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Chin. Phys. B, 2022, Vol. 31(9): 090306    DOI: 10.1088/1674-1056/ac7b1e
Special Issue: TOPICAL REVIEW — Celebrating 30 Years of Chinese Physics B
TOPICAL REVIEW—Celebrating 30 Years of Chinese Physics B Prev   Next  

An overview of quantum error mitigation formulas

Dayue Qin(秦大粤), Xiaosi Xu(徐晓思), and Ying Li(李颖)
Graduate School of China Academy of Engineering Physics, Beijing 100193, China
Abstract  Minimizing the effect of noise is essential for quantum computers. The conventional method to protect qubits against noise is through quantum error correction. However, for current quantum hardware in the so-called noisy intermediate-scale quantum (NISQ) era, noise presents in these systems and is too high for error correction to be beneficial. Quantum error mitigation is a set of alternative methods for minimizing errors, including error extrapolation, probabilistic error cancellation, measurement error mitigation, subspace expansion, symmetry verification, virtual distillation, etc. The requirement for these methods is usually less demanding than error correction. Quantum error mitigation is a promising way of reducing errors on NISQ quantum computers. This paper gives a comprehensive introduction to quantum error mitigation. The state-of-art error mitigation methods are covered and formulated in a general form, which provides a basis for comparing, combining and optimizing different methods in future work.
Keywords:  quantum error mitigation      quantum computing      quantum error correction      noisy intermediate-scale quantum  
Received:  24 April 2022      Revised:  21 June 2022      Accepted manuscript online:  22 June 2022
PACS:  03.67.Pp (Quantum error correction and other methods for protection against decoherence)  
Fund: This work is supported by the National Natural Science Foundation of China (Grant Nos. 11875050 and 12088101) and NSAF (Grant No. U1930403).
Corresponding Authors:  Ying Li     E-mail:

Cite this article: 

Dayue Qin(秦大粤), Xiaosi Xu(徐晓思), and Ying Li(李颖) An overview of quantum error mitigation formulas 2022 Chin. Phys. B 31 090306

