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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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).

[2] S. Lloyd, Science 273, 1073 (1996).

[3] D. S. Abrams and S. Lloyd, Physical Review Letters 83, 5162 (1999).

[4] A. Y. Kitaev, arXiv preprint arXiv:9511026 (1995).

[5] D. S. Abrams and S. Lloyd, Physical Review Letters 79, 2586 (1997).

[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).

[7] A. Aspuru-Guzik, A. D. Dutoi, P. J. Love, and M. Head-Gordon, Science 309, 1704 (2005).

[8] J. D. Whitfield, J. Biamonte, and A. Aspuru-Guzik, Molecular Physics 109, 735 (2011).

[9] D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings, and M. Troyer, Phys. Rev. A 90, 022305 (2014).

[10] H. F. Trotter, Proceedings of the American Mathematical Society 10, 545 (1959).

[11] M. Suzuki, Journal of Mathematical Physics 32, 400 (1991).

[12] D. Poulin, M. B. Hastings, D. Wecker, N. Wiebe, A. C. Doherty, and M. Troyer, Quantum Info. Comput. 15, 361 (2015).

[13] J. R. McClean, R. Babbush, P. J. Love, and A. Aspuru-Guzik, The Journal of Physical Chemistry Letters 5, 4368 (2014).

[14] R. Babbush, J. McClean, D. Wecker, A. Aspuru-Guzik, and N. Wiebe, Phys. Rev. A 91, 022311 (2015).

[15] M. B. Hastings, D. Wecker, B. Bauer, and M. Troyer, Quantum Info. Comput. 15, 1 (2015).

[16] F. Motzoi, M. Kaicher, and F. Wilhelm, Physical review letters 119, 160503 (2017).

[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).

[18] E. Campbell, Phys. Rev. Lett. 123, 070503 (2019).

[19] A. M. Childs and N. Wiebe, Quantum Info. Comput. 12, 901 (2012).

[20] D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, Phys. Rev. Lett. 114, 090502 (2015).

[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).

[22] G. H. Low and I. L. Chuang, Quantum 3, 163 (2019).

[23] G. H. Low and I. L. Chuang, Phys. Rev. Lett. 118, 010501 (2017).

[24] D. Poulin, A. Kitaev, D. S. Steiger, M. B. Hastings, and M. Troyer, Phys. Rev. Lett. 121, 010501 (2018).

[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).

[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).

[27] D. W. Berry, C. Gidney, M. Motta, J. R. McClean, and R. Babbush, Quantum 3, 208 (2019).

[28] R. Babbush, N. Wiebe, J. McClean, J. McClain, H. Neven, and G. K.-L. Chan, Phys. Rev. X 8, 011044 (2018b).

[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).

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

[31] C. Zalka, Fortschritte der Physik 46, 877 (1998).

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

[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).

[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).

[35] I. D. Kivlichan, N. Wiebe, R. Babbush, and A. Aspuru-Guzik, Journal of Physics A: Mathematical and Theoretical 50, 305301 (2017).

[36] R. Babbush, D. W. Berry, J. R. McClean, and H. Neven, npj Quantum Information 5, 92 (2019a).

[37] J. Hubbard, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 276, 238 (1963).

[38] P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006).

[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).

[40] G. Ortiz, J. Gubernatis, E. Knill, and R. Laflamme, Physical Review A 64, 022319 (2001).

[41] F. Verstraete and J. I. Cirac, Journal of Statistical Mechanics: Theory and Experiment 2005, P09012 (2005).

[42] F. Verstraete, J. I. Cirac, and J. I. Latorre, Phys. Rev. A 79, 032316 (2009).

[43] D. Wecker, M. B. Hastings, N. Wiebe, B. K. Clark, C. Nayak, and M. Troyer, Physical Review A 92, 062318 (2015a).

[44] Z. Jiang, K. J. Sung, K. Kechedzhi, V. N. Smelyanskiy, and S. Boixo, Physical Review Applied 9, 044036 (2018).

[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.

[46] A. M. Childs and Y. Su, Phys. Rev. Lett. 123, 050503 (2019).

[47] M. Reiher, N. Wiebe, K. M. Svore, D. Wecker, and M. Troyer, Proceedings of the National Academy of Sciences 114, 7555 (2017).

[48] R. Babbush, D. W. Berry, and H. Neven, Physical Review A 99, 040301 (2019b).

[49] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Physical Review A 86, 32324 (2012).

[50] C. Horsman, A. G. Fowler, S. Devitt, and R. V. Meter, New Journal of Physics 14, 123011 (2012).

