Quantum variational learning for quantum error-correcting codes

Chenfeng Cao1, Chao Zhang1, Zipeng Wu1, Markus Grassl2, and Bei Zeng1

1Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
2International Centre for Theory of Quantum Technologies, University of Gdansk, 80-309 Gdansk, Poland

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

Quantum error correction is believed to be a necessity for large-scale fault-tolerant quantum computation. In the past two decades, various constructions of quantum error-correcting codes (QECCs) have been developed, leading to many good code families. However, the majority of these codes are not suitable for near-term quantum devices. Here we present VarQEC, a noise-resilient variational quantum algorithm to search for quantum codes with a hardware-efficient encoding circuit. The cost functions are inspired by the most general and fundamental requirements of a QECC, the Knill-Laflamme conditions. Given the target noise channel (or the target code parameters) and the hardware connectivity graph, we optimize a shallow variational quantum circuit to prepare the basis states of an eligible code. In principle, VarQEC can find quantum codes for any error model, whether additive or non-additive, degenerate or non-degenerate, pure or impure. We have verified its effectiveness by (re)discovering some symmetric and asymmetric codes, e.g., $((n,2^{n-6},3))_2$ for $n$ from 7 to 14. We also found new $((6,2,3))_2$ and $((7,2,3))_2$ codes that are not equivalent to any stabilizer code, and extensive numerical evidence with VarQEC suggests that a $((7,3,3))_2$ code does not exist. Furthermore, we found many new channel-adaptive codes for error models involving nearest-neighbor correlated errors. Our work sheds new light on the understanding of QECC in general, which may also help to enhance near-term device performance with channel-adaptive error-correcting codes.

► BibTeX data

► References

[1] N. C. Jones, J. D. Whitfield, P. L. McMahon, M.-H. Yung, R. V. Meter, A. Aspuru-Guzik, and Y. Yamamoto, Faster quantum chemistry simulation on fault-tolerant quantum computers, New Journal of Physics 14, 115023 (2012).
https:/​/​doi.org/​10.1088/​1367-2630/​14/​11/​115023

[2] P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM J. Comput. 26, 1484–1509 (1997).
https:/​/​doi.org/​10.1137/​S0097539795293172

[3] A. W. Harrow, A. Hassidim, and S. Lloyd, Quantum algorithm for linear systems of equations, Phys. Rev. Lett. 103, 150502 (2009).
https:/​/​doi.org/​10.1103/​PhysRevLett.103.150502

[4] P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52, R2493 (1995).
https:/​/​doi.org/​10.1103/​PhysRevA.52.R2493

[5] D. Gottesman, Stabilizer codes and quantum error correction (California Institute of Technology, 1997).

[6] D. A. Lidar and T. A. Brun, Quantum error correction (Cambridge University Press, 2013).

[7] B. Zeng, X. Chen, D.-L. Zhou, and X.-G. Wen, Quantum information meets quantum matter: From quantum entanglement to topological phases of many-body systems (Springer, 2019).

[8] S. M. Girvin, Introduction to quantum error correction and fault tolerance (2021), arXiv:2111.08894.
arXiv:2111.08894

[9] F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill, Holographic quantum error-correcting codes: toy models for the bulk/​boundary correspondence, Journal of High Energy Physics 2015, 149 (2015).
https:/​/​doi.org/​10.1007/​JHEP06(2015)149

[10] E. Knill and R. Laflamme, Theory of quantum error-correcting codes, Phys. Rev. A 55, 900 (1997).
https:/​/​doi.org/​10.1103/​PhysRevA.55.900

[11] A. Y. Kitaev, Quantum computations: algorithms and error correction, Uspekhi Matematicheskikh Nauk 52, 53 (1997).

[12] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Surface codes: Towards practical large-scale quantum computation, Phys. Rev. A 86, 032324 (2012).
https:/​/​doi.org/​10.1103/​PhysRevA.86.032324

[13] A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Phys. Rev. A 54, 1098 (1996).
https:/​/​doi.org/​10.1103/​PhysRevA.54.1098

[14] A. Steane, Multiple-particle interference and quantum error correction, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 452, 2551 (1996a).
https:/​/​doi.org/​10.1098/​rspa.1996.0136

[15] A. Cross, G. Smith, J. A. Smolin, and B. Zeng, Codeword stabilized quantum codes, in 2008 IEEE International Symposium on Information Theory (2008) pp. 364–368.
https:/​/​doi.org/​10.1109/​ISIT.2008.4595009

[16] I. Chuang, A. Cross, G. Smith, J. Smolin, and B. Zeng, Codeword stabilized quantum codes: Algorithm and structure, Journal of Mathematical Physics 50, 042109 (2009).
https:/​/​doi.org/​10.1063/​1.3086833

[17] N. P. Breuckmann and J. N. Eberhardt, Quantum low-density parity-check codes, PRX Quantum 2, 040101 (2021).
https:/​/​doi.org/​10.1103/​PRXQuantum.2.040101

[18] P. Panteleev and G. Kalachev, Asymptotically good quantum and locally testable classical LDPC codes (2021), arXiv:2111.03654.
arXiv:2111.03654

[19] L. Egan, D. M. Debroy, C. Noel, A. Risinger, D. Zhu, D. Biswas, M. Newman, M. Li, K. R. Brown, M. Cetina, and C. Monroe, Fault-tolerant control of an error-corrected qubit, Nature 598, 281 (2021).
https:/​/​doi.org/​10.1038/​s41586-021-03928-y

[20] L. Postler, S. Heußen, I. Pogorelov, M. Rispler, T. Feldker, M. Meth, C. D. Marciniak, R. Stricker, M. Ringbauer, R. Blatt, P. Schindler, M. Müller, and T. Monz, Demonstration of fault-tolerant universal quantum gate operations (2021), arXiv:2111.12654.
arXiv:2111.12654

[21] C. M. Dawson, H. L. Haselgrove, and M. A. Nielsen, Noise thresholds for optical quantum computers, Phys. Rev. Lett. 96, 020501 (2006).
https:/​/​doi.org/​10.1103/​PhysRevLett.96.020501

[22] C. D. Wilen, S. Abdullah, N. A. Kurinsky, C. Stanford, L. Cardani, G. D'Imperio, C. Tomei, L. Faoro, L. B. Ioffe, C. H. Liu, A. Opremcak, B. G. Christensen, J. L. DuBois, and R. McDermott, Correlated charge noise and relaxation errors in superconducting qubits, Nature 594, 369 (2021).
https:/​/​doi.org/​10.1038/​s41586-021-03557-5

[23] Q. Guo, Y.-Y. Zhao, M. Grassl, X. Nie, G.-Y. Xiang, T. Xin, Z.-Q. Yin, and B. Zeng, Testing a quantum error-correcting code on various platforms, Science Bulletin 66, 29 (2021).
https:/​/​doi.org/​10.1016/​j.scib.2020.07.033

[24] S. Yu, Q. Chen, and C. H. Oh, Graphical quantum error-correcting codes (2007), arXiv:0709.1780.
arXiv:0709.1780

