混合量子-经典算法: 基础、设计与应用

陈然一鎏; 赵犇池; 宋旨欣; 赵炫强; 王琨; 王鑫

doi:10.7498/aps.70.20210985

摘要

量子计算作为一种新兴的计算范式, 有望解决在组合优化、量子化学、信息安全、人工智能领域中经典计算机难以解决的技术难题. 目前量子计算硬件与软件都在持续高速发展, 不过未来几年预计仍无法达到通用量子计算的标准. 因此短期内如何利用量子硬件解决实际问题成为了当前量子计算领域的一个研究热点, 探索近期量子硬件的应用对理解量子硬件的能力与推进量子计算的实用化进程有着重要意义. 针对近期量子硬件, 混合量子-经典算法(也称变分量子算法)是一个较为合理的模型. 混合量子-经典算法借助经典计算机尽可能发挥量子设备的计算能力, 结合量子计算与机器学习技术, 有望实现量子计算的首批实际应用, 在近期量子计算设备的算法研究中具有重要地位. 本文综述了混合量子-经典算法的设计框架以及在量子信息、组合优化、量子机器学习、量子纠错等领域的研究进展, 并对混合量子-经典算法的挑战以及未来研究方向进行了展望.

关键词:

Abstract

Quantum computing, as an emerging computing paradigm, is expected to tackle problems such as quantum chemistry, optimization, quantum chemistry, information security, and artificial intelligence, which are intractable with using classical computing. Quantum computing hardware and software continue to develop rapidly, but they are not expected to realize universal quantum computation in the next few years. Therefore, the use of quantum hardware to solve practical problems in the near term has become a hot topic in the field of quantum computing. Exploration of the applications of near-term quantum hardware is of great significance in understanding the capability of quantum hardware and promoting the practical process of quantum computing. Hybrid quantum-classical algorithm (also known as variational quantum algorithm) is an appropriate model for near-term quantum hardware. In the hybrid quantum-classical algorithm, classical computers are used to maximize the power of quantum devices. By combining quantum computing with machine learning, the hybrid quantum-classical algorithm is expected to achieve the first practical application of quantum computation and play an important role in the studying of quantum computing. In this review, we introduce the framework of hybrid quantum-classical algorithm and its applications in quantum chemistry, quantum information, combinatorial optimization, quantum machine learning, and other fields. We further discuss the challenges and future research directions of the hybrid quantum-classical algorithm.

Keywords:

作者及机构信息

百度研究院量子计算研究所, 北京　100193

通信作者: 王鑫, wangxin73@baidu.com

Authors and contacts

Institute for Quantum Computing, Baidu Research, Beijing 100193, China

Corresponding author: Wang Xin, wangxin73@baidu.com

文章全文 : translate this paragraph

参考文献

[1]	Shor P W 1994 FOCS: 35th Annual Symposium on Foundations of Computer Science Santa Fe, New Mexico, USA, November 20–22, 1994 pp124–134
[2]	Grover L K 1996 STOC: 28th Annual ACM Symposium on Theory of Computing Philadelphia, Pennsylvania, USA, May 22–24, 1996 pp212–219
[3]	Lloyd S 1996 Science 273 1073
[4]	Arute F, Arya K, Babbush R, et al. 2019 Nature 574 505 Google Scholar
[5]	Allen S, Kim J, Moehring D L, Monroe C R 2017 ICRC: IEEE International Conference on Rebooting Computing Washington DC, USA, November 8–9, 2017 pp1–3
[6]	Zhong H S, Wang H, Deng Y H, et al. 2020 Science 370 1460 Google Scholar
[7]	Gong M, Wang S, Zha C, et al. 2021 arXiv: 2102.02573 [quant-ph]
[8]	Preskill J 2018 Quantum 2 79 Google Scholar
[9]	Hadfield S, Wang Z, O’Gorman B, Rieffel E G, Venturelli D, Biswas R 2019 Algorithms 12 34 Google Scholar
[10]	Peruzzo A, McClean J, Shadbolt P, Yung M H, Zhou X Q, Love P J, Aspuru-Guzik A, O’brien J L 2014 Nat. Commun. 5 1 Google Scholar
[11]	Benedetti M, Lloyd E, Sack S, Fiorentini M 2019 Quantum Sci. Technol. 4 043001 Google Scholar
[12]	Zeng J, Wu Y, Liu J G, Wang L, Hu J 2019 Phys. Rev. A 99 052306 Google Scholar
[13]	Farhi E, Goldstone J, Gutmann S 2014 arXiv: 1411.4028 [quant-ph]
[14]	Liu J G, Zhang Y H, Wan Y, Wang L 2019 Phys. Rev. Res. 1 023025 Google Scholar
[15]	2020 Paddle Quantum URL https://github.com/Paddle Paddle/ Quantum
[16]	Bergholm V, Izaac J, Schuld M, et al. 2018 arXiv: 1811.04968 [quant-ph]
[17]	Broughton M, Verdon G, McCourt T, et al. 2020 arXiv: 2003.02989 [quant-ph]
[18]	https://github.com/paddlepaddle/paddle
[19]	Ma Y, Yu D, Wu T, Wang H 2019 Front. Data Comput. 1 105 Google Scholar
[20]	Cerezo M, Sharma K, Arrasmith A, Coles P J 2020 arXiv: 2004.01372 [quant-ph]
[21]	Cerezo M, Sone A, Volkoff T, Cincio L, Coles P J 2021 Nat. Commun. 12 1791 Google Scholar
[22]	Buhrman H, Cleve R, Watrous J, de Wolf R 2001 Phys. Rev. Lett. 87 167902 Google Scholar
[23]	Garcia-Escartin J C, Chamorro-Posada P 2013 Phys. Rev. A 87 052330 Google Scholar
[24]	Aharonov D, Jones V, Landau Z 2009 Algorithmica 55 395 Google Scholar
[25]	Sim S, Johnson P D, Aspuru-Guzik A 2019 Adv. Quantum Technol. 2 1900070 Google Scholar
[26]	Beer K, Bondarenko D, Farrelly T, Osborne T J, Salzmann R, Scheiermann D, Wolf R 2020 Nat. Commun. 11 1 Google Scholar
[27]	Sharma K, Cerezo M, Cincio L, Coles P J 2020 arXiv: 2005.12458 [quant-ph]
[28]	Oh S, Choi J, Kim J 2020 ICTC: International Conference on Information and Communication Technology Convergence Jeju Island, Korea, October 18–20, 2017 pp236–239
[29]	Zhang K, Hsieh M H, Liu L, Tao D 2020 arXiv: 2011.06258 [quant-ph]
[30]	Pesah A, Cerezo M, Wang S, Volkoff T, Sornborger A T, Coles P J 2020 arXiv: 2011.02966 [quant-ph]
[31]	Li G, Song Z, Wang X 2020 arXiv: 2012.08288 [quant-ph]
[32]	Huang H Y, Kueng R, Preskill J 2020 Nat. Phys. 16 1050 Google Scholar
[33]	Pan F, Zhang P 2021 arXiv: 2103.03074 [quant-ph]
[34]	Nocedal J, Wright S 2006 Numerical Optimization (New York: Springer Science & Business Media)
[35]	Stokes J, Izaac J, Killoran N, Carleo G 2020 Quantum 4 269 Google Scholar
[36]	Nakanishi K M, Fujii K, Todo S 2020 Phys. Rev. Res. 2 043158 Google Scholar
[37]	Ostaszewski M, Grant E, Benedetti M 2021 Quantum 5 391 Google Scholar
[38]	Parrish R M, Iosue J T, Ozaeta A, McMahon P L 2019 arXiv: 1904.03206 [quant-ph]
[39]	Mitarai K, Negoro M, Kitagawa M, Fujii K S 2018 Phys. Rev. A 98 032309 Google Scholar
[40]	Schuld M, Bergholm V, Gogolin C, Izaac J, Killoran N 2019 Phys. Rev. A 99 032331 Google Scholar
[41]	Farhi E, Neven H 2018 arXiv: 1802.06002 [quant-ph]
[42]	Wang Y, Li G, Wang X 2021 arXiv: 2103.01061 [quant-ph]
[43]	Nakanishi K M, Mitarai K, Fujii K 2019 Phys. Rev. Res. 1 033062 Google Scholar
[44]	LaRose R, Tikku A, O’Neel-Judy É, Cincio L, Coles P J 2019 npj Quantum Inf. 5 8 Google Scholar
[45]	Wang Y, Li G, Wang X 2020 arXiv: 2005.08797 [quant-ph]
[46]	Bravo-Prieto C, García-Martín D, Latorre J I 2020 Phys. Rev. A 101 062310 Google Scholar
[47]	Nielsen M A, Chuang I L 2010 Quantum Computation and Quantum Information (10th Anniversary Edition) (Cambridge: Cambridge University Press)
[48]	Życzkowski K, Horodecki P, Sanpera A, Lewenstein M 1998 Phys. Rev. A 58 883 Google Scholar
[49]	Chen R, Song Z, Zhao X, Wang X 2020 arXiv: 2012.05768 [quant-ph]
[50]	Watrous J 2008 arXiv: 0804.3401 [quant-ph]
[51]	Lucas A 2014 Front. Phys. 2 5 Google Scholar
[52]	Farhi E, Goldstone J, Gutmann S, Sipser M 2000 arXiv: quant-ph/0001106
[53]	Bauer B, Wang L, Pižorn I, Troyer M 2015 arXiv: 1501.06914 [cond-mat.dis-nn]
[54]	Schuld M, Bocharov A, Svore K M, Wiebe N 2020 Phys. Rev. A 101 032308 Google Scholar
[55]	Lloyd S, Weedbrook C 2018 Phys. Rev. Lett. 121 040502 Google Scholar
[56]	Benedetti M, Grant E, Wossnig L, Severini S 2019 New J. Phys. 21 043023 Google Scholar
[57]	Romero J, Olson J P, Aspuru-Guzik A 2017 Quantum Sci. Technol. 2 045001 Google Scholar
[58]	Cao C, Wang X 2021 arXiv: 2012.08331 [quant-ph]
[59]	Schuld M 2021 arXiv: 2101.11020 [quant-ph]
[60]	Havlíček V, Córcoles A D, Temme K, Harrow A W, Kandala A, Chow J M, Gambetta J M 2019 Nature 567 209 Google Scholar
[61]	Schuld M, Killoran N 2019 Phys. Rev. Lett. 122 040504 Google Scholar
