Improved Fault-Tolerant Quantum Simulation of Condensed-Phase Correlated Electrons via Trotterization

Ian D. Kivlichan1,2, Craig Gidney3, Dominic W. Berry4, Nathan Wiebe5, Jarrod McClean1, Wei Sun6, Zhang Jiang1, Nicholas Rubin1, Austin Fowler3, Alán Aspuru-Guzik7,8, Hartmut Neven1, and Ryan Babbush1

1Google Research, Venice, CA 90291, USA
2Department of Physics, Harvard University, Cambridge, MA 02138, USA
3Google Research, Santa Barbara, CA 93117, USA
4Department of Physics and Astronomy, Macquarie University, Sydney, NSW 2113, Australia
5Institute for Nuclear Theory, University of Washington, Seattle, WA 98195, USA
6Google Research, Mountain View, CA 94043, USA
7Department of Chemistry, University of Toronto, Toronto, Ontario M5G 1Z8, Canada
8Department of Computer Science, University of Toronto, Toronto, Ontario M5G 1Z8, Canada

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Abstract

Recent work has deployed linear combinations of unitaries techniques to reduce the cost of fault-tolerant quantum simulations of correlated electron models. Here, we show that one can sometimes improve upon those results with optimized implementations of Trotter-Suzuki-based product formulas. We show that low-order Trotter methods perform surprisingly well when used with phase estimation to compute relative precision quantities (e.g. energies per unit cell), as is often the goal for condensed-phase systems. In this context, simulations of the Hubbard and plane-wave electronic structure models with $N < 10^5$ fermionic modes can be performed with roughly ${\cal O}(1)$ and ${\cal O}(N^2)$ T complexities. We perform numerics revealing tradeoffs between the error and gate complexity of a Trotter step; e.g., we show that split-operator techniques have less Trotter error than popular alternatives. By compiling to surface code fault-tolerant gates and assuming error rates of one part per thousand, we show that one can error-correct quantum simulations of interesting, classically intractable instances with a few hundred thousand physical qubits.

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[1] R. P. Feynman, International Journal of Theoretical Physics 21, 467 (1982).
https:/​/​doi.org/​10.1007/​BF02650179

[2] S. Lloyd, Science 273, 1073 (1996).
https:/​/​doi.org/​10.1126/​science.273.5278.1073

[3] D. S. Abrams and S. Lloyd, Physical Review Letters 83, 5162 (1999).
https:/​/​doi.org/​10.1103/​PhysRevLett.83.5162

[4] A. Y. Kitaev, arXiv preprint arXiv:9511026 (1995).
arXiv:quant-ph/9511026

[5] D. S. Abrams and S. Lloyd, Physical Review Letters 79, 2586 (1997).
https:/​/​doi.org/​10.1103/​PhysRevLett.79.2586

[6] N. M. Tubman, C. Mejuto-Zaera, J. M. Epstein, D. Hait, D. S. Levine, W. Huggins, Z. Jiang, J. R. McClean, R. Babbush, M. Head-Gordon, and K. B. Whaley, arXiv preprint arXiv:1809.05523 (2018).
arXiv:1809.05523

[7] A. Aspuru-Guzik, A. D. Dutoi, P. J. Love, and M. Head-Gordon, Science 309, 1704 (2005).
https:/​/​doi.org/​10.1126/​science.1113479

[8] J. D. Whitfield, J. Biamonte, and A. Aspuru-Guzik, Molecular Physics 109, 735 (2011).
https:/​/​doi.org/​10.1080/​00268976.2011.552441

[9] D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings, and M. Troyer, Phys. Rev. A 90, 022305 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.90.022305

[10] H. F. Trotter, Proceedings of the American Mathematical Society 10, 545 (1959).
https:/​/​doi.org/​10.1090/​S0002-9939-1959-0108732-6
http:/​/​www.jstor.org/​stable/​2033649

[11] M. Suzuki, Journal of Mathematical Physics 32, 400 (1991).
https:/​/​doi.org/​10.1063/​1.529425

[12] D. Poulin, M. B. Hastings, D. Wecker, N. Wiebe, A. C. Doherty, and M. Troyer, Quantum Info. Comput. 15, 361 (2015).
https:/​/​doi.org/​10.5555/​2871401.2871402