[1] Nielsen M A and Chuang I L 2010 Quantum Computation and Quantum Information (Cambridge University Press) pp. 171-276
[2] Ladd T D, Jelezko F, Laflamme R, Nakamura Y, Monroe C and O'Brien J L 2010 Nature 464 45
[3] Bao F, Deng H and Ding D, et al. 2021 arXiv:2111.13504[quant-ph]
[4] Arute F, Arya K and Babbush R, et al. 2019 Nature 574 505
[5] Kim Y, Wood C J, Yoder T J, Merkel S T, Gambetta J M, Temme K and Kandala A 2021 arXiv:2108.09197[cond-mat, physics:quant-ph]
[6] Terhal B M 2015 Rev. Mod. Phys. 87 307
[7] Babbush R, Gidney C, Berry D W, Wiebe N, McClean J, Paler A, Fowler A and Neven H 2018 Phys. Rev. X 8 041015
[8] Preskill J 2018 Quantum 2 79
[9] Shor P W 1996 Proceedings of 37th conference on foundations of computer science, October 14-16, 1996, Burlington, VT, USA, p. 56
[10] Wu Y, Bao W S and Cao S, et al. 2021 Phys. Rev. Lett. 127 180501
[11] McArdle S, Endo S, Aspuru-Guzik A, Benjamin S C and Yuan X 2020 Rev. Mod. Phys. 92 015003
[12] Bharti K, Cervera-Lierta A, Kyaw T H, Haug T, Alperin-Lea S, Anand A, Degroote M, Heimonen H, Kottmann J S, Menke T, Mok W K, Sim S, Kwek L C and Aspuru-Guzik A 2022 Rev. Mod. Phys. 94 015004
[13] Li Y and Benjamin S C 2017 Phys. Rev. X 7 021050
[14] Temme K, Bravyi S and Gambetta J M 2017 Phys. Rev. Lett. 119 180509
[15] Endo S, Benjamin S C and Li Y 2018 Phys. Rev. X 8 031027
[16] Magesan E, Gambetta J M and Emerson J 2011 Phys. Rev. Lett. 106 180504
[17] Greenbaum D 2015 arXiv:1509.02921[quant-ph]
[18] Strikis A, Qin D, Chen Y, Benjamin S C and Li Y 2021 PRX Quantum 2 040330
[19] Czarnik P, Arrasmith A, Coles P J and Cincio L 2020 arXiv:2005.10189[quant-ph]
[20] McArdle S, Yuan X and Benjamin S 2019 Phys. Rev. Lett. 122 180501
[21] Bonet-Monroig X, Sagastizabal R, Singh M and O'Brien T E 2018 Phys. Rev. A 98 062339
[22] Koczor B 2021 Phys. Rev. X 11 031057
[23] Huggins W J, McArdle S, O'Brien T E, Lee J, Rubin N C, Boixo S, Whaley K B, Babbush R and McClean J R 2021 Phys. Rev. X 11 041036
[24] O'Brien T E, Polla S, Rubin N C, Huggins W J, McArdle S, Boixo S, McClean J R and Babbush R 2021 PRX Quantum 2 020317
[25] Huo M and Li Y 2022 Phys. Rev. A 105 022427
[26] McClean J R, Schwartz M E, Carter J and de Jong W A 2017 Phys. Rev. A 95 042308
[27] Colless J, Ramasesh V, Dahlen D, Blok M, Kimchi-Schwartz M, McClean J, Carter J, de Jong W and Siddiqi I 2018 Phys. Rev. X 8 011021
[28] Rubin N C, Babbush R and McClean J 2018 New J. Phys. 20 053020
[29] Peruzzo A, McClean J, Shadbolt P, Yung M H, Zhou X Q, Love P J, Aspuru-Guzik A and O'Brien J L 2014 Nat. Commun. 5 4213
[30] Wang K, Chen Y A and Wang X 2021 arXiv:2111.00691[math-ph, physics:quant-ph]
[31] Giurgica-Tiron T, Hindy Y, LaRose R, Mari A and Zeng W J 2020 IEEE International Conference on Quantum Computing and Engineering (QCE), October 12-16, 2020, Denver, CO, USA, pp. 306-316
[32] Cai Z 2021 npj Quantum Inf. 7 80
[33] Wang P, Luan C Y, Qiao M, Um M, Zhang J, Wang Y, Yuan X, Gu M, Zhang J and Kim K 2021 Nat. Commun. 12 233
[34] Krebsbach M, Trauzettel B and Calzona A 2022 arXiv:2201.08080[cond-mat, physics:quant-ph]
[35] Bishop C 2006 Pattern Recognition and Machine Learning (Springer) pp. 4-11
[36] Wallman J J and Emerson J 2016 Phys. Rev. A 94 052325
[37] Dumitrescu E F, McCaskey A J, Hagen G, Jansen G R, Morris T D, Papenbrock T, Pooser R C, Dean D J and Lougovski P 2018 Phys. Rev. Lett. 120 210501
[38] Klco N, Dumitrescu E F, McCaskey A J, Morris T D, Pooser R C, Sanz M, Solano E, Lougovski P and Savage M J 2018 Phys. Rev. A 98 032331