[62]	Blank C, Park D K, Rhee J K K, Petruccione F 2020 npj Quantum Inf. 6 1 Google Scholar
[63]	Schuld M, Fingerhuth M, Petruccione F 2017 EPL 119 60002 Google Scholar
[64]	Li T, Chakrabarti S, Wu X 2019 ICML: International Conference on Machine Learning Long Beach, USA, June 9–15, 2019 pp3815–3824
[65]	Wang K, Xiao L, Yi W, Ran S J, Xue P 2020 arXiv: 2003.08551 [quant-ph]
[66]	Goodfellow I J, Pouget-Abadie J, Mirza M, et al. 2014 arXiv: 1406.2661 [stat.ML]
[67]	Situ H, He Z, Wang Y, Li L, Zheng S 2020 Inf. Sci. 538 193 Google Scholar
[68]	Arjovsky M, Chintala S, Bottou L 2017 ICML: International Conference on Machine Learning Sydney, Australia, August 6–11, 2017 pp214–223
[69]	Chakrabarti S, Huang Y, Li T, Feizi S, Wu X 2019 arXiv: 1911.00111 [quant-ph]
[70]	Romero J, Aspuru-Guzik A 2021 Adv. Quantum Technol. 4 2000003 Google Scholar
[71]	Zeng J, Wu Y, Liu J G, Wang L, Hu J 2019 Phys. Rev. A 99 052306 Google Scholar
[72]	Zoufal C, Lucchi A, Woerner S 2019 npj Quantum Inf. 5 103 Google Scholar
[73]	Hu L, Wu S H, Cai W, et al. 2019 Sci. Adv. 5 eaav2761 Google Scholar
[74]	Du Y, Hsieh M H, Tao D 2019 arXiv: 1904.09602 [quant-ph]
[75]	Devitt S J, Munro W J, Nemoto K 2013 Rep. Prog. Phys. 76 076001 Google Scholar
[76]	Lidar D A, Brun T A 2013 Quantum Error Correction (Cambridge: Cambridge University Press)
[77]	Terhal B M 2015 Rev. Mod. Phys. 87 307 Google Scholar
[78]	Johnson P D, Romero J, Olson J, Cao Y, Aspuru-Guzik A 2017 arXiv: 1711.02249 [quant-ph]
[79]	Xu X, Benjamin S C, Yuan X 2021 Phys. Rev. Appl. 15 034068 Google Scholar
[80]	Breuckmann N P, Ni X 2018 Quantum 2 68 Google Scholar
[81]	Torlai G, Melko R G 2017 Phys. Rev. Lett. 119 030501 Google Scholar
[82]	Bausch J, Leditzky F 2020 New J. Phys. 22 023005 Google Scholar
[83]	Chen H, Vasmer M, Breuckmann N P, Grant E 2019 arXiv: 1912.10063 [quant-ph]
[84]	Nautrup H P, Delfosse N, Dunjko V, Briegel H J, Friis N 2019 Quantum 3 215 Google Scholar
[85]	Strikis A, Qin D, Chen Y, Benjamin S C, Li Y 2020 arXiv: 2005.07601 [quant-ph]
[86]	Montanaro A, Stanisic S 2021 arXiv: 2102.02120 [quant-ph]
[87]	Zlokapa A, Gheorghiu A 2020 arXiv: 2005.10811 [quant-ph]
[88]	Andreasson P, Johansson J, Liljestrand S, Granath M 2019 Quantum 3 183 Google Scholar
[89]	Dankert C, Cleve R, Emerson J, Livine E 2009 Phys. Rev. A 80 012304 Google Scholar
[90]	Nakata Y, Hirche C, Morgan C, Winter A 2017 J. Math. Phys. 58 052203 Google Scholar
[91]	Laflamme R, Miquel C, Paz J P, Zurek W H 1996 Phys. Rev. Lett. 77 198 Google Scholar
[92]	Bennett C H, DiVincenzo D P, Smolin J A, Wootters W K 1996 Phys. Rev. A 54 3824 Google Scholar
[93]	Gottesman D 1997 Ph. D. Dissertation (California: Caltech)
[94]	Steane A M 1996 Phys. Rev. Lett. 77 793 Google Scholar
[95]	Shor P W 1995 Phys. Rev. A 52 R2493 Google Scholar
[96]	Xu X, Sun J, Endo S, Li Y, Benjamin S C, Yuan X 2019 arXiv: 1909.03898 [quant-ph]
[97]	Huang H Y, Bharti K, Rebentrost P 2019 arXiv: 1909.07344 [quant-ph]
[98]	Bravo-Prieto C, LaRose R, Cerezo M, Subasi Y, Cincio L, Coles P J 2020 arXiv: 1909.05820 [quant-ph]
[99]	Wang K, Song Z, Zhao X, Wang Z, Wang X 2020 arXiv: 2012.14311 [quant-ph]
[100]	Pashayan H, Wallman J J, Bartlett S D 2015 Phys. Rev. Lett. 115 070501 Google Scholar
[101]	Horodecki M, Horodecki P, Horodecki R 1996 Phys. Lett. A 223 1 Google Scholar
[102]	Chowdhury A N, Low G H, Wiebe N 2020 arXiv: 2002.00055 [quant-ph]
[103]	Lehtovaara L, Toivanen J, Eloranta J 2007 J. Comput. Phys. 221 148 Google Scholar
[104]	McArdle S, Jones T, Endo S, Li Y, Benjamin S C, Yuan X 2019 npj Quantum Inf. 5 75 Google Scholar
[105]	Motta M, Sun C, Tan A T, O’Rourke M J, Ye E, Minnich A J, Brandao F G, Chan G K L 2020 Nat. Phys. 16 205 Google Scholar
[106]	Xue C, Chen Z Y, Wu Y C, Guo G P 2019 arXiv: 1909.02196 [quant-ph]
[107]	Marshall J, Wudarski F, Hadfield S, Hogg T 2020 IOP SciNotes 1 025208 Google Scholar
[108]	Gentini L, Cuccoli A, Pirandola S, Verrucchi P, Banchi L 2020 Phys. Rev. A 102 052414 Google Scholar
[109]	Wang S, Fontana E, Cerezo M, Sharma K, Sone A, Cincio L, Coles P J 2021 arXiv: 2007.14384 [quant-ph]
[110]	Sharma K, Khatri S, Cerezo M, Coles P J 2020 New J. Phys. 22 043006 Google Scholar
[111]	Fontana E, Cerezo M, Arrasmith A, Rungger I, Coles P J 2020 arXiv: 2011.08763 [quant-ph]
[112]	Franca D S, Garcia-Patron R 2020 arXiv: 2009.05532 [quant-ph]
[113]	Nakaji K, Yamamoto N 2021 Quantum 5 434 Google Scholar
[114]	Holmes Z, Sharma K, Cerezo M, Coles P J 2021 arXiv: 2101.02138 [quant-ph]
[115]	McClean J R, Boixo S, Smelyanskiy V N, Babbush R, Neven H 2018 Nat. Commun. 9 1 Google Scholar
[116]	Arrasmith A, Cerezo M, Czarnik P, Cincio L, Coles P J 2020 arXiv: 2011.12245 [quant-ph]
[117]	Harrow A W, Low R A 2009 Commun. Math. Phys. 291 257 Google Scholar
[118]	Grant E, Wossnig L, Ostaszewski M, Benedetti M 2019 Quantum 3 214 Google Scholar
[119]	Skolik A, McClean J R, Mohseni M, van der Smagt P, Leib M 2021 Quantum Mach. Intell. 3 5
[120]	Romero J, Babbush R, McClean J R, Hempel C, Love P J, Aspuru-Guzik A 2018 Quantum Sci. Technol. 4 014008 Google Scholar
[121]	Marrero C O, Kieferová M, Wiebe N 2020 arXiv: 2010.15968 [quant-ph]
[122]	You Y Z, Yang Z, Qi X L 2018 Phys. Rev. B 97 045153 Google Scholar
[123]	Gan W C, Shu F W 2017 Int. J. Mod. Phys. D 26 1743020 Google Scholar
[124]	Liu D, Ran S J, Wittek P, Peng C, García R B, Su G, Lewenstein M 2019 New J. Phys. 21 073059 Google Scholar
[125]	Han Z Y, Wang J, Fan H, Wang L, Zhang P 2018 Phys. Rev. X 8 031012 Google Scholar