[13] J. R. McClean, R. Babbush, P. J. Love, and A. Aspuru-Guzik, The Journal of Physical Chemistry Letters 5, 4368 (2014).
https:/​/​doi.org/​10.1021/​jz501649m

[14] R. Babbush, J. McClean, D. Wecker, A. Aspuru-Guzik, and N. Wiebe, Phys. Rev. A 91, 022311 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.91.022311

[15] M. B. Hastings, D. Wecker, B. Bauer, and M. Troyer, Quantum Info. Comput. 15, 1 (2015).
https:/​/​doi.org/​10.5555/​2685188.2685189

[16] F. Motzoi, M. Kaicher, and F. Wilhelm, Physical review letters 119, 160503 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.119.160503

[17] M. Motta, E. Ye, J. R. McClean, Z. Li, A. J. Minnich, R. Babbush, and G. K.-L. Chan, arXiv preprint arXiv:1808.02625 (2018).
arXiv:1808.02625

[18] E. Campbell, Phys. Rev. Lett. 123, 070503 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.070503

[19] A. M. Childs and N. Wiebe, Quantum Info. Comput. 12, 901 (2012).
https:/​/​doi.org/​10.5555/​2481569.2481570

[20] D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, Phys. Rev. Lett. 114, 090502 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.114.090502

[21] R. Babbush, D. W. Berry, I. D. Kivlichan, A. Y. Wei, P. J. Love, and A. Aspuru-Guzik, New Journal of Physics 18, 033032 (2016).
https:/​/​doi.org/​10.1088/​1367-2630/​18/​3/​033032

[22] G. H. Low and I. L. Chuang, Quantum 3, 163 (2019).
https:/​/​doi.org/​10.22331/​q-2019-07-12-163

[23] G. H. Low and I. L. Chuang, Phys. Rev. Lett. 118, 010501 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.118.010501

[24] D. Poulin, A. Kitaev, D. S. Steiger, M. B. Hastings, and M. Troyer, Phys. Rev. Lett. 121, 010501 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.121.010501

[25] D. W. Berry, M. Kieferová, A. Scherer, Y. R. Sanders, G. H. Low, N. Wiebe, C. Gidney, and R. Babbush, npj Quantum Information 4, 22 (2018).
https:/​/​doi.org/​10.1038/​s41534-018-0071-5

[26] R. Babbush, C. Gidney, D. W. Berry, N. Wiebe, J. McClean, A. Paler, A. Fowler, and H. Neven, Physical Review X 8, 041015 (2018a).
https:/​/​doi.org/​10.1103/​PhysRevX.8.041015

[27] D. W. Berry, C. Gidney, M. Motta, J. R. McClean, and R. Babbush, Quantum 3, 208 (2019).
https:/​/​doi.org/​10.22331/​q-2019-12-02-208

[28] R. Babbush, N. Wiebe, J. McClean, J. McClain, H. Neven, and G. K.-L. Chan, Phys. Rev. X 8, 011044 (2018b).
https:/​/​doi.org/​10.1103/​PhysRevX.8.011044

[29] I. D. Kivlichan, J. McClean, N. Wiebe, C. Gidney, A. Aspuru-Guzik, G. K.-L. Chan, and R. Babbush, Phys. Rev. Lett. 120, 110501 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.120.110501

[30] G. H. Low and N. Wiebe, arXiv preprint arXiv:1805.00675 (2018).
arXiv:1805.00675

[31] C. Zalka, Fortschritte der Physik 46, 877 (1998).
https:/​/​doi.org/​10.1002/​(SICI)1521-3978(199811)46:6/​8<877::AID-PROP877>3.0.CO;2-A

[32] B. Toloui and P. J. Love, arXiv preprint arXiv:1312.2579 (2013).
arXiv:1312.2579

[33] R. Babbush, D. W. Berry, Y. R. Sanders, I. D. Kivlichan, A. Scherer, A. Y. Wei, P. J. Love, and A. Aspuru-Guzik, Quantum Science and Technology 3, 015006 (2018c).
https:/​/​doi.org/​10.1088/​2058-9565/​aa9463

[34] I. Kassal, S. P. Jordan, P. J. Love, M. Mohseni, and A. Aspuru-Guzik, Proceedings of the National Academy of Sciences 105, 18681 (2008).
https:/​/​doi.org/​10.1073/​pnas.0808245105

