions on Quantum Computing,2022,3(2):1-28.
25	HUANG H Y , BHARTI K , REBENTROST P .Near-term quantum algorithms for linear systems of equations with regression loss functions[J].New Journal of Physics,2021,23(11):113021. DOI
26	ZHENG Yarui , SONG Chao , CHEN Mingcheng , et al.Solving systems of linear equations with a superconducting quantum processor[J].Physical Review Letters,2017,118(21):210504. DOI
27	PAN Jian , CAO Yudong , YAO Xiwei , et al.Experimental realization of quantum algorithm for solving linear systems of equations[J].Physical Review A,2014,89(2):022313. DOI
28	BARZ S , KASSAL I , RINGBAUER M , et al.A two-qubit photonic quantum processor and its application to solving systems of linear equations[J].Scientific Reports,2014,4(1):6115. DOI
29	WEN Jingwei , KONG Xiangyu , WEI Shijie , et al.Experimental realization of quantum algorithms for a linear system inspired by adiabatic quantum computing[J].Physical Review A,2019,99(1):012320. DOI
30	QIAN Peng, HUANG Weicong, LONG Guilu. A quantum algorithm for solving systems of nonlinear algebraic equations[EB/OL]. (2019-03-14)[2023-05-12]. https://doi.org/10.48550/arXiv.1903.05608.
31	RIZK-ALLAH R M .A quantum-based sine cosine algorithm for solving general systems of nonlinear equations[J].Artificial Intelligence Review,2021,54(5):3939-3990. DOI
32	XUE Cheng , WU Yuchun , GUO Guoping .Quantum newton's method for solving the system of nonlinear equations[J].SPIN,2021,11(3):2140004. DOI
33	JIN Lei , SHAN Zheng , HU Chunchao , et al.Higher-order quantum genetic algorithm and its application on non-linear equations[J].Journal of Physics: Conference Series,2021,1827(1):012159. DOI
34	XUE Cheng , XU Xiaofan , WU Yuchun , et al.Quantum algorithm for solving a quadratic nonlinear system of equations[J].Physical Review A,2022,106(3):032427. DOI
35	BERRY D W .High-order quantum algorithm for solving linear differential equations[J].Journal of Physics A: Mathematical and Theoretical,2014,47(10):105301. DOI
36	BIAMONTE J , WITTEK P , PANCOTTI N , et al.Quantum machine learning[J].Nature,2017,549(7671):195-202. DOI
37	HESTENES M R , STIEFEL E .Methods of conjugate gradients for solving linear systems[J].Journal of Research of the National Bureau of Standards,1952,49(6):409-436. DOI
38	CEREZO M , ARRASMITH A , BABBUSH R , et al.Variational quantum algorithms[J].Nature Reviews Physics,2021,3(9):625-644. DOI
39	朱凯博, 徐龙飞, 潘流俊, 等.JFNK方法在S N输运计算中的应用[J].计算物理,2021,38(4):381-392.
40	李帅龙, 崔国民, 陈家星, 等.一种采用种群多样性监控和实时更新策略的粒子群优化算法[J].计算物理,2017,34(3):344-354. DOI
41	FRANCIS J G F .The QR transformation-part 2[J].The Computer Journal,1962,4(4):332-345. DOI
42	KUBLANOVSKAYA V N .On some algorithms for the solution of the complete eigenvalue problem[J].USSR Computational Mathematics and Mathematical Physics,1962,1(3):637-657. DOI
43	JACOBI C G J .Vber ein leichtes Verfahren die in der Theorie der Säcularstörungen vorkommenden Gleichungen numerisch aufzulösen[J].Journal für Die Reine und Angewandte Mathematik,2009,1846(30):51-94.
44	GOLUB G H , VAN DER VORST H A .Eigenvalue computation in the 20th century[J].Journal of Computational and Applied Mathematics,2000,123(1/2):35-65.
45	ABRAMS D S , LLOYD S .Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors[J].Physical Review Letters,1999,83(24):5162-5165. DOI
46	WANG Daochen , HIGGOTT O , BRIERLEY S .Accelerated variational quantum eigensolver[J].Physical Review Letters,2019,122(14):140504. DOI
47	D'CUNHA R, OTTEN M, HERMES M R, et al. State preparation in quantum algorithms for fragment-based quantum chemistry[EB/OL]. (2023-06-07)[2023-05-11]. https://doi.org/10.48550/arXiv.2305.18110.