[25] D. Hu, W. Tang, M. Zhao, Q. Chen, S. Yu, and C. H. Oh, Graphical nonbinary quantum error-correcting codes, Phys. Rev. A 78, 012306 (2008).
https:/​/​doi.org/​10.1103/​PhysRevA.78.012306

[26] A. Jayashankar, A. M. Babu, H. K. Ng, and P. Mandayam, Finding good quantum codes using the cartan form, Phys. Rev. A 101, 042307 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.101.042307

[27] M. Li, M. Gutiérrez, S. E. David, A. Hernandez, and K. R. Brown, Fault tolerance with bare ancillary qubits for a [[7,1,3]] code, Phys. Rev. A 96, 032341 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.96.032341

[28] T. Fösel, P. Tighineanu, T. Weiss, and F. Marquardt, Reinforcement learning with neural networks for quantum feedback, Phys. Rev. X 8, 031084 (2018).
https:/​/​doi.org/​10.1103/​PhysRevX.8.031084

[29] P. Baireuther, T. E. O'Brien, B. Tarasinski, and C. W. J. Beenakker, Machine-learning-assisted correction of correlated qubit errors in a topological code, Quantum 2, 48 (2018).
https:/​/​doi.org/​10.22331/​q-2018-01-29-48

[30] P. Andreasson, J. Johansson, S. Liljestrand, and M. Granath, Quantum error correction for the toric code using deep reinforcement learning, Quantum 3, 183 (2019).
https:/​/​doi.org/​10.22331/​q-2019-09-02-183

[31] H. P. Nautrup, N. Delfosse, V. Dunjko, H. J. Briegel, and N. Friis, Optimizing quantum error correction codes with reinforcement learning, Quantum 3, 215 (2019).
https:/​/​doi.org/​10.22331/​q-2019-12-16-215

[32] M. Reimpell and R. F. Werner, Iterative optimization of quantum error correcting codes, Phys. Rev. Lett. 94, 080501 (2005).
https:/​/​doi.org/​10.1103/​PhysRevLett.94.080501

[33] A. S. Fletcher, P. W. Shor, and M. Z. Win, Optimum quantum error recovery using semidefinite programming, Phys. Rev. A 75, 012338 (2007).
https:/​/​doi.org/​10.1103/​PhysRevA.75.012338

[34] A. S. Fletcher, Channel-adapted quantum error correction (2007), arXiv:0706.3400.
arXiv:0706.3400

[35] R. Sweke, M. S. Kesselring, E. P. L. van Nieuwenburg, and J. Eisert, Reinforcement learning decoders for fault-tolerant quantum computation, Machine Learning: Science and Technology 2, 025005 (2020).
https:/​/​doi.org/​10.1088/​2632-2153/​abc609

[36] Y.-H. Liu and D. Poulin, Neural belief-propagation decoders for quantum error-correcting codes, Phys. Rev. Lett. 122, 200501 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.122.200501

[37] D. F. Locher, L. Cardarelli, and M. Müller, Quantum error correction with quantum autoencoders (2022), arXiv:2202.00555.
arXiv:2202.00555

[38] E. Knill and R. Laflamme, Concatenated quantum codes (1996), arXiv:quant-ph/​9608012.
arXiv:quant-ph/9608012

[39] M. Grassl, P. Shor, G. Smith, J. Smolin, and B. Zeng, Generalized concatenated quantum codes, Phys. Rev. A 79, 050306 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.79.050306

[40] D. Gottesman, An introduction to quantum error correction, in Proceedings of Symposia in Applied Mathematics, Vol. 58 (2002) pp. 221–236.

[41] P. Aliferis, F. Brito, D. P. DiVincenzo, J. Preskill, M. Steffen, and B. M. Terhal, Fault-tolerant computing with biased-noise superconducting qubits: a case study, New Journal of Physics 11, 013061 (2009).
https:/​/​doi.org/​10.1088/​1367-2630/​11/​1/​013061

[42] T. Jackson, M. Grassl, and B. Zeng, Concatenated codes for amplitude damping, in 2016 IEEE International Symposium on Information Theory (ISIT) (2016) pp. 2269–2273.
https:/​/​doi.org/​10.1109/​ISIT.2016.7541703

[43] D. W. Leung, M. A. Nielsen, I. L. Chuang, and Y. Yamamoto, Approximate quantum error correction can lead to better codes, Phys. Rev. A 56, 2567 (1997).
https:/​/​doi.org/​10.1103/​PhysRevA.56.2567

[44] B. Schumacher and M. D. Westmoreland, Approximate quantum error correction, Quantum Information Processing 1, 5 (2002).
https:/​/​doi.org/​10.1023/​A:1019653202562

[45] F. G. S. L. Brandão, E. Crosson, M. B. Şahinoğlu, and J. Bowen, Quantum error correcting codes in eigenstates of translation-invariant spin chains, Phys. Rev. Lett. 123, 110502 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.110502

[46] C. Bény and O. Oreshkov, General conditions for approximate quantum error correction and near-optimal recovery channels, Phys. Rev. Lett. 104, 120501 (2010).
https:/​/​doi.org/​10.1103/​PhysRevLett.104.120501

[47] D. Bures, An extension of Kakutani's theorem on infinite product measures to the tensor product of semifinite w*-algebras, Transactions of the American Mathematical Society 135, 199 (1969).
https:/​/​doi.org/​10.2307/​1995012

[48] M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Variational quantum algorithms, Nature Reviews Physics 3, 625 (2021a).
https:/​/​doi.org/​10.1038/​s42254-021-00348-9

[49] K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and A. Aspuru-Guzik, Noisy intermediate-scale quantum algorithms, Rev. Mod. Phys. 94, 015004 (2022).
https:/​/​doi.org/​10.1103/​RevModPhys.94.015004

[50] A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O'Brien, A variational eigenvalue solver on a photonic quantum processor, Nature Communications 5, 4213 (2014).
https:/​/​doi.org/​10.1038/​ncomms5213

[51] A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets, Nature 549, 242 (2017).
https:/​/​doi.org/​10.1038/​nature23879

[52] Y. Nam, J.-S. Chen, N. C. Pisenti, K. Wright, C. Delaney, D. Maslov, K. R. Brown, S. Allen, J. M. Amini, J. Apisdorf, K. M. Beck, A. Blinov, V. Chaplin, M. Chmielewski, C. Collins, S. Debnath, K. M. Hudek, A. M. Ducore, M. Keesan, S. M. Kreikemeier, J. Mizrahi, P. Solomon, M. Williams, J. D. Wong-Campos, D. Moehring, C. Monroe, and J. Kim, Ground-state energy estimation of the water molecule on a trapped-ion quantum computer, npj Quantum Information 6, 33 (2020).
https:/​/​doi.org/​10.1038/​s41534-020-0259-3

[53] C. Cao, Y. Yu, Z. Wu, N. Shannon, B. Zeng, and R. Joynt, Mitigating algorithmic errors in quantum optimization through energy extrapolation (2021), arXiv:2109.08132.
arXiv:2109.08132