[35] I. D. Kivlichan, N. Wiebe, R. Babbush, and A. Aspuru-Guzik, Journal of Physics A: Mathematical and Theoretical 50, 305301 (2017).
https:/​/​doi.org/​10.1088/​1751-8121/​aa77b8

[36] R. Babbush, D. W. Berry, J. R. McClean, and H. Neven, npj Quantum Information 5, 92 (2019a).
https:/​/​doi.org/​10.1038/​s41534-019-0199-y

[37] J. Hubbard, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 276, 238 (1963).
https:/​/​doi.org/​10.1098/​rspa.1963.0204

[38] P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006).
https:/​/​doi.org/​10.1103/​RevModPhys.78.17

[39] J. P. F. LeBlanc, A. E. Antipov, F. Becca, I. W. Bulik, G. K.-L. Chan, C.-M. Chung, Y. Deng, M. Ferrero, T. M. Henderson, C. A. Jiménez-Hoyos, E. Kozik, X.-W. Liu, A. J. Millis, N. V. Prokof'ev, M. Qin, G. E. Scuseria, H. Shi, B. V. Svistunov, L. F. Tocchio, I. S. Tupitsyn, S. R. White, S. Zhang, B.-X. Zheng, Z. Zhu, and E. Gull (Simons Collaboration on the Many-Electron Problem), Phys. Rev. X 5, 041041 (2015).
https:/​/​doi.org/​10.1103/​PhysRevX.5.041041

[40] G. Ortiz, J. Gubernatis, E. Knill, and R. Laflamme, Physical Review A 64, 022319 (2001).
https:/​/​doi.org/​10.1103/​PhysRevA.64.022319

[41] F. Verstraete and J. I. Cirac, Journal of Statistical Mechanics: Theory and Experiment 2005, P09012 (2005).
https:/​/​doi.org/​10.1088/​1742-5468/​2005/​09/​P09012

[42] F. Verstraete, J. I. Cirac, and J. I. Latorre, Phys. Rev. A 79, 032316 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.79.032316

[43] D. Wecker, M. B. Hastings, N. Wiebe, B. K. Clark, C. Nayak, and M. Troyer, Physical Review A 92, 062318 (2015a).
https:/​/​doi.org/​10.1103/​PhysRevA.92.062318

[44] Z. Jiang, K. J. Sung, K. Kechedzhi, V. N. Smelyanskiy, and S. Boixo, Physical Review Applied 9, 044036 (2018).
https:/​/​doi.org/​10.1103/​PhysRevApplied.9.044036

[45] J. Haah, M. Hastings, R. Kothari, and G. H. Low, in 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, 2018) pp. 350–360.
https:/​/​doi.org/​10.1109/​FOCS.2018.00041

[46] A. M. Childs and Y. Su, Phys. Rev. Lett. 123, 050503 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.050503

[47] M. Reiher, N. Wiebe, K. M. Svore, D. Wecker, and M. Troyer, Proceedings of the National Academy of Sciences 114, 7555 (2017).
https:/​/​doi.org/​10.1073/​pnas.1619152114

[48] R. Babbush, D. W. Berry, and H. Neven, Physical Review A 99, 040301 (2019b).
https:/​/​doi.org/​10.1103/​PhysRevA.99.040301

[49] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Physical Review A 86, 32324 (2012).
https:/​/​doi.org/​10.1103/​PhysRevA.86.032324

[50] C. Horsman, A. G. Fowler, S. Devitt, and R. V. Meter, New Journal of Physics 14, 123011 (2012).
https:/​/​doi.org/​10.1088/​1367-2630/​14/​12/​123011

[51] A. G. Fowler and C. Gidney, arXiv preprint arXiv:1808.06709 (2018).
arXiv:1808.06709

[52] A. M. Childs, D. Maslov, Y. Nam, N. J. Ross, and Y. Su, Proceedings of the National Academy of Sciences 115, 9456 (2018).
https:/​/​doi.org/​10.1073/​pnas.1801723115

[53] Y. Nam and D. Maslov, npj Quantum Information 5, 44 (2019).
https:/​/​doi.org/​10.1038/​s41534-019-0152-0

[54] P. Corboz, Phys. Rev. B 93, 045116 (2016).
https:/​/​doi.org/​10.1103/​PhysRevB.93.045116

[55] J. J. Shepherd and A. Grüneis, Phys. Rev. Lett. 110, 226401 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.110.226401

[56] J. J. Shepherd, T. M. Henderson, and G. E. Scuseria, Phys. Rev. Lett. 112, 133002 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.112.133002

