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] 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).

[3] Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu, "Theory of Trotter Error with Commutator Scaling", Physical Review X 11 1, 011020 (2021).

[4] Nicholas P. Bauman, Hongbin Liu, Eric J. Bylaska, Sriram Krishnamoorthy, Guang Hao Low, Christopher E. Granade, Nathan Wiebe, Nathan A. Baker, Bo Peng, Martin Roetteler, Matthias Troyer, and Karol Kowalski, "Toward Quantum Computing for High-Energy Excited States in Molecular Systems: Quantum Phase Estimations of Core-Level States", Journal of Chemical Theory and Computation 17 1, 201 (2021).

[5] 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).

[6] Yuval R. Sanders, Dominic W. Berry, Pedro C.S. Costa, Louis W. Tessler, Nathan Wiebe, Craig Gidney, Hartmut Neven, and Ryan Babbush, "Compilation of Fault-Tolerant Quantum Heuristics for Combinatorial Optimization", PRX Quantum 1 2, 020312 (2020).

[7] Tiffany M. Mintz, Alexander J. McCaskey, Eugene F. Dumitrescu, Shirley V. Moore, Sarah Powers, and Pavel Lougovski, "QCOR", ACM Journal on Emerging Technologies in Computing Systems 16 2, 1 (2020).

[8] 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).

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

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

[11] 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).

[12] 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).

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

[14] 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).

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

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

[17] 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).

[18] 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.

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

[20] 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).

[21] 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).

[22] 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.

[23] Yuan Su, Hsin-Yuan Huang, and Earl T. Campbell, "Nearly tight Trotterization of interacting electrons", arXiv:2012.09194.

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

[25] 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.

[26] Jessica Lemieux, Guillaume Duclos-Cianci, David Sénéchal, and David Poulin, "Resource estimate for quantum many-body ground state preparation on a quantum computer", arXiv:2006.04650.

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

[28] 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.

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

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

[31] Sam McArdle, "Learning from physics experiments, with quantum computers: Applications in muon spectroscopy", arXiv:2012.06602.

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

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