[54] J. Romero, J. P. Olson, and A. Aspuru-Guzik, Quantum autoencoders for efficient compression of quantum data, Quantum Science and Technology 2, 045001 (2017).
https:/​/​iopscience.iop.org/​article/​10.1088/​2058-9565/​aa8072

[55] C. Cao and X. Wang, Noise-assisted quantum autoencoder, Phys. Rev. Applied 15, 054012 (2021).
https:/​/​doi.org/​10.1103/​PhysRevApplied.15.054012

[56] K. Sharma, S. Khatri, M. Cerezo, and P. J. Coles, Noise resilience of variational quantum compiling, New Journal of Physics 22, 043006 (2020).
https:/​/​doi.org/​10.1088/​1367-2630/​ab784c

[57] X. Xu, S. C. Benjamin, and X. Yuan, Variational circuit compiler for quantum error correction, Phys. Rev. Applied 15, 034068 (2021).
https:/​/​doi.org/​10.1103/​PhysRevApplied.15.034068

[58] K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, Quantum circuit learning, Phys. Rev. A 98, 032309 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.98.032309

[59] H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nature Physics 16, 1050 (2020).
https:/​/​doi.org/​10.1038/​s41567-020-0932-7

[60] M. J. D. Powell, An efficient method for finding the minimum of a function of several variables without calculating derivatives, The Computer Journal 7, 155 (1964), https:/​/​academic.oup.com/​comjnl/​article-pdf/​7/​2/​155/​959784/​070155.pdf.
https:/​/​doi.org/​10.1093/​comjnl/​7.2.155
arXiv:https://academic.oup.com/comjnl/article-pdf/7/2/155/959784/070155.pdf

[61] T. Haug, K. Bharti, and M. Kim, Capacity and quantum geometry of parametrized quantum circuits, PRX Quantum 2, 040309 (2021).
https:/​/​doi.org/​10.1103/​PRXQuantum.2.040309

[62] P. D. Johnson, J. Romero, J. Olson, Y. Cao, and A. Aspuru-Guzik, QVECTOR: an algorithm for device-tailored quantum error correction (2017), arXiv:1711.02249.
arXiv:1711.02249

[63] R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, Perfect quantum error correcting code, Phys. Rev. Lett. 77, 198 (1996).
https:/​/​doi.org/​10.1103/​PhysRevLett.77.198

[64] E. M. Rains, R. H. Hardin, P. W. Shor, and N. J. A. Sloane, A nonadditive quantum code, Phys. Rev. Lett. 79, 953 (1997).
https:/​/​doi.org/​10.1103/​PhysRevLett.79.953

[65] A. M. Steane, Simple quantum error-correcting codes, Phys. Rev. A 54, 4741 (1996b).
https:/​/​doi.org/​10.1103/​PhysRevA.54.4741

[66] L. Ioffe and M. Mézard, Asymmetric quantum error-correcting codes, Phys. Rev. A 75, 032345 (2007).
https:/​/​doi.org/​10.1103/​PhysRevA.75.032345

[67] P. K. Sarvepalli, A. Klappenecker, and M. Rotteler, Asymmetric quantum LDPC codes, in 2008 IEEE International Symposium on Information Theory (2008) pp. 305–309.
https:/​/​doi.org/​10.1109/​ISIT.2008.4594997

[68] P. K. Sarvepalli, A. Klappenecker, and M. Rötteler, Asymmetric quantum codes: constructions, bounds and performance, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, 1645 (2009).
https:/​/​doi.org/​10.1098/​rspa.2008.0439

[69] M. F. Ezerman, S. Ling, and P. Sole, Additive asymmetric quantum codes, IEEE Transactions on Information Theory 57, 5536 (2011).
https:/​/​doi.org/​10.1109/​TIT.2011.2159040

[70] M. F. Ezerman, S. Jitman, S. Ling, and D. V. Pasechnik, CSS-like constructions of asymmetric quantum codes, IEEE Transactions on Information Theory 59, 6732 (2013).
https:/​/​doi.org/​10.1109/​TIT.2013.2272575

[71] T. Jackson, M. Grassl, and B. Zeng, Codeword stabilized quantum codes for asymmetric channels, in 2016 IEEE International Symposium on Information Theory (ISIT) (2016) pp. 2264–2268.
https:/​/​doi.org/​10.1109/​ISIT.2016.7541702

[72] J. P. Bonilla Ataides, D. K. Tuckett, S. D. Bartlett, S. T. Flammia, and B. J. Brown, The xzzx surface code, Nature Communications 12, 2172 (2021).
https:/​/​doi.org/​10.1038/​s41467-021-22274-1

[73] P. Prabhu and B. W. Reichardt, Distance-four quantum codes with combined postselection and error correction (2021), arXiv:2112.03785.
arXiv:2112.03785

[74] A. Calderbank, E. Rains, P. Shor, and N. Sloane, Quantum error correction via codes over GF(4), IEEE Transactions on Information Theory 44, 1369 (1998).
https:/​/​doi.org/​10.1109/​18.681315

[75] Y. Hama, Quantum circuits for collective amplitude damping in two-qubit systems, (2020), arXiv:2012.02410.
arXiv:2012.02410

[76] M. Grassl, L. Kong, Z. Wei, Z.-Q. Yin, and B. Zeng, Quantum error-correcting codes for qudit amplitude damping, IEEE Transactions on Information Theory 64, 4674 (2018).

[77] P. Shor and R. Laflamme, Quantum analog of the macwilliams identities for classical coding theory, Phys. Rev. Lett. 78, 1600 (1997).
https:/​/​doi.org/​10.1103/​PhysRevLett.78.1600

[78] ``VarQEC GitHub repository". https:/​/​github.com/​caochenfeng/​VarQEC-public (2022).
https:/​/​github.com/​caochenfeng/​VarQEC-public

[79] Z. Chen, K. J. Satzinger, J. Atalaya, A. N. Korotkov, A. Dunsworth, D. Sank, C. Quintana, M. McEwen, R. Barends, P. V. Klimov, S. Hong, C. Jones, A. Petukhov, D. Kafri, S. Demura, B. Burkett, C. Gidney, A. G. Fowler, A. Paler, H. Putterman, I. Aleiner, F. Arute, K. Arya, R. Babbush, J. C. Bardin, A. Bengtsson, A. Bourassa, M. Broughton, B. B. Buckley, D. A. Buell, N. Bushnell, B. Chiaro, R. Collins, W. Courtney, A. R. Derk, D. Eppens, C. Erickson, E. Farhi, B. Foxen, M. Giustina, A. Greene, J. A. Gross, M. P. Harrigan, S. D. Harrington, J. Hilton, A. Ho, T. Huang, W. J. Huggins, L. B. Ioffe, S. V. Isakov, E. Jeffrey, Z. Jiang, K. Kechedzhi, S. Kim, A. Kitaev, F. Kostritsa, D. Landhuis, P. Laptev, E. Lucero, O. Martin, J. R. McClean, T. McCourt, X. Mi, K. C. Miao, M. Mohseni, S. Montazeri, W. Mruczkiewicz, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Newman, M. Y. Niu, T. E. O'Brien, A. Opremcak, E. Ostby, B. Pató, N. Redd, P. Roushan, N. C. Rubin, V. Shvarts, D. Strain, M. Szalay, M. D. Trevithick, B. Villalonga, T. White, Z. J. Yao, P. Yeh, J. Yoo, A. Zalcman, H. Neven, S. Boixo, V. Smelyanskiy, Y. Chen, A. Megrant, J. Kelly, and Google Quantum AI, Exponential suppression of bit or phase errors with cyclic error correction, Nature 595, 383 (2021).
https:/​/​doi.org/​10.1038/​s41586-021-03588-y