[57] P.-F. Loos and P. M. W. Gill, Wiley Interdisciplinary Reviews: Computational Molecular Science 6, 410 (2016).
https:/​/​doi.org/​10.1002/​wcms.1257

[58] J. McClain, J. Lischner, T. Watson, D. A. Matthews, E. Ronca, S. G. Louie, T. C. Berkelbach, and G. K.-L. Chan, Phys. Rev. B 93, 235139 (2016).
https:/​/​doi.org/​10.1103/​PhysRevB.93.235139

[59] D. W. Berry, B. L. Higgins, S. D. Bartlett, M. W. Mitchell, G. J. Pryde, and H. M. Wiseman, Phys. Rev. A 80, 052114 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.80.052114

[60] B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde, Nature 450, 393 (2007).
https:/​/​doi.org/​10.1038/​nature06257

[61] C. Gidney, Quantum 2, 74 (2018).
https:/​/​doi.org/​10.22331/​q-2018-06-18-74

[62] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957).
https:/​/​doi.org/​10.1103/​PhysRev.108.1175

[63] P. Jordan and E. Wigner, Zeitschrift für Physik 47, 631 (1928).
https:/​/​doi.org/​10.1007/​BF01331938

[64] R. D. Somma, G. Ortiz, J. Gubernatis, E. Knill, and R. Laflamme, Phys. Rev. A 65, 17 (2002).
https:/​/​doi.org/​10.1103/​PhysRevA.65.042323

[65] S. Bravyi and A. Kitaev, Annals of Physics 298, 210 (2002).
https:/​/​doi.org/​10.1006/​aphy.2002.6254

[66] J. T. Seeley, M. J. Richard, and P. J. Love, Journal of Chemical Physics 137, 224109 (2012).
https:/​/​doi.org/​10.1063/​1.4768229

[67] A. Tranter, S. Sofia, J. Seeley, M. Kaicher, J. McClean, R. Babbush, P. V. Coveney, F. Mintert, F. Wilhelm, and P. J. Love, International Journal of Quantum Chemistry 115, 1431 (2015).
https:/​/​doi.org/​10.1002/​qua.24969

[68] S. Bravyi, J. M. Gambetta, A. Mezzacapo, and K. Temme, arXiv preprint arXiv:1701.08213 (2017).
arXiv:1701.08213

[69] J. D. Whitfield, V. Havlíček, and M. Troyer, Physical Review A 94, 030301 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.94.030301

[70] V. Havlíček, M. Troyer, and J. D. Whitfield, Phys. Rev. A 95, 032332 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.95.032332

[71] Z. Jiang, J. McClean, R. Babbush, and H. Neven, Phys. Rev. Applied 12, 064041 (2019).
https:/​/​doi.org/​10.1103/​PhysRevApplied.12.064041

[72] A. J. Ferris, Phys. Rev. Lett. 113, 010401 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.113.010401

[73] A. Chandran, J. Carrasquilla, I. H. Kim, D. A. Abanin, and G. Vidal, Physical Review B 92, 024201 (2015).
https:/​/​doi.org/​10.1103/​PhysRevB.92.024201

[74] M. C. Tran, A. Y. Guo, Y. Su, J. R. Garrison, Z. Eldredge, M. Foss-Feig, A. M. Childs, and A. V. Gorshkov, Phys. Rev. X 9, 031006 (2019).
https:/​/​doi.org/​10.1103/​PhysRevX.9.031006

[75] A. M. Childs, A. Ostrander, and Y. Su, Quantum 3, 182 (2019).
https:/​/​doi.org/​10.22331/​q-2019-09-02-182

[76] S. R. White, The Journal of Chemical Physics 147, 244102 (2017).
https:/​/​doi.org/​10.1063/​1.5007066

[77] J. McClean, N. Rubin, K. Sung, I. D. Kivlichan, X. Bonet-Monroig, Y. Cao, C. Dai, E. S. Fried, C. Gidney, B. Gimby, P. Gokhale, T. Haner, T. Hardikar, V. Havlíček, O. Higgott, C. Huang, J. Izaac, Z. Jiang, X. Liu, S. McArdle, M. Neeley, T. O'Brien, B. O'Gorman, I. Ozfidan, M. D. Radin, J. Romero, N. P. D. Sawaya, B. Senjean, K. Setia, S. Sim, D. S. Steiger, M. Steudtner, Q. Sun, W. Sun, D. Wang, F. Zhang, and R. Babbush, Quantum Science and Technology (2020), 10.1088/​2058-9565/​ab8ebc.
https:/​/​doi.org/​10.1088/​2058-9565/​ab8ebc