48	DOBŠÍČGEK M , JOHANSSON G , SHUMEIKO V , et al.Arbitrary accuracy iterative quantum phase estimation algorithm using a single ancillary qubit: A two-qubit benchmark[J].Physical Review A,2007,76(3):030306. DOI
49	WANG Hefeng , WU Lianao , LIU Yuxi , et al.Measurement-based quantum phase estimation algorithm for finding eigenvalues of non-unitary matrices[J].Physical Review A,2010,82(6):062303. DOI
50	吴绍君. 量子强化学习算法及其在特征值问题上的应用[D]. 成都: 电子科技大学, 2021.
51	赵乾坤.量子迭代相位估计算法在IBM超导平台的实现[J].福建电脑,2023,39(2):6-10.
52	RYABINKIN I G , GENIN S N , IZMAYLOV A F .Constrained variational quantum eigensolver: Quantum computer search engine in the Fock space[J].Journal of Chemical Theory and Computation,2019,15(1):249-255. DOI
53	LIANG Jinmin , SHEN Shuqian , LI Ming , et al.Variational quantum algorithms for dimensionality reduction and classification[J].Physical Review A,2020,101(3):032323. DOI
54	WEI Shijie , LI Hang , LONG Guilu .A full quantum eigensolver for quantum chemistry simulations[J].Research,2020,2020,1486935.
55	ZHANG Danbo , CHEN Binlin , YUAN Zhanhao , et al.Variational quantum eigensolvers by variance minimization[J].Chinese Physics B,2022,31(12):120301. DOI
56	张欣, 赵国忠, 李宏.局部间断Petrov-Galerkin方法在大气污染模型中的应用[J].计算物理,2021,38(2):171-182.
57	KIANI B T , DE PALMA G , ENGLUND D , et al.Quantum advantage for differential equation analysis[J].Physical Review A,2022,105(2):022415. DOI
58	KIM S , KARRILA S J .Microhydrodynamics: Principles and selected applications[M].Boston: Butterworth-Heinemann,2013.
59	BERRY D W , CHILDS A M , OSTRANDER A , et al.Quantum algorithm for linear differential equations with exponentially improved dependence on precision[J].Communications in Mathematical Physics,2017,356(3):1057-1081. DOI
60	SOMMA R .Quantum linear systems algorithm with exponentially improved dependence on precision[J].SIAM Journal on Computing,2015,46(6):1920-1950.
61	LEYTON S K, OSBORNE T J. A quantum algorithm to solve nonlinear differential equations[EB/OL]. (2008-12-23)[2023-05-11]. https://doi.org/10.48550/arXiv.0812.4423.
62	LIU Jinpeng , KOLDEN H Ø , KROVI H K , et al.Efficient quantum algorithm for dissipative nonlinear differential equations[J].Proceedings of the National Academy of Sciences of the United States of America,2021,118(35):e2026805118.
63	CARLEMAN T .Application de la théorie des équations intégrales linéaires aux systèmes d'équations différentielles non linéaires[J].Acta Mathematica,1932,59(1):63-87.
64	CHEN Guoting , DELLA DORA J .An algorithm for computing a new normal form for dynamical systems[J].Journal of Symbolic Computation,2000,29(3):393-418. DOI
65	LLOYD S, DE PALMA G, GOKLER C, et al. Quantum algorithm for nonlinear differential equations[EB/OL]. (2020-12-21)[2023-05-11]. https://doi.org/10.48550/arXiv.2011.06571.
66	BENEDIKTER N , PORTA M , SCHLEIN B .Effective evolution equations from quantum dynamics[M].Cham: Springer,2016.
67	TROTTER H F .On the product of semi-groups of operators[J].Proceedings of the American Mathematical Society,1959,10(4):545-551. DOI
68	COSTA P C S , JORDAN S , OSTRANDER A .Quantum algorithm for simulating the wave equation[J].Physical Review A,2019,99(1):012323. DOI
69	LUBASCH M , JOO J , MOINIER P , et al.Variational quantum algorithms for nonlinear problems[J].Physical Review A,2020,101(1):010301. DOI
70	KYRⅡENKO O , PAINE A E , ELFVING V E .Solving nonlinear differential equations with differentiable quantum circuits[J].Physical Review A,2021,103(5):052416. DOI
71	MARTOŇÁK R , SANTORO G E , TOSATTI E .Quantum annealing of the traveling-salesman problem[J].Physical Review E,2004,70(5):057701. DOI
72	PELOFSKE E, HAHN G, DJIDJEV H. Solving large maximum clique problems on a quantum annealer[C]//QTOP 2019: Quantum Technology and Optimization Problems. Munich, Germany: Springer, 2019: 123-135.