[80] A. M. Dalzell, N. Hunter-Jones, and F. G. S. L. Brandão, Random quantum circuits transform local noise into global white noise (2021), arXiv:2111.14907.
arXiv:2111.14907

[81] A. Deshpande, B. Fefferman, A. V. Gorshkov, M. J. Gullans, P. Niroula, and O. Shtanko, Tight bounds on the convergence of noisy random circuits to uniform (2021), arXiv:2112.00716.
arXiv:2112.00716

[82] W. J. Huggins, S. McArdle, T. E. O'Brien, J. Lee, N. C. Rubin, S. Boixo, K. B. Whaley, R. Babbush, and J. R. McClean, Virtual distillation for quantum error mitigation, Phys. Rev. X 11, 041036 (2021).
https:/​/​doi.org/​10.1103/​PhysRevX.11.041036

[83] B. Koczor, Exponential error suppression for near-term quantum devices, Phys. Rev. X 11, 031057 (2021).
https:/​/​doi.org/​10.1103/​PhysRevX.11.031057

[84] J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nature Communications 9, 4812 (2018).
https:/​/​doi.org/​10.1038/​s41467-018-07090-4

[85] M. Cerezo, A. Sone, T. Volkoff, L. Cincio, and P. J. Coles, Cost function dependent barren plateaus in shallow parametrized quantum circuits, Nature Communications 12, 1791 (2021b).
https:/​/​doi.org/​10.1038/​s41467-021-21728-w

[86] S. Wang, E. Fontana, M. Cerezo, K. Sharma, A. Sone, L. Cincio, and P. J. Coles, Noise-induced barren plateaus in variational quantum algorithms, Nature Communications 12, 6961 (2021).
https:/​/​doi.org/​10.1038/​s41467-021-27045-6

[87] T. L. Patti, K. Najafi, X. Gao, and S. F. Yelin, Entanglement devised barren plateau mitigation, Phys. Rev. Research 3, 033090 (2021).
https:/​/​doi.org/​10.1103/​PhysRevResearch.3.033090

[88] S. H. Sack, R. A. Medina, A. A. Michailidis, R. Kueng, and M. Serbyn, Avoiding barren plateaus using classical shadows, PRX Quantum 3, 020365 (2022).
https:/​/​doi.org/​10.1103/​PRXQuantum.3.020365

[89] 5-qubit backend: IBM Q team, ``IBM Q 5 Quito backend specification V1.1.34". Retrieved from https:/​/​quantum-computing.ibm.com (2022).
https:/​/​quantum-computing.ibm.com

[90] M. Grassl, S. Lu, and B. Zeng, Codes for simultaneous transmission of quantum and classical information, in 2017 IEEE International Symposium on Information Theory (ISIT) (2017) pp. 1718–1722.
https:/​/​doi.org/​10.1109/​ISIT.2017.8006823

[91] R. Duan, Super-activation of zero-error capacity of noisy quantum channels (2009), arXiv:0906.2527.
arXiv:0906.2527

[92] X.-D. Yu, T. Simnacher, N. Wyderka, H. C. Nguyen, and O. Gühne, A complete hierarchy for the pure state marginal problem in quantum mechanics, Nature Communications 12, 1012 (2021).
https:/​/​doi.org/​10.1038/​s41467-020-20799-5

[93] R. Orús, Tensor networks for complex quantum systems, Nature Reviews Physics 1, 538 (2019).
https:/​/​doi.org/​10.1038/​s42254-019-0086-7

[94] J. I. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys. 93, 045003 (2021).
https:/​/​doi.org/​10.1103/​RevModPhys.93.045003

[95] S. Cheng, C. Cao, C. Zhang, Y. Liu, S.-Y. Hou, P. Xu, and B. Zeng, Simulating noisy quantum circuits with matrix product density operators, Phys. Rev. Research 3, 023005 (2021).
https:/​/​doi.org/​10.1103/​PhysRevResearch.3.023005

[96] G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science 355, 602 (2017).
https:/​/​doi.org/​10.1126/​science.aag2302

[97] C. W. Helstrom, Quantum detection and estimation theory, Journal of Statistical Physics 1, 231 (1969).
https:/​/​doi.org/​10.1007/​BF01007479

[98] D. Šafránek, Simple expression for the quantum Fisher information matrix, Phys. Rev. A 97, 042322 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.97.042322

[99] J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Quantum fisher information matrix and multiparameter estimation, Journal of Physics A: Mathematical and Theoretical 53, 023001 (2019).
https:/​/​doi.org/​10.1088/​1751-8121/​ab5d4d

[100] J. J. Meyer, Fisher Information in Noisy Intermediate-Scale Quantum Applications, Quantum 5, 539 (2021).
https:/​/​doi.org/​10.22331/​q-2021-09-09-539

[101] J. Milnor and J. D. Stasheff, Characteristic Classes. Annals of Mathematics Studies, volume 76 (Princeton University Press, 2016).

[1] N. Cody Jones, James D. Whitfield, Peter L. McMahon, Man-Hong Yung, Rodney Van Meter, Alán Aspuru-Guzik, and Yoshihisa Yamamoto. ``Faster quantum chemistry simulation on fault-tolerant quantum computers''. New Journal of Physics 14, 115023 (2012).
https:/​/​doi.org/​10.1088/​1367-2630/​14/​11/​115023

[2] Peter W. Shor. ``Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer''. SIAM J. Comput. 26, 1484–1509 (1997).
https:/​/​doi.org/​10.1137/​S0097539795293172

[3] Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. ``Quantum algorithm for linear systems of equations''. Phys. Rev. Lett. 103, 150502 (2009).
https:/​/​doi.org/​10.1103/​PhysRevLett.103.150502

[4] Peter W. Shor. ``Scheme for reducing decoherence in quantum computer memory''. Phys. Rev. A 52, R2493–R2496 (1995).
https:/​/​doi.org/​10.1103/​PhysRevA.52.R2493

[5] Daniel Gottesman. ``Stabilizer codes and quantum error correction'' (1997).
arXiv:quant-ph/9705052

[6] Daniel A. Lidar and Todd A. Brun. ``Quantum error correction''. Cambridge University Press. (2013).
https:/​/​doi.org/​10.1017/​CBO9781139034807

[7] Bei Zeng, Xie Chen, Duan-Lu Zhou, and Xiao-Gang Wen. ``Quantum information meets quantum matter: From quantum entanglement to topological phases of many-body systems''. Springer. (2019).
https:/​/​doi.org/​10.1007/​978-1-4939-9084-9