[78] D. Wecker, M. B. Hastings, and M. Troyer, Phys. Rev. A 92, 042303 (2015b).
https:/​/​doi.org/​10.1103/​PhysRevA.92.042303

[79] N. Wiebe and C. Granade, Phys. Rev. Lett. 117, 010503 (2016).
https:/​/​doi.org/​10.1103/​PhysRevLett.117.010503

[80] A. Bocharov, M. Roetteler, and K. M. Svore, Phys. Rev. Lett. 114, 080502 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.114.080502

[81] C. Gidney and A. G. Fowler, Quantum 3, 135 (2019).
https:/​/​doi.org/​10.22331/​q-2019-04-30-135

[82] B. Tanatar and D. M. Ceperley, Phys. Rev. B 39, 5005 (1989).
https:/​/​doi.org/​10.1103/​physrevb.39.5005
https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.39.5005

[83] J. J. Shepherd, G. Booth, A. Grüneis, and A. Alavi, Phys. Rev. B 85, 081103 (2012).
https:/​/​doi.org/​10.1103/​PhysRevB.85.081103

[84] A. G. Fowler, arXiv preprint arXiv:1310.0863 (2013).
arXiv:1310.0863

[85] D. Litinski, Quantum 3, 128 (2019).
https:/​/​doi.org/​10.22331/​q-2019-03-05-128

[86] J. E. Savage, Models of computation, Vol. 136 (Addison-Wesley Reading, MA, 1998).

[87] J. Nordstrom, Logical Methods in Computer Science Volume 9, Issue 3 (2013), 10.2168/​LMCS-9(3:15)2013.
https:/​/​doi.org/​10.2168/​LMCS-9(3:15)2013

[88] R. Bhatia and C. Davis, Linear and Multilinear Algebra 15, 71 (1984).
https:/​/​doi.org/​10.1080/​03081088408817578

[89] Q. Sun, T. C. Berkelbach, N. S. Blunt, G. H. Booth, S. Guo, Z. Li, J. Liu, J. D. McClain, E. R. Sayfutyarova, S. Sharma, et al., Wiley Interdisciplinary Reviews: Computational Molecular Science 8, e1340 (2018).
https:/​/​doi.org/​10.1002/​wcms.1340

[90] T. Chachiyo, The Journal of Chemical Physics 145, 021101 (2016).
https:/​/​doi.org/​10.1063/​1.4958669

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[5] Dmitri Maslov, Jin-Sung Kim, Sergey Bravyi, Theodore J. Yoder, and Sarah Sheldon, "Quantum advantage for computations with limited space", Nature Physics 17 8, 894 (2021).

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[9] Alec Dinerstein, Caroline S Gorham, and Eugene F Dumitrescu, "The hybrid topological longitudinal transmon qubit", Materials for Quantum Technology 1 2, 021001 (2021).

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[17] Christopher Chamberland and Kyungjoo Noh, "Very low overhead fault-tolerant magic state preparation using redundant ancilla encoding and flag qubits", npj Quantum Information 6 1, 91 (2020).

[18] Changhao Yi and Elizabeth Crosson, "Spectral analysis of product formulas for quantum simulation", npj Quantum Information 8 1, 37 (2022).

[19] Jie Liu, Yi Fan, Zhenyu Li, and Jinlong Yang, "Quantum algorithms for electronic structures: basis sets and boundary conditions", Chemical Society Reviews 51 8, 3263 (2022).

[20] Earl T Campbell, "Early fault-tolerant simulations of the Hubbard model", Quantum Science and Technology 7 1, 015007 (2022).

[21] Pablo A. M. Casares, Roberto Campos, and M. A. Martin-Delgado, "TFermion: A non-Clifford gate cost assessment library of quantum phase estimation algorithms for quantum chemistry", Quantum 6, 768 (2022).

[22] Christopher Chamberland and Earl T. Campbell, "Universal Quantum Computing with Twist-Free and Temporally Encoded Lattice Surgery", PRX Quantum 3 1, 010331 (2022).