施引文献

图 1 混合量子-经典算法流程

Fig. 1. Diagram for hybrid quantum-classical algorithms.

下载: 全尺寸图片幻灯片

图 2 (a) 交换测试电路; (b)破坏性交换测试电路; (c) 哈达玛测试电路

Fig. 2. (a) Swap test circuit; (b) destructive swap test circuit; (c) Hadamard test circuit.

下载: 全尺寸图片幻灯片

图 3 (a) 量子感知机; (b)量子卷积神经网络; (c)影子电路

Fig. 3. (a) Quantum perceptron; (b) quantum convolutional neural network; (c) shadow circuit.

下载: 全尺寸图片幻灯片

图 4 量子纠错基本框架: 量子态$\left| \varPsi \right\rangle $使用编码信道${\cal{U}}$编码, 经过噪声信道${\cal{N}}$后使用纠错信道${\cal{W}}$纠正错误, 最后使用解码信道${\cal{U}}^\dagger$解码, 还原输入量子态

Fig. 4. Framework of quantum error correction. The quantum state $\left| \varPsi \right\rangle $ first is encoded by the encoding channel ${\cal{U}}$, then passes the noise channel ${\cal{N}}$, and then is corrected by the correction channel ${\cal{W}}$, finally is recovered by the decoding channel ${\cal{U}}^\dagger$.

下载: 全尺寸图片幻灯片

图 5 无噪(左)与有噪(右)条件下损失函数景观对比

Fig. 5. Comparison between noise-free (left) and noisy (right) cost function landscape.