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

[52] A. M. Childs, D. Maslov, Y. Nam, N. J. Ross, and Y. Su, Proceedings of the National Academy of Sciences 115, 9456 (2018).

[53] Y. Nam and D. Maslov, npj Quantum Information 5, 44 (2019).

[54] P. Corboz, Phys. Rev. B 93, 045116 (2016).

[55] J. J. Shepherd and A. Grüneis, Phys. Rev. Lett. 110, 226401 (2013).

[56] J. J. Shepherd, T. M. Henderson, and G. E. Scuseria, Phys. Rev. Lett. 112, 133002 (2014).

[57] P.-F. Loos and P. M. W. Gill, Wiley Interdisciplinary Reviews: Computational Molecular Science 6, 410 (2016).

[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).

[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).

[60] B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde, Nature 450, 393 (2007).

[61] C. Gidney, Quantum 2, 74 (2018).

[62] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957).

[63] P. Jordan and E. Wigner, Zeitschrift für Physik 47, 631 (1928).

[64] R. D. Somma, G. Ortiz, J. Gubernatis, E. Knill, and R. Laflamme, Phys. Rev. A 65, 17 (2002).

[65] S. Bravyi and A. Kitaev, Annals of Physics 298, 210 (2002).

[66] J. T. Seeley, M. J. Richard, and P. J. Love, Journal of Chemical Physics 137, 224109 (2012).

[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).

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

[69] J. D. Whitfield, V. Havlíček, and M. Troyer, Physical Review A 94, 030301 (2016).

[70] V. Havlíček, M. Troyer, and J. D. Whitfield, Phys. Rev. A 95, 032332 (2017).

[71] Z. Jiang, J. McClean, R. Babbush, and H. Neven, Phys. Rev. Applied 12, 064041 (2019).

[72] A. J. Ferris, Phys. Rev. Lett. 113, 010401 (2014).

[73] A. Chandran, J. Carrasquilla, I. H. Kim, D. A. Abanin, and G. Vidal, Physical Review B 92, 024201 (2015).

[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).

[75] A. M. Childs, A. Ostrander, and Y. Su, Quantum 3, 182 (2019).

[76] S. R. White, The Journal of Chemical Physics 147, 244102 (2017).

[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.

[78] D. Wecker, M. B. Hastings, and M. Troyer, Phys. Rev. A 92, 042303 (2015b).

[79] N. Wiebe and C. Granade, Phys. Rev. Lett. 117, 010503 (2016).

[80] A. Bocharov, M. Roetteler, and K. M. Svore, Phys. Rev. Lett. 114, 080502 (2015).

[81] C. Gidney and A. G. Fowler, Quantum 3, 135 (2019).

[82] B. Tanatar and D. M. Ceperley, Phys. Rev. B 39, 5005 (1989).

[83] J. J. Shepherd, G. Booth, A. Grüneis, and A. Alavi, Phys. Rev. B 85, 081103 (2012).

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

[85] D. Litinski, Quantum 3, 128 (2019).

[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.

[88] R. Bhatia and C. Davis, Linear and Multilinear Algebra 15, 71 (1984).

[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).

[90] T. Chachiyo, The Journal of Chemical Physics 145, 021101 (2016).

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[2] Sam McArdle, Suguru Endo, Alan Aspuru-Guzik, Simon Benjamin, and Xiao Yuan, "Quantum computational chemistry", arXiv:1808.10402, Reviews of Modern Physics 92 1, 015003 (2018).

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

[10] 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).

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[14] Christopher Chamberland and Kyungjoo Noh, "Very low overhead fault-tolerant magic state preparation using redundant ancilla encoding and flag qubits", arXiv:2003.03049.

[15] 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).

[16] Yingkai Ouyang, David R. White, and Earl T. Campbell, "Compilation by stochastic Hamiltonian sparsification", arXiv:1910.06255.

[17] Laura Clinton, Johannes Bausch, and Toby Cubitt, "Hamiltonian Simulation Algorithms for Near-Term Quantum Hardware", arXiv:2003.06886.

[18] Vera von Burg, Guang Hao Low, Thomas Häner, Damian S. Steiger, Markus Reiher, Martin Roetteler, and Matthias Troyer, "Quantum computing enhanced computational catalysis", arXiv:2007.14460.

[19] Thi Ha Kyaw, Tim Menke, Sukin Sim, Nicolas P. D. Sawaya, William D. Oliver, Gian Giacomo Guerreschi, and Alán Aspuru-Guzik, "Quantum computer-aided design: digital quantum simulation of quantum processors", arXiv:2006.03070.

[20] 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.

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

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

[23] Bhupesh Bishnoi, "Quantum-Computation and Applications", arXiv:2006.02799.

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