[8] Steven M. Girvin. ``Introduction to quantum error correction and fault tolerance'' (2021). arXiv:2111.08894.
arXiv:2111.08894

[9] Fernando Pastawski, Beni Yoshida, Daniel Harlow, and John Preskill. ``Holographic quantum error-correcting codes: toy models for the bulk/​boundary correspondence''. Journal of High Energy Physics 2015, 149 (2015).
https:/​/​doi.org/​10.1007/​JHEP06(2015)149

[10] Emanuel Knill and Raymond Laflamme. ``Theory of quantum error-correcting codes''. Phys. Rev. A 55, 900–911 (1997).
https:/​/​doi.org/​10.1103/​PhysRevA.55.900

[11] A. Yu Kitaev. ``Quantum computations: algorithms and error correction''. Russian Mathematical Surveys 52, 1191–1249 (1997).
https:/​/​doi.org/​10.1070/​rm1997v052n06abeh002155

[12] Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland. ``Surface codes: Towards practical large-scale quantum computation''. Phys. Rev. A 86, 032324 (2012).
https:/​/​doi.org/​10.1103/​PhysRevA.86.032324

[13] A. R. Calderbank and Peter W. Shor. ``Good quantum error-correcting codes exist''. Phys. Rev. A 54, 1098–1105 (1996).
https:/​/​doi.org/​10.1103/​PhysRevA.54.1098

[14] Andrew Steane. ``Multiple-particle interference and quantum error correction''. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 452, 2551–2577 (1996).
https:/​/​doi.org/​10.1098/​rspa.1996.0136

[15] Andrew Cross, Graeme Smith, John A. Smolin, and Bei Zeng. ``Codeword stabilized quantum codes''. In 2008 IEEE International Symposium on Information Theory. Pages 364–368. (2008).
https:/​/​doi.org/​10.1109/​ISIT.2008.4595009

[16] Isaac Chuang, Andrew Cross, Graeme Smith, John Smolin, and Bei Zeng. ``Codeword stabilized quantum codes: Algorithm and structure''. Journal of Mathematical Physics 50, 042109 (2009).
https:/​/​doi.org/​10.1063/​1.3086833

[17] Nikolas P. Breuckmann and Jens Niklas Eberhardt. ``Quantum low-density parity-check codes''. PRX Quantum 2, 040101 (2021).
https:/​/​doi.org/​10.1103/​PRXQuantum.2.040101

[18] Pavel Panteleev and Gleb Kalachev. ``Asymptotically good quantum and locally testableclassical ldpc codes''. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing. Pages 375–388. Association for Computing Machinery (2022).
https:/​/​doi.org/​10.1145/​3519935.3520017

[19] Laird Egan, Dripto M. Debroy, Crystal Noel, Andrew Risinger, Daiwei Zhu, Debopriyo Biswas, Michael Newman, Muyuan Li, Kenneth R. Brown, Marko Cetina, and Christopher Monroe. ``Fault-tolerant control of an error-corrected qubit''. Nature 598, 281–286 (2021).
https:/​/​doi.org/​10.1038/​s41586-021-03928-y

[20] Lukas Postler, Sascha Heußen, Ivan Pogorelov, Manuel Rispler, Thomas Feldker, Michael Meth, Christian D. Marciniak, Roman Stricker, Martin Ringbauer, Rainer Blatt, Philipp Schindler, Markus Müller, and Thomas Monz. ``Demonstration of fault-tolerant universal quantum gate operations''. Nature 605, 675–680 (2022).
https:/​/​doi.org/​10.1038/​s41586-022-04721-1

[21] Christopher M. Dawson, Henry L. Haselgrove, and Michael A. Nielsen. ``Noise thresholds for optical quantum computers''. Phys. Rev. Lett. 96, 020501 (2006).
https:/​/​doi.org/​10.1103/​PhysRevLett.96.020501

[22] C. D. Wilen, S. Abdullah, N. A. Kurinsky, C. Stanford, L. Cardani, G. D'Imperio, C. Tomei, L. Faoro, L. B. Ioffe, C. H. Liu, A. Opremcak, B. G. Christensen, J. L. DuBois, and R. McDermott. ``Correlated charge noise and relaxation errors in superconducting qubits''. Nature 594, 369–373 (2021).
https:/​/​doi.org/​10.1038/​s41586-021-03557-5

[23] Qihao Guo, Yuan-Yuan Zhao, Markus Grassl, Xinfang Nie, Guo-Yong Xiang, Tao Xin, Zhang-Qi Yin, and Bei Zeng. ``Testing a quantum error-correcting code on various platforms''. Science Bulletin 66, 29–35 (2021).
https:/​/​doi.org/​10.1016/​j.scib.2020.07.033

[24] Sixia Yu, Qing Chen, and C. H. Oh. ``Graphical quantum error-correcting codes'' (2007). arXiv:0709.1780.
arXiv:0709.1780

[25] Dan Hu, Weidong Tang, Meisheng Zhao, Qing Chen, Sixia Yu, and C. H. Oh. ``Graphical nonbinary quantum error-correcting codes''. Phys. Rev. A 78, 012306 (2008).
https:/​/​doi.org/​10.1103/​PhysRevA.78.012306

[26] Akshaya Jayashankar, Anjala M. Babu, Hui Khoon Ng, and Prabha Mandayam. ``Finding good quantum codes using the cartan form''. Phys. Rev. A 101, 042307 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.101.042307

[27] Muyuan Li, Mauricio Gutiérrez, Stanley E. David, Alonzo Hernandez, and Kenneth R. Brown. ``Fault tolerance with bare ancillary qubits for a [[7,1,3]] code''. Phys. Rev. A 96, 032341 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.96.032341

[28] Thomas Fösel, Petru Tighineanu, Talitha Weiss, and Florian Marquardt. ``Reinforcement learning with neural networks for quantum feedback''. Phys. Rev. X 8, 031084 (2018).
https:/​/​doi.org/​10.1103/​PhysRevX.8.031084

[29] Paul Baireuther, Thomas E. O'Brien, Brian Tarasinski, and Carlo W. J. Beenakker. ``Machine-learning-assisted correction of correlated qubit errors in a topological code''. Quantum 2, 48 (2018).
https:/​/​doi.org/​10.22331/​q-2018-01-29-48

[30] Philip Andreasson, Joel Johansson, Simon Liljestrand, and Mats Granath. ``Quantum error correction for the toric code using deep reinforcement learning''. Quantum 3, 183 (2019).
https:/​/​doi.org/​10.22331/​q-2019-09-02-183

[31] Hendrik Poulsen Nautrup, Nicolas Delfosse, Vedran Dunjko, Hans J. Briegel, and Nicolai Friis. ``Optimizing quantum error correction codes with reinforcement learning''. Quantum 3, 215 (2019).
https:/​/​doi.org/​10.22331/​q-2019-12-16-215