[23] Yasunari Suzuki, Yoshiaki Kawase, Yuya Masumura, Yuria Hiraga, Masahiro Nakadai, Jiabao Chen, Ken M. Nakanishi, Kosuke Mitarai, Ryosuke Imai, Shiro Tamiya, Takahiro Yamamoto, Tennin Yan, Toru Kawakubo, Yuya O. Nakagawa, Yohei Ibe, Youyuan Zhang, Hirotsugu Yamashita, Hikaru Yoshimura, Akihiro Hayashi, and Keisuke Fujii, "Qulacs: a fast and versatile quantum circuit simulator for research purpose", Quantum 5, 559 (2021).

[24] Christopher Chamberland, Kyungjoo Noh, Patricio Arrangoiz-Arriola, Earl T. Campbell, Connor T. Hann, Joseph Iverson, Harald Putterman, Thomas C. Bohdanowicz, Steven T. Flammia, Andrew Keller, Gil Refael, John Preskill, Liang Jiang, Amir H. Safavi-Naeini, Oskar Painter, and Fernando G.S.L. Brandão, "Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes", arXiv:2012.04108, PRX Quantum 3 1, 010329 (2022).

[25] Jules Tilly, Hongxiang Chen, Shuxiang Cao, Dario Picozzi, Kanav Setia, Ying Li, Edward Grant, Leonard Wossnig, Ivan Rungger, George H. Booth, and Jonathan Tennyson, "The Variational Quantum Eigensolver: A review of methods and best practices", Physics Reports 986, 1 (2022).

[26] Yi-Tong Zou, Yu-Jiao Bo, and Ji-Chong Yang, "Optimize quantum simulation using a force-gradient integrator", EPL (Europhysics Letters) 135 1, 10004 (2021).

[27] 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, and Julian Kelly, "Exponential suppression of bit or phase errors with cyclic error correction", Nature 595 7867, 383 (2021).

[28] Yingkai Ouyang, David R. White, and Earl T. Campbell, "Compilation by stochastic Hamiltonian sparsification", arXiv:1910.06255, Quantum 4, 235 (2020).

[29] Jessica Lemieux, Guillaume Duclos-Cianci, David Sénéchal, and David Poulin, "Resource estimate for quantum many-body ground-state preparation on a quantum computer", Physical Review A 103 5, 052408 (2021).

[30] Yudong Cao, Jonathan Romero, Jonathan P. Olson, Matthias Degroote, Peter D. Johnson, Mária Kieferová, Ian D. Kivlichan, Tim Menke, Borja Peropadre, Nicolas P. D. Sawaya, Sukin Sim, Libor Veis, and Alán Aspuru-Guzik, "Quantum Chemistry in the Age of Quantum Computing", Chemical Reviews 119 19, 10856 (2019).

[31] Yasunari Suzuki, Suguru Endo, Keisuke Fujii, and Yuuki Tokunaga, "Quantum Error Mitigation as a Universal Error Reduction Technique: Applications from the NISQ to the Fault-Tolerant Quantum Computing Eras", PRX Quantum 3 1, 010345 (2022).

[32] Bela Bauer, Sergey Bravyi, Mario Motta, and Garnet Kin-Lic Chan, "Quantum Algorithms for Quantum Chemistry and Quantum Materials Science", Chemical Reviews 120 22, 12685 (2020).

[33] Jie Liu, Lingyun Wan, Zhenyu Li, and Jinlong Yang, "Simulating Periodic Systems on a Quantum Computer Using Molecular Orbitals", Journal of Chemical Theory and Computation 16 11, 6904 (2020).

[34] Masuo Suzuki, "Quantum analysis on the convergence speed of exponential product formulas — differential-subtraction and exchange-integration method on concise norm bounds", Journal of Mathematical Physics 62 6, 062202 (2021).

[35] Poulami Das, Aditya Locharla, and Cody Jones, Proceedings of the 27th ACM International Conference on Architectural Support for Programming Languages and Operating Systems 541 (2022) ISBN:9781450392051.

[36] Erik Gustafson, Yingyue Zhu, Patrick Dreher, Norbert M. Linke, and Yannick Meurice, "Real-time quantum calculations of phase shifts using wave packet time delays", Physical Review D 104 5, 054507 (2021).

[37] Jérôme F. Gonthier, Maxwell D. Radin, Corneliu Buda, Eric J. Doskocil, Clena M. Abuan, and Jhonathan Romero, "Measurements as a roadblock to near-term practical quantum advantage in chemistry: Resource analysis", Physical Review Research 4 3, 033154 (2022).

