Hybrid quantum-classical algorithms: Foundation, design and applications

Chen Ran-Yi-Liu; Zhao Ben-Chi; Song Zhi-Xin; Zhao Xuan-Qiang; Wang Kun; Wang Xin

doi:10.7498/aps.70.20210985

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:

Authors and contacts

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

Corresponding author: Wang Xin, wangxin73@baidu.com

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References

[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

Cited By

图 1 混合量子-经典算法流程

Figure 1. Diagram for hybrid quantum-classical algorithms.

DownLoad: Full-Size Img PowerPoint

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

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

DownLoad: Full-Size Img PowerPoint

图 3 (a) 量子感知机; (b)量子卷积神经网络; (c)影子电路

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

DownLoad: Full-Size Img PowerPoint

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

Figure 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 ${\cal{U}}^\dagger$.

DownLoad: Full-Size Img PowerPoint

图 5 无噪(左)与有噪(右)条件下损失函数景观对比

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

DownLoad: Full-Size Img PowerPoint

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

DownLoad: 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

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