下载: 全尺寸图片幻灯片

表 1 主要符号表

Table 1. Notations

定义	符号
厄米特算符	H
含参数酉算符	${\boldsymbol U}({\boldsymbol{\theta} }), {\boldsymbol V}({\boldsymbol{\theta} })$
不含参数酉算符	W
可调参数	θ
量子态	${\boldsymbol \rho}, {\boldsymbol \sigma}$
量子比特数	n
电路层数	L
损失函数	$ C, C({\boldsymbol{\theta}})$
能量	E
泡利算符	P
迹	Tr

下载: 导出CSV

[1]	Shor P W 1994 FOCS: 35th Annual Symposium on Foundations of Computer Science Santa Fe, New Mexico, USA, November 20–22, 1994 pp124–134
[2]	Grover L K 1996 STOC: 28th Annual ACM Symposium on Theory of Computing Philadelphia, Pennsylvania, USA, May 22–24, 1996 pp212–219
[3]	Lloyd S 1996 Science 273 1073
[4]	Arute F, Arya K, Babbush R, et al. 2019 Nature 574 505 Google Scholar
[5]	Allen S, Kim J, Moehring D L, Monroe C R 2017 ICRC: IEEE International Conference on Rebooting Computing Washington DC, USA, November 8–9, 2017 pp1–3
[6]	Zhong H S, Wang H, Deng Y H, et al. 2020 Science 370 1460 Google Scholar
[7]	Gong M, Wang S, Zha C, et al. 2021 arXiv: 2102.02573 [quant-ph]
[8]	Preskill J 2018 Quantum 2 79 Google Scholar
[9]	Hadfield S, Wang Z, O’Gorman B, Rieffel E G, Venturelli D, Biswas R 2019 Algorithms 12 34 Google Scholar
[10]	Peruzzo A, McClean J, Shadbolt P, Yung M H, Zhou X Q, Love P J, Aspuru-Guzik A, O’brien J L 2014 Nat. Commun. 5 1 Google Scholar
[11]	Benedetti M, Lloyd E, Sack S, Fiorentini M 2019 Quantum Sci. Technol. 4 043001 Google Scholar
[12]	Zeng J, Wu Y, Liu J G, Wang L, Hu J 2019 Phys. Rev. A 99 052306 Google Scholar
[13]	Farhi E, Goldstone J, Gutmann S 2014 arXiv: 1411.4028 [quant-ph]
[14]	Liu J G, Zhang Y H, Wan Y, Wang L 2019 Phys. Rev. Res. 1 023025 Google Scholar
[15]	2020 Paddle Quantum URL https://github.com/Paddle Paddle/ Quantum
[16]	Bergholm V, Izaac J, Schuld M, et al. 2018 arXiv: 1811.04968 [quant-ph]
[17]	Broughton M, Verdon G, McCourt T, et al. 2020 arXiv: 2003.02989 [quant-ph]
[18]	https://github.com/paddlepaddle/paddle
[19]	Ma Y, Yu D, Wu T, Wang H 2019 Front. Data Comput. 1 105 Google Scholar
[20]	Cerezo M, Sharma K, Arrasmith A, Coles P J 2020 arXiv: 2004.01372 [quant-ph]
[21]	Cerezo M, Sone A, Volkoff T, Cincio L, Coles P J 2021 Nat. Commun. 12 1791 Google Scholar
[22]	Buhrman H, Cleve R, Watrous J, de Wolf R 2001 Phys. Rev. Lett. 87 167902 Google Scholar
[23]	Garcia-Escartin J C, Chamorro-Posada P 2013 Phys. Rev. A 87 052330 Google Scholar
[24]	Aharonov D, Jones V, Landau Z 2009 Algorithmica 55 395 Google Scholar
[25]	Sim S, Johnson P D, Aspuru-Guzik A 2019 Adv. Quantum Technol. 2 1900070 Google Scholar
[26]	Beer K, Bondarenko D, Farrelly T, Osborne T J, Salzmann R, Scheiermann D, Wolf R 2020 Nat. Commun. 11 1 Google Scholar
[27]	Sharma K, Cerezo M, Cincio L, Coles P J 2020 arXiv: 2005.12458 [quant-ph]
[28]	Oh S, Choi J, Kim J 2020 ICTC: International Conference on Information and Communication Technology Convergence Jeju Island, Korea, October 18–20, 2017 pp236–239
[29]	Zhang K, Hsieh M H, Liu L, Tao D 2020 arXiv: 2011.06258 [quant-ph]
[30]	Pesah A, Cerezo M, Wang S, Volkoff T, Sornborger A T, Coles P J 2020 arXiv: 2011.02966 [quant-ph]
[31]	Li G, Song Z, Wang X 2020 arXiv: 2012.08288 [quant-ph]
[32]	Huang H Y, Kueng R, Preskill J 2020 Nat. Phys. 16 1050 Google Scholar
[33]	Pan F, Zhang P 2021 arXiv: 2103.03074 [quant-ph]
[34]	Nocedal J, Wright S 2006 Numerical Optimization (New York: Springer Science & Business Media)
[35]	Stokes J, Izaac J, Killoran N, Carleo G 2020 Quantum 4 269 Google Scholar
[36]	Nakanishi K M, Fujii K, Todo S 2020 Phys. Rev. Res. 2 043158 Google Scholar
[37]	Ostaszewski M, Grant E, Benedetti M 2021 Quantum 5 391 Google Scholar
[38]	Parrish R M, Iosue J T, Ozaeta A, McMahon P L 2019 arXiv: 1904.03206 [quant-ph]