73	CALUDE C S , DINNEEN M J , HUA R .QUBO formulations for the graph isomorphism problem and related problems[J].Theoretical Computer Science,2017,701,54-69. DOI
74	USHIJIMA-MWESIGWA H, NEGRE C F A, MNISZEWSKI S M. Graph partitioning using quantum annealing on the D-Wave system[C]//Proceedings of the Second International Workshop on Post Moores Era Supercomputing. Denver, CO, USA: Association for Computing Machinery, 2017: 22-29.
75	HARIKRISHNAKUMAR R, NANNAPANENI S, NGUYEN N H, et al. A quantum annealing approach for dynamic multi-depot capacitated vehicle routing problem[EB/OL]. (2020-05-27)[2023-05-11]. https://doi.org/10.48550/arXiv.2005.12478.
76	SRINIVASAN K, SATYAJIT S, BEHERA B K, et al. Efficient quantum algorithm for solving travelling salesman problem: An IBM quantum experience[EB/OL]. (2018-05-28)[2023-05-11]. https://doi.org/10.48550/arXiv.1805.10928.
77	NATH R K , THAPLIYAL H , HUMBLE T S .A review of machine learning classification using quantum annealing for real-world applications[J].SN Computer Science,2021,2(5):365. DOI
78	FARHI E , GUTMANN S .Quantum computation and decision trees[J].Physical Review A,1998,58(2):915-928. DOI
79	AMBAINIS A .Quantum walks and their algorithmic applications[J].International Journal of Quantum Information,2003,1(4):507-518. DOI
80	CHILDS A M , FARHI E , GUTMANN S .An example of the difference between quantum and classical random walks[J].Quantum Information Processing,2002,1(1):35-43.
81	CHILDS A M, CLEVE R, DEOTTO E, et al. Exponential algorithmic speedup by a quantum walk[C]//Proceedings of the thirty-fifth annual ACM symposium on Theory of computing. San Diego, CA, USA: Association for Computing Machinery, 2003: 59-68.
82	BRASSARD G , HØYER P , MOSCA M , et al.Quantum amplitude amplification and estimation[J].Quantum Computation and Information,2002,305,53-74.
83	AMBAINIS A .Quantum walk algorithm for element distinctness[J].SIAM Journal on Computing,2007,37(1):210-239. DOI
84	BUHRMAN H , DVRR C , HEILIGMAN M , et al.Quantum algorithms for element distinctness[J].SIAM Journal on Computing,2005,34(6):1324-1330. DOI
85	MAGNIEZ F , SANTHA M , SZEGEDY M .Quantum algorithms for the triangle problem[J].SIAM Journal on Computing,2007,37(2):413-424. DOI
86	CHILDS A M , EISENBERG J M .Quantum algorithms for subset finding[J].Quantum Information & Computation,2005,5(7):593-604.
87	DVRR C , HEILIGMAN M , HØYER P .Quantum query complexity of some graph problems[J].SIAM Journal on Computing,2006,35(6):1310-1328. DOI
88	WIEBE N , BERRY D W , HØYER P , et al.Simulating quantum dynamics on a quantum computer[J].Journal of Physics A: Mathematical and Theoretical,2011,44(44):445308. DOI
89	LLOYD S , MOHSENI M , REBENTROST P .Quantum principal component analysis[J].Nature Physics,2014,10(9):631-633. DOI
90	REBENTROST P , MOHSENI M , LLOYD S .Quantum support vector machine for big data classification[J].Physical Review Letters,2014,113(13):130503. DOI
91	CONG I , DUAN Luming .Quantum discriminant analysis for dimensionality reduction and classification[J].New Journal of Physics,2016,18(7):073011. DOI
92	REBENTROST P , BROMLEY T R , WEEDBROOK C , et al.Quantum hopfield neural network[J].Physical Review A,2018,98(4):042308. DOI
93	DALLAIRE-DEMERS P L , KILLORAN N .Quantum generative adversarial networks[J].Physical Review A,2018,98(1):012324. DOI
94	KERENIDIS I, PRAKASH A. Quantum recommendation systems[C]//8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Berkeley: Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017: 49: 1-49: 21.
