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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.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.
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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${\cal{U}}^\dagger$ .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 -
[1] [2] [3] Lloyd S 1996 Science 273 1073
[4] Arute F, Arya K, Babbush R, et al. 2019 Nature 574 505Google 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 1460Google Scholar
[7] Gong M, Wang S, Zha C, et al. 2021 arXiv: 2102.02573 [quant-ph]
[8] Preskill J 2018 Quantum 2 79Google Scholar
[9] Hadfield S, Wang Z, O’Gorman B, Rieffel E G, Venturelli D, Biswas R 2019 Algorithms 12 34Google 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 1Google Scholar
[11] Benedetti M, Lloyd E, Sack S, Fiorentini M 2019 Quantum Sci. Technol. 4 043001Google Scholar
[12] Zeng J, Wu Y, Liu J G, Wang L, Hu J 2019 Phys. Rev. A 99 052306Google 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 023025Google 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] [19] Ma Y, Yu D, Wu T, Wang H 2019 Front. Data Comput. 1 105Google 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 1791Google Scholar
[22] Buhrman H, Cleve R, Watrous J, de Wolf R 2001 Phys. Rev. Lett. 87 167902Google Scholar
[23] Garcia-Escartin J C, Chamorro-Posada P 2013 Phys. Rev. A 87 052330Google Scholar
[24] Aharonov D, Jones V, Landau Z 2009 Algorithmica 55 395Google Scholar
[25] Sim S, Johnson P D, Aspuru-Guzik A 2019 Adv. Quantum Technol. 2 1900070Google Scholar
[26] Beer K, Bondarenko D, Farrelly T, Osborne T J, Salzmann R, Scheiermann D, Wolf R 2020 Nat. Commun. 11 1Google 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]
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[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 269Google Scholar
[36] Nakanishi K M, Fujii K, Todo S 2020 Phys. Rev. Res. 2 043158Google Scholar
[37] Ostaszewski M, Grant E, Benedetti M 2021 Quantum 5 391Google 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 032309Google Scholar
[40] Schuld M, Bergholm V, Gogolin C, Izaac J, Killoran N 2019 Phys. Rev. A 99 032331Google 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 033062Google Scholar
[44] LaRose R, Tikku A, O’Neel-Judy É, Cincio L, Coles P J 2019 npj Quantum Inf. 5 8Google 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 062310Google 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 883Google 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 5Google 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 032308Google Scholar
[55] Lloyd S, Weedbrook C 2018 Phys. Rev. Lett. 121 040502Google Scholar
[56] Benedetti M, Grant E, Wossnig L, Severini S 2019 New J. Phys. 21 043023Google Scholar
[57] Romero J, Olson J P, Aspuru-Guzik A 2017 Quantum Sci. Technol. 2 045001Google 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 209Google Scholar
[61] Schuld M, Killoran N 2019 Phys. Rev. Lett. 122 040504Google Scholar
[62] Blank C, Park D K, Rhee J K K, Petruccione F 2020 npj Quantum Inf. 6 1Google Scholar
[63] Schuld M, Fingerhuth M, Petruccione F 2017 EPL 119 60002Google 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 193Google 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 2000003Google Scholar
[71] Zeng J, Wu Y, Liu J G, Wang L, Hu J 2019 Phys. Rev. A 99 052306Google Scholar
[72] Zoufal C, Lucchi A, Woerner S 2019 npj Quantum Inf. 5 103Google Scholar
[73] Hu L, Wu S H, Cai W, et al. 2019 Sci. Adv. 5 eaav2761Google 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 076001Google 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 307Google 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 034068Google Scholar
[80] Breuckmann N P, Ni X 2018 Quantum 2 68Google Scholar
[81] Torlai G, Melko R G 2017 Phys. Rev. Lett. 119 030501Google Scholar
[82] Bausch J, Leditzky F 2020 New J. Phys. 22 023005Google 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 215Google 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 183Google Scholar
[89] Dankert C, Cleve R, Emerson J, Livine E 2009 Phys. Rev. A 80 012304Google Scholar
[90] Nakata Y, Hirche C, Morgan C, Winter A 2017 J. Math. Phys. 58 052203Google Scholar
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[93] Gottesman D 1997 Ph. D. Dissertation (California: Caltech)
[94] Steane A M 1996 Phys. Rev. Lett. 77 793Google Scholar
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[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]
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