[38] Kade Head-Marsden, Johannes Flick, Christopher J. Ciccarino, and Prineha Narang, "Quantum Information and Algorithms for Correlated Quantum Matter", Chemical Reviews 121 5, 3061 (2021).

[39] Lindsay Bassman, Roel Van Beeumen, Ed Younis, Ethan Smith, Costin Iancu, and Wibe A. de Jong, "Constant-depth circuits for dynamic simulations of materials on quantum computers", Materials Theory 6 1, 13 (2022).

[40] Benchen Huang, Marco Govoni, and Giulia Galli, "Simulating the Electronic Structure of Spin Defects on Quantum Computers", PRX Quantum 3 1, 010339 (2022).

[41] Yuan Su, Dominic W. Berry, Nathan Wiebe, Nicholas Rubin, and Ryan Babbush, "Fault-Tolerant Quantum Simulations of Chemistry in First Quantization", PRX Quantum 2 4, 040332 (2021).

[42] Natalie Klco, Alessandro Roggero, and Martin J Savage, "Standard model physics and the digital quantum revolution: thoughts about the interface", Reports on Progress in Physics 85 6, 064301 (2022).

[43] John S. Van Dyke, George S. Barron, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou, "Preparing Bethe Ansatz Eigenstates on a Quantum Computer", PRX Quantum 2 4, 040329 (2021).

[44] Joonho Lee, Dominic W. Berry, Craig Gidney, William J. Huggins, Jarrod R. McClean, Nathan Wiebe, and Ryan Babbush, "Even More Efficient Quantum Computations of Chemistry Through Tensor Hypercontraction", PRX Quantum 2 3, 030305 (2021).

[45] Erik J. Gustafson and Henry Lamm, "Toward quantum simulations of Z2 gauge theory without state preparation", Physical Review D 103 5, 054507 (2021).

[46] Trevor Keen, Bo Peng, Karol Kowalski, Pavel Lougovski, and Steven Johnston, "Hybrid quantum-classical approach for coupled-cluster Green&apos;s function theory", Quantum 6, 675 (2022).

[47] Ryan Babbush, Jarrod R. McClean, Michael Newman, Craig Gidney, Sergio Boixo, and Hartmut Neven, "Focus beyond Quadratic Speedups for Error-Corrected Quantum Advantage", PRX Quantum 2 1, 010103 (2021).

[48] Christian Vorwerk, Nan Sheng, Marco Govoni, Benchen Huang, and Giulia Galli, "Quantum embedding theories to simulate condensed systems on quantum computers", Nature Computational Science 2 7, 424 (2022).

[49] Qingfeng Wang, Ming Li, Christopher Monroe, and Yunseong Nam, "Resource-Optimized Fermionic Local-Hamiltonian Simulation on a Quantum Computer for Quantum Chemistry", Quantum 5, 509 (2021).

[50] Mario Motta, Erika Ye, Jarrod R. McClean, Zhendong Li, Austin J. Minnich, Ryan Babbush, and Garnet Kin-Lic Chan, "Low rank representations for quantum simulation of electronic structure", npj Quantum Information 7 1, 83 (2021).

[51] Laura Clinton, Johannes Bausch, and Toby Cubitt, "Hamiltonian simulation algorithms for near-term quantum hardware", Nature Communications 12 1, 4989 (2021).

[52] Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C. Benjamin, and Xiao Yuan, "Quantum computational chemistry", arXiv:1808.10402, Reviews of Modern Physics 92 1, 015003 (2020).

[53] Sam McArdle, Xiao Yuan, and Simon Benjamin, "Error-Mitigated Digital Quantum Simulation", Physical Review Letters 122 18, 180501 (2019).

[54] Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu, "A Theory of Trotter Error", arXiv:1912.08854.

[55] Bela Bauer, Sergey Bravyi, Mario Motta, and Garnet Kin-Lic Chan, "Quantum algorithms for quantum chemistry and quantum materials science", arXiv:2001.03685.

[56] Tyler Takeshita, Nicholas C. Rubin, Zhang Jiang, Eunseok Lee, Ryan Babbush, and Jarrod R. McClean, "Increasing the Representation Accuracy of Quantum Simulations of Chemistry without Extra Quantum Resources", Physical Review X 10 1, 011004 (2020).