95	GRANT E , BENEDETTI M , CAO Shuxiang , et al.Hierarchical quantum classifiers[J].NPJ Quantum Information,2018,4(1):65. DOI
96	SCHULD M , BOCHAROV A , SVORE K M , et al.Circuit-centric quantum classifiers[J].Physical Review A,2020,101(3):032308. DOI
97	HUANG Rui , TAN Xiaoqing , XU Qingshan .Variational quantum tensor networks classifiers[J].Neurocomputing,2021,452,89-98. DOI
98	PARK G , HUH J , PARK D K .Variational quantum one-class classifier[J].Machine Learning: Science and Technology,2023,4(1):015006. DOI
99	DILIP R , LIU Yujie , SMITH A , et al.Data compression for quantum machine learning[J].Physical Review Research,2022,4(4):043007. DOI
100	PÉREZ-SALINAS A , CERVERA-LIERTA A , GIL-FUSTER E , et al.Data re-uploading for a universal quantum classifier[J].Quantum,2020,4,226. DOI
101	LAROSE R , COYLE B .Robust data encodings for quantum classifiers[J].Physical Review A,2020,102(3):032420. DOI
102	CHEN S Y C , HUANG C M , HSING C W , et al.An end-to-end trainable hybrid classical-quantum classifier[J].Machine Learning: Science and Technology,2021,2(4):045021. DOI
103	LIU Junhua , LIM K H , WOOD K L , et al.Hybrid quantum-classical convolutional neural networks[J].Science China Physics, Mechanics & Astronomy,2021,64(9):290311.
104	HUR T , KIM L , PARK D K .Quantum convolutional neural network for classical data classification[J].Quantum Machine Intelligence,2022,4(1):3. DOI
105	SCHULD M , FINGERHUTH M , PETRUCCIONE F .Implementing a distance-based classifier with a quantum interference circuit[J].Europhysics Letters,2017,119(6):60002. DOI
106	JOHRI S , DEBNATH S , MOCHERLA A , et al.Nearest centroid classification on a trapped ion quantum computer[J].NPJ Quantum Information,2021,7(1):122. DOI
107	CINCIO L , SUBASI Y , SORNBORGER A T , et al.Learning the quantum algorithm for state overlap[J].New Journal of Physics,2018,20(11):113022. DOI
108	CEREZO M , POREMBA A , CINCIO L , et al.Variational quantum fidelity estimation[J].Quantum,2020,4,248. DOI
109	CHEN H , WOSSNIG L , SEVERINI S , et al.Universal discriminative quantum neural networks[J].Quantum Machine Intelligence,2020,3(1):1.
110	KARDASHIN A S , VLASOVA A V , PERVISHKO A A , et al.Quantum-machine-learning channel discrimination[J].Physical Review A,2022,106(3):032409. DOI
111	OHZEKI M , NISHIMORI H .Quantum annealing: an introduction and new developments[J].Journal of Computational and Theoretical Nanoscience,2011,8(6):963-971. DOI
112	FARHI E , GOLDSTONE J , GUTMANN S , et al.A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem[J].Science,2001,292(5516):472-475. DOI
113	NEVEN H, DENCHEV V S, DREW-BROOK M, et al. Nips 2009 demonstration: Binary classification using hardware implementation of quantum annealing[EB/OL]. [2023-05-12]. https://api.semanticscholar.org/CorpusID:59263234.
114	FENG Yuan , YU Nengkun , YING Mingsheng .Model checking quantum Markov chains[J].Journal of Computer and System Sciences,2013,79(7):1181-1198. DOI
115	TANG E. A quantum-inspired classical algorithm for recommendation systems[C]//Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing. Phoenix, AZ, USA: Association for Computing Machinery, 2019: 217-228.
116	TANG E .Quantum principal component analysis only achieves an exponential speedup because of its state preparation assumptions[J].Physical Review Letters,2021,127(6):060503. DOI
117	CHIA N H, LIN H H, WANG Chunhao. Quantum-inspired sublinear classical algorithms for solving low-rank linear systems[EB/OL]. (2018-11-12)[2023-05-12]. https://doi.org/10.48550/arXiv.1811.04852.
118	WANG Guoming .Efficient quantum algorithms for analyzing large sparse electrical networks[J].Quantum Information & Computation,2017,17(11/12):987-1026.
119	BELOVS A. Quantum walks and electric networks[EB/OL]. (2013-02-13)[2023-05-12]. https://doi.org/10.48550/arXiv.1302.3143.