[39]	Mitarai K, Negoro M, Kitagawa M, Fujii K S 2018 Phys. Rev. A 98 032309 Google Scholar
[40]	Schuld M, Bergholm V, Gogolin C, Izaac J, Killoran N 2019 Phys. Rev. A 99 032331 Google Scholar
[41]	Farhi E, Neven H 2018 arXiv: 1802.06002 [quant-ph]
[42]	Wang Y, Li G, Wang X 2021 arXiv: 2103.01061 [quant-ph]
[43]	Nakanishi K M, Mitarai K, Fujii K 2019 Phys. Rev. Res. 1 033062 Google Scholar
[44]	LaRose R, Tikku A, O’Neel-Judy É, Cincio L, Coles P J 2019 npj Quantum Inf. 5 8 Google Scholar
[45]	Wang Y, Li G, Wang X 2020 arXiv: 2005.08797 [quant-ph]
[46]	Bravo-Prieto C, García-Martín D, Latorre J I 2020 Phys. Rev. A 101 062310 Google Scholar
[47]	Nielsen M A, Chuang I L 2010 Quantum Computation and Quantum Information (10th Anniversary Edition) (Cambridge: Cambridge University Press)
[48]	Życzkowski K, Horodecki P, Sanpera A, Lewenstein M 1998 Phys. Rev. A 58 883 Google Scholar
[49]	Chen R, Song Z, Zhao X, Wang X 2020 arXiv: 2012.05768 [quant-ph]
[50]	Watrous J 2008 arXiv: 0804.3401 [quant-ph]
[51]	Lucas A 2014 Front. Phys. 2 5 Google Scholar
[52]	Farhi E, Goldstone J, Gutmann S, Sipser M 2000 arXiv: quant-ph/0001106
[53]	Bauer B, Wang L, Pižorn I, Troyer M 2015 arXiv: 1501.06914 [cond-mat.dis-nn]
[54]	Schuld M, Bocharov A, Svore K M, Wiebe N 2020 Phys. Rev. A 101 032308 Google Scholar
[55]	Lloyd S, Weedbrook C 2018 Phys. Rev. Lett. 121 040502 Google Scholar
[56]	Benedetti M, Grant E, Wossnig L, Severini S 2019 New J. Phys. 21 043023 Google Scholar
[57]	Romero J, Olson J P, Aspuru-Guzik A 2017 Quantum Sci. Technol. 2 045001 Google Scholar
[58]	Cao C, Wang X 2021 arXiv: 2012.08331 [quant-ph]
[59]	Schuld M 2021 arXiv: 2101.11020 [quant-ph]
[60]	Havlíček V, Córcoles A D, Temme K, Harrow A W, Kandala A, Chow J M, Gambetta J M 2019 Nature 567 209 Google Scholar
[61]	Schuld M, Killoran N 2019 Phys. Rev. Lett. 122 040504 Google Scholar
[62]	Blank C, Park D K, Rhee J K K, Petruccione F 2020 npj Quantum Inf. 6 1 Google Scholar
[63]	Schuld M, Fingerhuth M, Petruccione F 2017 EPL 119 60002 Google Scholar
[64]	Li T, Chakrabarti S, Wu X 2019 ICML: International Conference on Machine Learning Long Beach, USA, June 9–15, 2019 pp3815–3824
[65]	Wang K, Xiao L, Yi W, Ran S J, Xue P 2020 arXiv: 2003.08551 [quant-ph]
[66]	Goodfellow I J, Pouget-Abadie J, Mirza M, et al. 2014 arXiv: 1406.2661 [stat.ML]
[67]	Situ H, He Z, Wang Y, Li L, Zheng S 2020 Inf. Sci. 538 193 Google Scholar
[68]	Arjovsky M, Chintala S, Bottou L 2017 ICML: International Conference on Machine Learning Sydney, Australia, August 6–11, 2017 pp214–223
[69]	Chakrabarti S, Huang Y, Li T, Feizi S, Wu X 2019 arXiv: 1911.00111 [quant-ph]
[70]	Romero J, Aspuru-Guzik A 2021 Adv. Quantum Technol. 4 2000003 Google Scholar
[71]	Zeng J, Wu Y, Liu J G, Wang L, Hu J 2019 Phys. Rev. A 99 052306 Google Scholar
[72]	Zoufal C, Lucchi A, Woerner S 2019 npj Quantum Inf. 5 103 Google Scholar
[73]	Hu L, Wu S H, Cai W, et al. 2019 Sci. Adv. 5 eaav2761 Google Scholar
[74]	Du Y, Hsieh M H, Tao D 2019 arXiv: 1904.09602 [quant-ph]
[75]	Devitt S J, Munro W J, Nemoto K 2013 Rep. Prog. Phys. 76 076001 Google Scholar
[76]	Lidar D A, Brun T A 2013 Quantum Error Correction (Cambridge: Cambridge University Press)
[77]	Terhal B M 2015 Rev. Mod. Phys. 87 307 Google Scholar
[78]	Johnson P D, Romero J, Olson J, Cao Y, Aspuru-Guzik A 2017 arXiv: 1711.02249 [quant-ph]
[79]	Xu X, Benjamin S C, Yuan X 2021 Phys. Rev. Appl. 15 034068 Google Scholar
[80]	Breuckmann N P, Ni X 2018 Quantum 2 68 Google Scholar
[81]	Torlai G, Melko R G 2017 Phys. Rev. Lett. 119 030501 Google Scholar
[82]	Bausch J, Leditzky F 2020 New J. Phys. 22 023005 Google Scholar
[83]	Chen H, Vasmer M, Breuckmann N P, Grant E 2019 arXiv: 1912.10063 [quant-ph]