[32] M. Reimpell and R. F. Werner. ``Iterative optimization of quantum error correcting codes''. Phys. Rev. Lett. 94, 080501 (2005).
https:/​/​doi.org/​10.1103/​PhysRevLett.94.080501

[33] Andrew S. Fletcher, Peter W. Shor, and Moe Z. Win. ``Optimum quantum error recovery using semidefinite programming''. Phys. Rev. A 75, 012338 (2007).
https:/​/​doi.org/​10.1103/​PhysRevA.75.012338

[34] Andrew S. Fletcher. ``Channel-adapted quantum error correction'' (2007). arXiv:0706.3400.
arXiv:0706.3400

[35] Ryan Sweke, Markus S. Kesselring, Evert P. L. van Nieuwenburg, and Jens Eisert. ``Reinforcement learning decoders for fault-tolerant quantum computation''. Machine Learning: Science and Technology 2, 025005 (2020).
https:/​/​doi.org/​10.1088/​2632-2153/​abc609

[36] Ye-Hua Liu and David Poulin. ``Neural belief-propagation decoders for quantum error-correcting codes''. Phys. Rev. Lett. 122, 200501 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.122.200501

[37] David F. Locher, Lorenzo Cardarelli, and Markus Müller. ``Quantum error correction with quantum autoencoders'' (2022). arXiv:2202.00555.
arXiv:2202.00555

[38] Emanuel Knill and Raymond Laflamme. ``Concatenated quantum codes'' (1996). arXiv:quant-ph/​9608012.
arXiv:quant-ph/9608012

[39] Markus Grassl, Peter Shor, Graeme Smith, John Smolin, and Bei Zeng. ``Generalized concatenated quantum codes''. Phys. Rev. A 79, 050306 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.79.050306

[40] Daniel Gottesman. ``An introduction to quantum error correction''. In Proceedings of Symposia in Applied Mathematics. Volume 58, pages 221–236. (2002).

[41] P. Aliferis, F. Brito, D. P. DiVincenzo, J. Preskill, M. Steffen, and B. M. Terhal. ``Fault-tolerant computing with biased-noise superconducting qubits: a case study''. New Journal of Physics 11, 013061 (2009).
https:/​/​doi.org/​10.1088/​1367-2630/​11/​1/​013061

[42] Tyler Jackson, Markus Grassl, and Bei Zeng. ``Concatenated codes for amplitude damping''. In 2016 IEEE International Symposium on Information Theory (ISIT). Pages 2269–2273. (2016).
https:/​/​doi.org/​10.1109/​ISIT.2016.7541703

[43] Debbie W. Leung, M. A. Nielsen, Isaac L. Chuang, and Yoshihisa Yamamoto. ``Approximate quantum error correction can lead to better codes''. Phys. Rev. A 56, 2567–2573 (1997).
https:/​/​doi.org/​10.1103/​PhysRevA.56.2567

[44] Benjamin Schumacher and Michael D. Westmoreland. ``Approximate quantum error correction''. Quantum Information Processing 1, 5–12 (2002).
https:/​/​doi.org/​10.1023/​A:1019653202562

[45] Fernando G. S. L. Brandão, Elizabeth Crosson, M. Burak Şahinoğlu, and John Bowen. ``Quantum error correcting codes in eigenstates of translation-invariant spin chains''. Phys. Rev. Lett. 123, 110502 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.110502

[46] Cédric Bény and Ognyan Oreshkov. ``General conditions for approximate quantum error correction and near-optimal recovery channels''. Phys. Rev. Lett. 104, 120501 (2010).
https:/​/​doi.org/​10.1103/​PhysRevLett.104.120501

[47] Donald Bures. ``An extension of Kakutani's theorem on infinite product measures to the tensor product of semifinite w*-algebras''. Transactions of the American Mathematical Society 135, 199–212 (1969).
https:/​/​doi.org/​10.2307/​1995012

[48] M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles. ``Variational quantum algorithms''. Nature Reviews Physics 3, 625–644 (2021).
https:/​/​doi.org/​10.1038/​s42254-021-00348-9

[49] Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Alán Aspuru-Guzik. ``Noisy intermediate-scale quantum algorithms''. Rev. Mod. Phys. 94, 015004 (2022).
https:/​/​doi.org/​10.1103/​RevModPhys.94.015004

[50] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Alán Aspuru-Guzik, and Jeremy L. O'Brien. ``A variational eigenvalue solver on a photonic quantum processor''. Nature Communications 5, 4213 (2014).
https:/​/​doi.org/​10.1038/​ncomms5213

[51] Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M. Chow, and Jay M. Gambetta. ``Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets''. Nature 549, 242–246 (2017).
https:/​/​doi.org/​10.1038/​nature23879

[52] Yunseong Nam, Jwo-Sy Chen, Neal C. Pisenti, Kenneth Wright, Conor Delaney, Dmitri Maslov, Kenneth R. Brown, Stewart Allen, Jason M. Amini, Joel Apisdorf, Kristin M. Beck, Aleksey Blinov, Vandiver Chaplin, Mika Chmielewski, Coleman Collins, Shantanu Debnath, Kai M. Hudek, Andrew M. Ducore, Matthew Keesan, Sarah M. Kreikemeier, Jonathan Mizrahi, Phil Solomon, Mike Williams, Jaime David Wong-Campos, David Moehring, Christopher Monroe, and Jungsang Kim. ``Ground-state energy estimation of the water molecule on a trapped-ion quantum computer''. npj Quantum Information 6, 33 (2020).
https:/​/​doi.org/​10.1038/​s41534-020-0259-3

[53] Chenfeng Cao, Yunlong Yu, Zipeng Wu, Nic Shannon, Bei Zeng, and Robert Joynt. ``Mitigating algorithmic errors in quantum optimization through energy extrapolation''. Quantum Science and Technology (2022).
https:/​/​doi.org/​10.1088/​2058-9565/​ac969c

[54] Jonathan Romero, Jonathan P Olson, and Alan Aspuru-Guzik. ``Quantum autoencoders for efficient compression of quantum data''. Quantum Science and Technology 2, 045001 (2017).
https:/​/​doi.org/​10.1088/​2058-9565/​aa8072

[55] Chenfeng Cao and Xin Wang. ``Noise-assisted quantum autoencoder''. Phys. Rev. Applied 15, 054012 (2021).
https:/​/​doi.org/​10.1103/​PhysRevApplied.15.054012

[56] Kunal Sharma, Sumeet Khatri, M. Cerezo, and Patrick J. Coles. ``Noise resilience of variational quantum compiling''. New Journal of Physics 22, 043006 (2020).
https:/​/​doi.org/​10.1088/​1367-2630/​ab784c

[57] Xiaosi Xu, Simon C. Benjamin, and Xiao Yuan. ``Variational circuit compiler for quantum error correction''. Phys. Rev. Applied 15, 034068 (2021).
https:/​/​doi.org/​10.1103/​PhysRevApplied.15.034068

[58] K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii. ``Quantum circuit learning''. Phys. Rev. A 98, 032309 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.98.032309