120	Google AI Quantum and Collaborators , ARUTE F , ARYA K , et al.Hartree-Fock on a superconducting qubit quantum computer[J].Science,2020,369(6507):1084-1089. DOI
121	CHIACCHIO E I R. Implementing real-world optimisation use cases in state-of-the-art quantum devices[EB/OL]. (2022-11-07)[2023-05-12]. https://entropicalabs.com/news/2022-11-implementing-real-world-optimisation-use-cases-in-state-of-the-art-quantum-devices/.
122	ANSCHUETZ E, OLSON J, ASPURU-GUZIK A, et al. Variational quantum factoring[C]//QTOP 2019: Quantum Technology and Optimization Problems. Munich, Germany: Springer, 2019: 74-85.
123	LLOYD S .Universal quantum simulators[J].Science,1996,273(5278):1073-1078. DOI
124	AHARONOV D, TA-SHMA A. Adiabatic quantum state generation and statistical zero knowledge[C]//Proceedings of the thirty-fifth annual ACM symposium on Theory of computing. San Diego, CA, USA: Association for Computing Machinery, 2003: 20-29.
125	BERRY D W , AHOKAS G , CLEVE R , et al.Efficient quantum algorithms for simulating sparse Hamiltonians[J].Communications in Mathematical Physics,2007,270(2):359-371. DOI
126	COLE R , VISHKIN U .Deterministic coin tossing with applications to optimal parallel list ranking[J].Information and Control,1986,70(1):32-53. DOI
127	LINIAL N .Locality in distributed graph algorithms[J].SIAM Journal on Computing,1992,21(1):193-201. DOI
128	BERRY D W , CHILDS A M .Black-box Hamiltonian simulation and unitary implementation[J].Quantum Information & Computation,2012,12(1/2):29-62.
129	BERRY D W, CHILDS A M, CLEVE R, et al. Exponential improvement in precision for simulating sparse Hamiltonians[C]//Proceedings of the forty-sixth annual ACM symposium on Theory of computing. New York: Association for Computing Machinery, 2014: 283-292.
130	KIRBY W M , LOVE P J .Variational quantum eigensolvers for sparse hamiltonians[J].Physical Review Letters,2021,127(11):110503. DOI
131	HUANG C, NI Xiaotong, ZHANG Fang, et al. Alibaba cloud quantum development platform: Surface code simulations with crosstalk[EB/OL]. (2020-02-20)[2023-05-12]. https://doi.org/10.48550/arXiv.2002.08918.
132	ZHANG Fang, HUANG C, NEWMAN M, et al. Alibaba cloud quantum development platform: Large-scale classical simulation of quantum circuits[EB/OL]. (2019-09-05)[2023-05-12]. https://doi.org/10.48550/arXiv.1907.11217.
133	ZHANG Shixin , ALLCOCK J , WAN Zhouquan , et al.TensorCircuit: A quantum software framework for the NISQ era[J].Quantum,2023,7,912. DOI
134	DOU Menghan, ZOU Tianrui, FANG Yuan, et al. QPanda: High-performance quantum computing framework for multiple application scenarios[EB/OL]. (2022-12-29)[2023-05-12]. https://doi.org/10.48550/arXiv.2212.14201.
135	GEMME G , GROSSI M , FERRARO D , et al.IBM quantum platforms: A quantum battery perspective[J].Batteries,2022,8(5):43. DOI
136	饶欣欣, 李卓瑛, 宋潇, 等.离子阱量子计算机的发展现状与趋势[J].世界科技研究与发展,2022,44(2):157-171.
137	ZHONG Hansen , WANG Hui , DENG Yuhao , et al.Quantum computational advantage using photons[J].Science,2020,370(6523):1460-1463. DOI
138	KIM Y , EDDINS A , ANAND S , et al.Evidence for the utility of quantum computing before fault tolerance[J].Nature,2023,618(7965):500-505. DOI

[1]	马颖, 田维坚, 樊养余. 基于云模型的自适应量子免疫克隆算法[J]. 计算物理, 2013, 30(4): 627-632.
[2]	李盼池, 王琪超, 施光尧. 基于Bloch球面搜索的量子粒子群优化算法[J]. 计算物理, 2013, 30(3): 454-462.
[3]	李盼池, 李士勇. 加权量子搜索算法及其相位匹配条件研究[J]. 计算物理, 2008, 25(5): 623-630.