[84]	Nautrup H P, Delfosse N, Dunjko V, Briegel H J, Friis N 2019 Quantum 3 215 Google Scholar
[85]	Strikis A, Qin D, Chen Y, Benjamin S C, Li Y 2020 arXiv: 2005.07601 [quant-ph]
[86]	Montanaro A, Stanisic S 2021 arXiv: 2102.02120 [quant-ph]
[87]	Zlokapa A, Gheorghiu A 2020 arXiv: 2005.10811 [quant-ph]
[88]	Andreasson P, Johansson J, Liljestrand S, Granath M 2019 Quantum 3 183 Google Scholar
[89]	Dankert C, Cleve R, Emerson J, Livine E 2009 Phys. Rev. A 80 012304 Google Scholar
[90]	Nakata Y, Hirche C, Morgan C, Winter A 2017 J. Math. Phys. 58 052203 Google Scholar
[91]	Laflamme R, Miquel C, Paz J P, Zurek W H 1996 Phys. Rev. Lett. 77 198 Google Scholar
[92]	Bennett C H, DiVincenzo D P, Smolin J A, Wootters W K 1996 Phys. Rev. A 54 3824 Google Scholar
[93]	Gottesman D 1997 Ph. D. Dissertation (California: Caltech)
[94]	Steane A M 1996 Phys. Rev. Lett. 77 793 Google Scholar
[95]	Shor P W 1995 Phys. Rev. A 52 R2493 Google Scholar
[96]	Xu X, Sun J, Endo S, Li Y, Benjamin S C, Yuan X 2019 arXiv: 1909.03898 [quant-ph]
[97]	Huang H Y, Bharti K, Rebentrost P 2019 arXiv: 1909.07344 [quant-ph]
[98]	Bravo-Prieto C, LaRose R, Cerezo M, Subasi Y, Cincio L, Coles P J 2020 arXiv: 1909.05820 [quant-ph]
[99]	Wang K, Song Z, Zhao X, Wang Z, Wang X 2020 arXiv: 2012.14311 [quant-ph]
[100]	Pashayan H, Wallman J J, Bartlett S D 2015 Phys. Rev. Lett. 115 070501 Google Scholar
[101]	Horodecki M, Horodecki P, Horodecki R 1996 Phys. Lett. A 223 1 Google Scholar
[102]	Chowdhury A N, Low G H, Wiebe N 2020 arXiv: 2002.00055 [quant-ph]
[103]	Lehtovaara L, Toivanen J, Eloranta J 2007 J. Comput. Phys. 221 148 Google Scholar
[104]	McArdle S, Jones T, Endo S, Li Y, Benjamin S C, Yuan X 2019 npj Quantum Inf. 5 75 Google Scholar
[105]	Motta M, Sun C, Tan A T, O’Rourke M J, Ye E, Minnich A J, Brandao F G, Chan G K L 2020 Nat. Phys. 16 205 Google Scholar
[106]	Xue C, Chen Z Y, Wu Y C, Guo G P 2019 arXiv: 1909.02196 [quant-ph]
[107]	Marshall J, Wudarski F, Hadfield S, Hogg T 2020 IOP SciNotes 1 025208 Google Scholar
[108]	Gentini L, Cuccoli A, Pirandola S, Verrucchi P, Banchi L 2020 Phys. Rev. A 102 052414 Google Scholar
[109]	Wang S, Fontana E, Cerezo M, Sharma K, Sone A, Cincio L, Coles P J 2021 arXiv: 2007.14384 [quant-ph]
[110]	Sharma K, Khatri S, Cerezo M, Coles P J 2020 New J. Phys. 22 043006 Google Scholar
[111]	Fontana E, Cerezo M, Arrasmith A, Rungger I, Coles P J 2020 arXiv: 2011.08763 [quant-ph]
[112]	Franca D S, Garcia-Patron R 2020 arXiv: 2009.05532 [quant-ph]
[113]	Nakaji K, Yamamoto N 2021 Quantum 5 434 Google Scholar
[114]	Holmes Z, Sharma K, Cerezo M, Coles P J 2021 arXiv: 2101.02138 [quant-ph]
[115]	McClean J R, Boixo S, Smelyanskiy V N, Babbush R, Neven H 2018 Nat. Commun. 9 1 Google Scholar
[116]	Arrasmith A, Cerezo M, Czarnik P, Cincio L, Coles P J 2020 arXiv: 2011.12245 [quant-ph]
[117]	Harrow A W, Low R A 2009 Commun. Math. Phys. 291 257 Google Scholar
[118]	Grant E, Wossnig L, Ostaszewski M, Benedetti M 2019 Quantum 3 214 Google Scholar
[119]	Skolik A, McClean J R, Mohseni M, van der Smagt P, Leib M 2021 Quantum Mach. Intell. 3 5
[120]	Romero J, Babbush R, McClean J R, Hempel C, Love P J, Aspuru-Guzik A 2018 Quantum Sci. Technol. 4 014008 Google Scholar
[121]	Marrero C O, Kieferová M, Wiebe N 2020 arXiv: 2010.15968 [quant-ph]
[122]	You Y Z, Yang Z, Qi X L 2018 Phys. Rev. B 97 045153 Google Scholar
[123]	Gan W C, Shu F W 2017 Int. J. Mod. Phys. D 26 1743020 Google Scholar
[124]	Liu D, Ran S J, Wittek P, Peng C, García R B, Su G, Lewenstein M 2019 New J. Phys. 21 073059 Google Scholar
[125]	Han Z Y, Wang J, Fan H, Wang L, Zhang P 2018 Phys. Rev. X 8 031012 Google Scholar