[59] Hsin-Yuan Huang, Richard Kueng, and John Preskill. ``Predicting many properties of a quantum system from very few measurements''. Nature Physics 16, 1050–1057 (2020).
https:/​/​doi.org/​10.1038/​s41567-020-0932-7

[60] M. J. D. Powell. ``An efficient method for finding the minimum of a function of several variables without calculating derivatives''. The Computer Journal 7, 155–162 (1964).
https:/​/​doi.org/​10.1093/​comjnl/​7.2.155

[61] Tobias Haug, Kishor Bharti, and M. S. Kim. ``Capacity and quantum geometry of parametrized quantum circuits''. PRX Quantum 2, 040309 (2021).
https:/​/​doi.org/​10.1103/​PRXQuantum.2.040309

[62] Peter D. Johnson, Jonathan Romero, Jonathan Olson, Yudong Cao, and Alán Aspuru-Guzik. ``QVECTOR: an algorithm for device-tailored quantum error correction'' (2017). arXiv:1711.02249.
arXiv:1711.02249

[63] Raymond Laflamme, Cesar Miquel, Juan Pablo Paz, and Wojciech Hubert Zurek. ``Perfect quantum error correcting code''. Phys. Rev. Lett. 77, 198–201 (1996).
https:/​/​doi.org/​10.1103/​PhysRevLett.77.198

[64] Eric M. Rains, R. H. Hardin, Peter W. Shor, and N. J. A. Sloane. ``A nonadditive quantum code''. Phys. Rev. Lett. 79, 953–954 (1997).
https:/​/​doi.org/​10.1103/​PhysRevLett.79.953

[65] A. M. Steane. ``Simple quantum error-correcting codes''. Phys. Rev. A 54, 4741–4751 (1996).
https:/​/​doi.org/​10.1103/​PhysRevA.54.4741

[66] Lev Ioffe and Marc Mézard. ``Asymmetric quantum error-correcting codes''. Phys. Rev. A 75, 032345 (2007).
https:/​/​doi.org/​10.1103/​PhysRevA.75.032345

[67] Pradeep Kiran Sarvepalli, Andreas Klappenecker, and Martin Rotteler. ``Asymmetric quantum LDPC codes''. In 2008 IEEE International Symposium on Information Theory. Pages 305–309. (2008).
https:/​/​doi.org/​10.1109/​ISIT.2008.4594997

[68] Pradeep Kiran Sarvepalli, Andreas Klappenecker, and Martin Rötteler. ``Asymmetric quantum codes: constructions, bounds and performance''. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, 1645–1672 (2009).
https:/​/​doi.org/​10.1098/​rspa.2008.0439

[69] Martianus Frederic Ezerman, San Ling, and Patrick Sole. ``Additive asymmetric quantum codes''. IEEE Transactions on Information Theory 57, 5536–5550 (2011).
https:/​/​doi.org/​10.1109/​TIT.2011.2159040

[70] Martianus Frederic Ezerman, Somphong Jitman, San Ling, and Dmitrii V. Pasechnik. ``CSS-like constructions of asymmetric quantum codes''. IEEE Transactions on Information Theory 59, 6732–6754 (2013).
https:/​/​doi.org/​10.1109/​TIT.2013.2272575

[71] Tyler Jackson, Markus Grassl, and Bei Zeng. ``Codeword stabilized quantum codes for asymmetric channels''. In 2016 IEEE International Symposium on Information Theory (ISIT). Pages 2264–2268. (2016).
https:/​/​doi.org/​10.1109/​ISIT.2016.7541702

[72] J. Pablo Bonilla Ataides, David K. Tuckett, Stephen D. Bartlett, Steven T. Flammia, and Benjamin J. Brown. ``The xzzx surface code''. Nature Communications 12, 2172 (2021).
https:/​/​doi.org/​10.1038/​s41467-021-22274-1

[73] Prithviraj Prabhu and Ben W. Reichardt. ``Distance-four quantum codes with combined postselection and error correction'' (2021). arXiv:2112.03785.
arXiv:2112.03785

[74] A. R. Calderbank, E. M. Rains, P. M. Shor, and N. J. A. Sloane. ``Quantum error correction via codes over GF(4)''. IEEE Transactions on Information Theory 44, 1369–1387 (1998).
https:/​/​doi.org/​10.1109/​18.681315

[75] Yusuke Hama. ``Quantum circuits for collective amplitude damping in two-qubit systems'' (2020). arXiv:2012.02410.
arXiv:2012.02410

[76] Markus Grassl, Linghang Kong, Zhaohui Wei, Zhang-Qi Yin, and Bei Zeng. ``Quantum error-correcting codes for qudit amplitude damping''. IEEE Transactions on Information Theory 64, 4674–4685 (2018).
https:/​/​doi.org/​10.1109/​TIT.2018.2790423

[77] Peter Shor and Raymond Laflamme. ``Quantum analog of the macwilliams identities for classical coding theory''. Phys. Rev. Lett. 78, 1600–1602 (1997).
https:/​/​doi.org/​10.1103/​PhysRevLett.78.1600

[78] Chenfeng Cao. ``VarQEC GitHub repository''. https:/​/​github.com/​caochenfeng/​VarQEC-public (2022).
https:/​/​github.com/​caochenfeng/​VarQEC-public

[79] Zijun Chen, Kevin J. Satzinger, Juan Atalaya, Alexander N. Korotkov, Andrew Dunsworth, Daniel Sank, Chris Quintana, Matt McEwen, Rami Barends, Paul V. Klimov, Sabrina Hong, Cody Jones, Andre Petukhov, Dvir Kafri, Sean Demura, Brian Burkett, Craig Gidney, Austin G. Fowler, Alexandru Paler, Harald Putterman, Igor Aleiner, Frank Arute, Kunal Arya, Ryan Babbush, Joseph C. Bardin, Andreas Bengtsson, Alexandre Bourassa, Michael Broughton, Bob B. Buckley, David A. Buell, Nicholas Bushnell, Benjamin Chiaro, Roberto Collins, William Courtney, Alan R. Derk, Daniel Eppens, Catherine Erickson, Edward Farhi, Brooks Foxen, Marissa Giustina, Ami Greene, Jonathan A. Gross, Matthew P. Harrigan, Sean D. Harrington, Jeremy Hilton, Alan Ho, Trent Huang, William J. Huggins, L. B. Ioffe, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Kostyantyn Kechedzhi, Seon Kim, Alexei Kitaev, Fedor Kostritsa, David Landhuis, Pavel Laptev, Erik Lucero, Orion Martin, Jarrod R. McClean, Trevor McCourt, Xiao Mi, Kevin C. Miao, Masoud Mohseni, Shirin Montazeri, Wojciech Mruczkiewicz, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Michael Newman, Murphy Yuezhen Niu, Thomas E. O'Brien, Alex Opremcak, Eric Ostby, Bálint Pató, Nicholas Redd, Pedram Roushan, Nicholas C. Rubin, Vladimir Shvarts, Doug Strain, Marco Szalay, Matthew D. Trevithick, Benjamin Villalonga, Theodore White, Z. Jamie Yao, Ping Yeh, Juhwan Yoo, Adam Zalcman, Hartmut Neven, Sergio Boixo, Vadim Smelyanskiy, Yu Chen, Anthony Megrant, Julian Kelly, and Google Quantum AI. ``Exponential suppression of bit or phase errors with cyclic error correction''. Nature 595, 383–387 (2021).
https:/​/​doi.org/​10.1038/​s41586-021-03588-y