[57] Thomas E. O'Brien, Bruno Senjean, Ramiro Sagastizabal, Xavier Bonet-Monroig, Alicja Dutkiewicz, Francesco Buda, Leonardo DiCarlo, and Lucas Visscher, "Calculating energy derivatives for quantum chemistry on a quantum computer", npj Quantum Information 5, 113 (2019).

[58] Bhupesh Bishnoi, "Quantum Computation", arXiv:2006.02799.

[59] Ryan Babbush, Dominic W. Berry, Jarrod R. McClean, and Hartmut Neven, "Quantum simulation of chemistry with sublinear scaling in basis size", npj Quantum Information 5, 92 (2019).

[60] Alessandro Roggero, Andy C. Y. Li, Joseph Carlson, Rajan Gupta, and Gabriel N. Perdue, "Quantum computing for neutrino-nucleus scattering", Physical Review D 101 7, 074038 (2020).

[61] Thi Ha Kyaw, Tim Menke, Sukin Sim, Abhinav Anand, Nicolas P. D. Sawaya, William D. Oliver, Gian Giacomo Guerreschi, and Alán Aspuru-Guzik, "Quantum Computer-Aided Design: Digital Quantum Simulation of Quantum Processors", Physical Review Applied 16 4, 044042 (2021).

[62] Zhenyu Cai, "Resource Estimation for Quantum Variational Simulations of the Hubbard Model", Physical Review Applied 14 1, 014059 (2020).

[63] John Golden, Andreas Bärtschi, Stephan Eidenbenz, and Daniel O'Malley, "Evidence for Super-Polynomial Advantage of QAOA over Unstructured Search", arXiv:2202.00648.

[64] David Headley, Thorge Müller, Ana Martin, Enrique Solano, Mikel Sanz, and Frank K. Wilhelm, "Approximating the Quantum Approximate Optimisation Algorithm", arXiv:2002.12215.

[65] Armin Rahmani, Kevin J. Sung, Harald Putterman, Pedram Roushan, Pouyan Ghaemi, and Zhang Jiang, "Creating and manipulating a Laughlin-type $\nu=1/3$ fractional quantum Hall state on a quantum computer with linear depth circuits", arXiv:2005.02399.

[66] Dominic W. Berry, Craig Gidney, Mario Motta, Jarrod R. McClean, and Ryan Babbush, "Qubitization of Arbitrary Basis Quantum Chemistry Leveraging Sparsity and Low Rank Factorization", arXiv:1902.02134.

[67] Alexey N. Pyrkov, Yurii Zotov, Jiangyu Cui, and Manhong Yung, "Global sensitivity analysis for optimization of the Trotter-Suzuki decomposition", arXiv:2101.03349.

[68] Ian D. Kivlichan, Christopher E. Granade, and Nathan Wiebe, "Phase estimation with randomized Hamiltonians", arXiv:1907.10070.

[69] Sam McArdle, "Learning from Physics Experiments with Quantum Computers: Applications in Muon Spectroscopy", PRX Quantum 2 2, 020349 (2021).

[70] Adam R. Brown, "Polynomial Equivalence of Complexity Geometries", arXiv:2205.04485.

[71] Alexandru Paler and Austin G. Fowler, "Pipelined correlated minimum weight perfect matching of the surface code", arXiv:2205.09828.

[72] Bill Poirier, "Efficient Evaluation of Exponential and Gaussian Functions on a Quantum Computer", arXiv:2110.05653.

[73] Sam Pallister, "A Jordan-Wigner gadget that reduces T count by more than 6x for quantum chemistry applications", arXiv:2004.05117.

[74] Daniel Burgarth, Paolo Facchi, Giovanni Gramegna, and Saverio Pascazio, "Generalized product formulas and quantum control", Journal of Physics A Mathematical General 52 43, 435301 (2019).

[75] Bill Poirier and Jonathan Jerke, "Full-dimensional Schrödinger wavefunction calculations using tensors and quantum computers: the Cartesian component-separated approach", Physical Chemistry Chemical Physics (Incorporating Faraday Transactions) 24 7, 4437 (2022).

The above citations are from Crossref's cited-by service (last updated successfully 2022-10-01 22:40:17) and SAO/NASA ADS (last updated successfully 2022-10-01 22:40:18). The list may be incomplete as not all publishers provide suitable and complete citation data.