[39] He A, Nachman B, de Jong W A and Bauer C W 2020 Phys. Rev. A 102 012426
[40] Schultz K, LaRose R, Mari A, Quiroz G, Shammah N, Clader B D and Zeng W J 2022 arXiv:2201.11792[quant-ph]
[41] Garmon J W O, Pooser R C and Dumitrescu E F 2020 Phys. Rev. A 101 042308
[42] Kandala A, Temme K, Córcoles A D, Mezzacapo A, Chow J M and Gambetta J M 2019 Nature 567 491
[43] McKay D C, Alexander T, Bello L, Biercuk M J, Bishop L, Chen J, Chow J M, Corcoles A D, Egger D, Filipp S, Gomez J, Hush M, Javadi-Abhari A, Moreda D, Nation P, Paulovicks B, Winston E, Wood C J, Wootton J and Gambetta J M 2018 arXiv:1809.03452[quant-ph]
[45] Otten M and Gray S K 2019 Phys. Rev. A 99 012338
[46] Otten M and Gray S K 2019 npj Quantum Inf. 5 11
[47] Takagi R 2021 Phys. Rev. Research 3 033178
[48] Piveteau C, Sutter D and Woerner S 2022 npj Quantum Inf. 8 12
[49] Jiang J, Wang K and Wang X 2021 Quantum 5 600
[50] Sun J, Yuan X, Tsunoda T, Vedral V, Benjamin S C and Endo S 2021 Phys. Rev. Applied 15 034026
[51] Hakoshima H, Matsuzaki Y and Endo S 2021 Phys. Rev. A 103 012611
[52] Chen Y, Farahzad M, Yoo S and Wei T C 2019 Phys. Rev. A 100 052315
[53] Kwon H and Bae J 2021 IEEE Transactions on Computers 70 1401
[54] Maciejewski F B, Zimboás Z and Oszmaniec M 2020 Quantum 4 257
[55] Geller M R and Sun M 2021 Quantum Sci. Technol. 6 025009
[56] Bravyi S, Sheldon S, Kandala A, Mckay D C and Gambetta J M 2021 Phys. Rev. A 103 042605
[57] Berthiaume A, Deutsch D and Jozsa R 1994 Proceedings Workshop on Physics and Computation. PhysComp'94, November 17-20, 1994, Dallas, TX, USA, pp. 60-62
[58] Barenco A, Berthiaume A, Deutsch D, Ekert A, Jozsa R and Macchiavello C 1997 SIAM J. Comput. 26 1541
[59] Cotler J, Choi S, Lukin A, Gharibyan H, Grover T, Tai M E, Rispoli M, Schittko R, Preiss P M, Kaufman A M, Greiner M, Pichler H and Hayden P 2019 Phys. Rev. X 9 031013
[60] Cai Z 2021 arXiv:2107.07279[quant-ph]
[61] Czarnik P, Arrasmith A, Cincio L and Coles P J 2021 arXiv:2102.06056[quant-ph]
[62] Koczor B 2021 New J. Phys. 23 123047
[63] McClean J R, Jiang Z, Rubin N C, Babbush R and Neven H 2020 Nat. Commun. 11 636
[64] Cai Z 2021 Quantum 5 548
[65] Aaronson S and Gottesman D 2004 Phys. Rev. A 70 052328
[66] Qin D, Chen Y and Li Y 2021 arXiv:2112.06255[quant-ph]
[67] Vovrosh J, Khosla K E, Greenaway S, Self C, Kim M S and Knolle J 2021 Phys. Rev. E 104 035309
[68] Urbanek M, Nachman B, Pascuzzi V R, He A, Bauer C W and de Jong W A 2021 Phys. Rev. Lett. 127 270502
[69] Wang X, Feng X, Funcke L, Hartung T, Jansen K, Kühn S, Polykratis G and Stornati P 2021 arXiv:2111.15522[hep-lat, physics:quant-ph]
[70] Wang Z, Chen Y, Song Z, Qin D, Li H, Guo Q, Wang H, Song C and Li Y 2021 Phys. Rev. Lett. 126 080501
[71] Dankert C, Cleve R, Emerson J and Livine E 2009 Phys. Rev. A 80 012304
[72] Dalzell A M, Hunter-Jones N and Brandão F G S L 2021 arXiv:2111.14907[quant-ph]
[73] Mari A, Shammah N and Zeng W J 2021 Phys. Rev. A 104 052607
[74] Yoshioka N, Hakoshima H, Matsuzaki Y, Tokunaga Y, Suzuki Y and Endo S 2022 Phys. Rev. Lett. 129 020502
[75] Bultrini D, Gordon M H, Czarnik P, Arrasmith A, Coles P J and Cincio L 2021 arXiv:2107.13470[quant-ph]
[76] Xiong Y, Chandra D, Ng S X and Hanzo L 2020 IEEE Access 8 228967
[77] Piveteau C, Sutter D, Bravyi S, Gambetta J M and Temme K 2021 Phys. Rev. Lett. 127 200505
[78] Lostaglio M and Ciani A 2021 Phys. Rev. Lett. 127 200506
[79] Suzuki Y, Endo S, Fujii K and Tokunaga Y 2022 PRX Quantum 3 010345
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