[80] Alexander M. Dalzell, Nicholas Hunter-Jones, and Fernando G. S. L. Brandão. ``Random quantum circuits transform local noise into global white noise'' (2021). arXiv:2111.14907.
arXiv:2111.14907

[81] Abhinav Deshpande, Bill Fefferman, Alexey V. Gorshkov, Michael J. Gullans, Pradeep Niroula, and Oles Shtanko. ``Tight bounds on the convergence of noisy random circuits to uniform'' (2021). arXiv:2112.00716.
arXiv:2112.00716

[82] William J. Huggins, Sam McArdle, Thomas E. O'Brien, Joonho Lee, Nicholas C. Rubin, Sergio Boixo, K. Birgitta Whaley, Ryan Babbush, and Jarrod R. McClean. ``Virtual distillation for quantum error mitigation''. Phys. Rev. X 11, 041036 (2021).
https:/​/​doi.org/​10.1103/​PhysRevX.11.041036

[83] Bálint Koczor. ``Exponential error suppression for near-term quantum devices''. Phys. Rev. X 11, 031057 (2021).
https:/​/​doi.org/​10.1103/​PhysRevX.11.031057

[84] Jarrod R. McClean, Sergio Boixo, Vadim N. Smelyanskiy, Ryan Babbush, and Hartmut Neven. ``Barren plateaus in quantum neural network training landscapes''. Nature Communications 9, 4812 (2018).
https:/​/​doi.org/​10.1038/​s41467-018-07090-4

[85] M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J. Coles. ``Cost function dependent barren plateaus in shallow parametrized quantum circuits''. Nature Communications 12, 1791 (2021).
https:/​/​doi.org/​10.1038/​s41467-021-21728-w

[86] Samson Wang, Enrico Fontana, M. Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio, and Patrick J. Coles. ``Noise-induced barren plateaus in variational quantum algorithms''. Nature Communications 12, 6961 (2021).
https:/​/​doi.org/​10.1038/​s41467-021-27045-6

[87] Taylor L. Patti, Khadijeh Najafi, Xun Gao, and Susanne F. Yelin. ``Entanglement devised barren plateau mitigation''. Phys. Rev. Research 3, 033090 (2021).
https:/​/​doi.org/​10.1103/​PhysRevResearch.3.033090

[88] Stefan H. Sack, Raimel A. Medina, Alexios A. Michailidis, Richard Kueng, and Maksym Serbyn. ``Avoiding barren plateaus using classical shadows''. PRX Quantum 3, 020365 (2022).
https:/​/​doi.org/​10.1103/​PRXQuantum.3.020365

[89] 5 qubit backend: IBM Q team. ``IBM Q 5 Quito backend specification v1.1.34''. Retrieved from https:/​/​quantum-computing.ibm.com (2022).
https:/​/​quantum-computing.ibm.com

[90] Markus Grassl, Sirui Lu, and Bei Zeng. ``Codes for simultaneous transmission of quantum and classical information''. In 2017 IEEE International Symposium on Information Theory (ISIT). Pages 1718–1722. (2017).
https:/​/​doi.org/​10.1109/​ISIT.2017.8006823

[91] Runyao Duan. ``Super-activation of zero-error capacity of noisy quantum channels'' (2009). arXiv:0906.2527.
arXiv:0906.2527

[92] Xiao-Dong Yu, Timo Simnacher, Nikolai Wyderka, H. Chau Nguyen, and Otfried Gühne. ``A complete hierarchy for the pure state marginal problem in quantum mechanics''. Nature Communications 12, 1012 (2021).
https:/​/​doi.org/​10.1038/​s41467-020-20799-5

[93] Román Orús. ``Tensor networks for complex quantum systems''. Nature Reviews Physics 1, 538–550 (2019).
https:/​/​doi.org/​10.1038/​s42254-019-0086-7

[94] J. Ignacio Cirac, David Pérez-García, Norbert Schuch, and Frank Verstraete. ``Matrix product states and projected entangled pair states: Concepts, symmetries, theorems''. Rev. Mod. Phys. 93, 045003 (2021).
https:/​/​doi.org/​10.1103/​RevModPhys.93.045003

[95] Song Cheng, Chenfeng Cao, Chao Zhang, Yongxiang Liu, Shi-Yao Hou, Pengxiang Xu, and Bei Zeng. ``Simulating noisy quantum circuits with matrix product density operators''. Phys. Rev. Research 3, 023005 (2021).
https:/​/​doi.org/​10.1103/​PhysRevResearch.3.023005

[96] Giuseppe Carleo and Matthias Troyer. ``Solving the quantum many-body problem with artificial neural networks''. Science 355, 602–606 (2017).
https:/​/​doi.org/​10.1126/​science.aag2302

[97] Carl W. Helstrom. ``Quantum detection and estimation theory''. Journal of Statistical Physics 1, 231–252 (1969).
https:/​/​doi.org/​10.1007/​BF01007479

[98] Dominik Šafránek. ``Simple expression for the quantum Fisher information matrix''. Phys. Rev. A 97, 042322 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.97.042322

[99] Jing Liu, Haidong Yuan, Xiao-Ming Lu, and Xiaoguang Wang. ``Quantum fisher information matrix and multiparameter estimation''. Journal of Physics A: Mathematical and Theoretical 53, 023001 (2019).
https:/​/​doi.org/​10.1088/​1751-8121/​ab5d4d

[100] Johannes Jakob Meyer. ``Fisher Information in Noisy Intermediate-Scale Quantum Applications''. Quantum 5, 539 (2021).
https:/​/​doi.org/​10.22331/​q-2021-09-09-539

[101] John Milnor and James D Stasheff. ``Characteristic classes. annals of mathematics studies, volume 76''. Princeton University Press. (2016).

Cited by

[1] Akshaya Jayashankar and Prabha Mandayam, "Quantum Error Correction: Noise-Adapted Techniques and Applications", arXiv:2208.00365, Journal of the Indian Institute of Science (2022).

[2] Chenfeng Cao, Yunlong Yu, Zipeng Wu, Nic Shannon, Bei Zeng, and Robert Joynt, "Mitigating algorithmic errors in quantum optimization through energy extrapolation", arXiv:2109.08132, Quantum Science and Technology 8 1, 015004 (2023).

[3] Shi-Yao Hou, Zipeng Wu, Jinfeng Zeng, Ningping Cao, Chenfeng Cao, Youning Li, and Bei Zeng, "Maximum entropy methods for quantum state compatibility problems", arXiv:2207.11645.

The above citations are from Crossref's cited-by service (last updated successfully 2022-11-30 03:59:30) and SAO/NASA ADS (last updated successfully 2022-11-30 03:59:31). The list may be incomplete as not all publishers provide suitable and complete citation data.