[1]	姜达, 余东洋, 郑沾, 曹晓超, 林强, 刘伍明. 面向量子计算的拓扑超导体材料、物理和器件研究. 物理学报, 2022, 71(16): 160302. doi: 10.7498/aps.71.20220596
[2]	王美红, 郝树宏, 秦忠忠, 苏晓龙. 连续变量量子计算和量子纠错研究进展. 物理学报, 2022, 71(16): 160305. doi: 10.7498/aps.71.20220635
[3]	王晨旭, 贺冉, 李睿睿, 陈炎, 房鼎, 崔金明, 黄运锋, 李传锋, 郭光灿. 量子计算与量子模拟中离子阱结构研究进展. 物理学报, 2022, 71(13): 133701. doi: 10.7498/aps.71.20220224
[4]	周宗权. 量子存储式量子计算机与无噪声光子回波. 物理学报, 2022, 71(7): 070305. doi: 10.7498/aps.71.20212245
[5]	王宁, 王保传, 郭国平. 硅基半导体量子计算研究进展. 物理学报, 2022, 71(23): 230301. doi: 10.7498/aps.71.20221900
[6]	王一诺, 宋昭阳, 马玉林, 华南, 马鸿洋. 基于DNA编码与交替量子随机行走的彩色图像加密算法. 物理学报, 2021, 70(23): 230302. doi: 10.7498/aps.70.20211255
[7]	张结印, 高飞, 张建军. 硅和锗量子计算材料研究进展. 物理学报, 2021, 70(21): 217802. doi: 10.7498/aps.70.20211492
[8]	张诗豪, 张向东, 李绿周. 基于测量的量子计算研究进展. 物理学报, 2021, 70(21): 210301. doi: 10.7498/aps.70.20210923
[9]	何映萍, 洪健松, 刘雄军. 马约拉纳零能模的非阿贝尔统计及其在拓扑量子计算的应用. 物理学报, 2020, 69(11): 110302. doi: 10.7498/aps.69.20200812
[10]	史保森, 丁冬生, 张伟, 李恩泽. 基于拉曼协议的量子存储. 物理学报, 2019, 68(3): 034203. doi: 10.7498/aps.68.20182215
[11]	窦建鹏, 李航, 庞晓玲, 张超妮, 杨天怀, 金贤敏. 量子存储研究进展. 物理学报, 2019, 68(3): 030307. doi: 10.7498/aps.68.20190039
[12]	范桁. 量子计算与量子模拟. 物理学报, 2018, 67(12): 120301. doi: 10.7498/aps.67.20180710
[13]	李明, 陈阳, 郭光灿, 任希锋. 表面等离激元量子信息应用研究进展. 物理学报, 2017, 66(14): 144202. doi: 10.7498/aps.66.144202
[14]	李卓, 邢莉娟. 差错基、量子码与群代数. 物理学报, 2013, 62(13): 130306. doi: 10.7498/aps.62.130306
[15]	邢莉娟, 李卓, 张武军. 加强的量子汉明限. 物理学报, 2011, 60(5): 050304. doi: 10.7498/aps.60.050304
[16]	邢莉娟, 李卓, 白宝明, 王新梅. 量子卷积码的编译码方法. 物理学报, 2008, 57(8): 4695-4699. doi: 10.7498/aps.57.4695
[17]	李卓, 邢莉娟. 量子Generalized Reed-Solomon码. 物理学报, 2008, 57(1): 28-30. doi: 10.7498/aps.57.28
[18]	叶宾, 须文波, 顾斌杰. 量子Harper模型的量子计算鲁棒性与耗散退相干. 物理学报, 2008, 57(2): 689-695. doi: 10.7498/aps.57.689
[19]	叶宾, 谷瑞军, 须文波. 周期驱动的Harper模型的量子计算鲁棒性与量子混沌. 物理学报, 2007, 56(7): 3709-3718. doi: 10.7498/aps.56.3709
[20]	张权, 唐朝京, 高峰. 量子Turbo码. 物理学报, 2002, 51(1): 15-20. doi: 10.7498/aps.51.15

计量

文章访问数: 13899
PDF下载量: 676
被引次数: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

搜索

留言板