Exploiting anticommutation in Hamiltonian simulation

Qi Zhao1 and Xiao Yuan2,3

1Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, Maryland 20742, USA
2Center on Frontiers of Computing Studies, Department of Computer Science, Peking University, Beijing 100871, China
3Stanford Institute for Theoretical Physics, Stanford University, Stanford California 94305, USA

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

Abstract

Quantum computing can efficiently simulate Hamiltonian dynamics of many-body quantum physics, a task that is generally intractable with classical computers. The hardness lies at the ubiquitous anti-commutative relations of quantum operators, in corresponding with the notorious negative sign problem in classical simulation. Intuitively, Hamiltonians with more commutative terms are also easier to simulate on a quantum computer, and anti-commutative relations generally cause more errors, such as in the product formula method. Here, we theoretically explore the role of anti-commutative relation in Hamiltonian simulation. We find that, contrary to our intuition, anti-commutative relations could also reduce the hardness of Hamiltonian simulation. Specifically, Hamiltonians with mutually anti-commutative terms are easy to simulate, as what happens with ones consisting of mutually commutative terms. Such a property is further utilized to reduce the algorithmic error or the gate complexity in the truncated Taylor series quantum algorithm for general problems. Moreover, we propose two modified linear combinations of unitaries methods tailored for Hamiltonians with different degrees of anti-commutation. We numerically verify that the proposed methods exploiting anti-commutative relations could significantly improve the simulation accuracy of electronic Hamiltonians. Our work sheds light on the roles of commutative and anti-commutative relations in simulating quantum systems.

► BibTeX data

► References

[1] Alán Aspuru-Guzik, Anthony D Dutoi, Peter J Love, and Martin Head-Gordon. Simulated quantum computation of molecular energies. Science, 309 (5741): 1704–1707, 2005. https:/​/​doi.org/​10.1126/​science.1113479. URL https:/​/​science.sciencemag.org/​content/​309/​5741/​1704.
https:/​/​doi.org/​10.1126/​science.1113479
https:/​/​science.sciencemag.org/​content/​309/​5741/​1704

[2] Ryan Babbush, Peter J Love, and Alán Aspuru-Guzik. Adiabatic quantum simulation of quantum chemistry. Scientific reports, 4: 6603, 2014. https:/​/​doi.org/​10.1038/​srep06603. URL https:/​/​rdcu.be/​cv1RR.
https:/​/​doi.org/​10.1038/​srep06603
https:/​/​rdcu.be/​cv1RR

[3] Ryan Babbush, Jarrod McClean, Dave Wecker, Alán Aspuru-Guzik, and Nathan Wiebe. Chemical basis of trotter-suzuki errors in quantum chemistry simulation. Phys. Rev. A, 91: 022311, Feb 2015. https:/​/​doi.org/​10.1103/​PhysRevA.91.022311. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.91.022311.
https:/​/​doi.org/​10.1103/​PhysRevA.91.022311

[4] R Barends, L Lamata, J Kelly, L García-Álvarez, AG Fowler, A Megrant, E Jeffrey, TC White, D Sank, JY Mutus, et al. Digital quantum simulation of fermionic models with a superconducting circuit. Nature communications, 6: 7654, 2015. https:/​/​doi.org/​10.1038/​ncomms8654. URL https:/​/​rdcu.be/​cv1R9.
https:/​/​doi.org/​10.1038/​ncomms8654
https:/​/​rdcu.be/​cv1R9

[5] Dominic W Berry and Andrew M Childs. Black-box hamiltonian simulation and unitary implementation. Quantum Information & Computation, 12 (1-2): 29–62, 2012. URL https:/​/​dl.acm.org/​doi/​10.5555/​2231036.2231040.
https:/​/​dl.acm.org/​doi/​10.5555/​2231036.2231040

[6] Dominic W Berry, Graeme Ahokas, Richard Cleve, and Barry C Sanders. Efficient quantum algorithms for simulating sparse hamiltonians. Communications in Mathematical Physics, 270 (2): 359–371, 2007. https:/​/​doi.org/​10.1007/​s00220-006-0150-x.
https:/​/​doi.org/​10.1007/​s00220-006-0150-x

[7] Dominic W Berry, Andrew M Childs, Richard Cleve, Robin Kothari, and Rolando D Somma. Exponential improvement in precision for simulating sparse hamiltonians. In Proceedings of the forty-sixth annual ACM symposium on Theory of computing, pages 283–292, 2014. https:/​/​doi.org/​10.1145/​2591796.2591854.
https:/​/​doi.org/​10.1145/​2591796.2591854

[8] Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma. Simulating hamiltonian dynamics with a truncated taylor series. Phys. Rev. Lett., 114: 090502, Mar 2015. https:/​/​doi.org/​10.1103/​PhysRevLett.114.090502. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.114.090502.
https:/​/​doi.org/​10.1103/​PhysRevLett.114.090502

[9] Sergey Bravyi. Monte carlo simulation of stoquastic hamiltonians. Quantum Information & Computation, 15 (13-14): 1122–1140, 2015. URL https:/​/​dl.acm.org/​doi/​10.5555/​2871363.2871366.
https:/​/​dl.acm.org/​doi/​10.5555/​2871363.2871366

[10] Sergey Bravyi and David Gosset. Polynomial-time classical simulation of quantum ferromagnets. Phys. Rev. Lett., 119: 100503, Sep 2017. https:/​/​doi.org/​10.1103/​PhysRevLett.119.100503. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.119.100503.
https:/​/​doi.org/​10.1103/​PhysRevLett.119.100503

[11] Kenneth R. Brown, Robert J. Clark, and Isaac L. Chuang. Limitations of quantum simulation examined by simulating a pairing hamiltonian using nuclear magnetic resonance. Phys. Rev. Lett., 97: 050504, Aug 2006. https:/​/​doi.org/​10.1103/​PhysRevLett.97.050504. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.97.050504.
https:/​/​doi.org/​10.1103/​PhysRevLett.97.050504

[12] Earl Campbell. Shorter gate sequences for quantum computing by mixing unitaries. Phys. Rev. A, 95: 042306, Apr 2017. https:/​/​doi.org/​10.1103/​PhysRevA.95.042306. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.95.042306.
https:/​/​doi.org/​10.1103/​PhysRevA.95.042306

[13] Earl Campbell. Random compiler for fast hamiltonian simulation. Phys. Rev. Lett., 123: 070503, Aug 2019. https:/​/​doi.org/​10.1103/​PhysRevLett.123.070503. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.123.070503.
https:/​/​doi.org/​10.1103/​PhysRevLett.123.070503

[14] Chi-Fang Chen, Richard Kueng, Joel A Tropp, et al. Concentration for random product formulas. arXiv preprint arXiv:2008.11751, 2020.
arXiv:2008.11751

[15] Andrew M. Childs and Yuan Su. Nearly optimal lattice simulation by product formulas. Phys. Rev. Lett., 123: 050503, Aug 2019. https:/​/​doi.org/​10.1103/​PhysRevLett.123.050503. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.123.050503.
https:/​/​doi.org/​10.1103/​PhysRevLett.123.050503

[16] Andrew M Childs and Nathan Wiebe. Hamiltonian simulation using linear combinations of unitary operations. Quantum Information & Computation, 12 (11-12): 901–924, 2012. URL https:/​/​dl.acm.org/​doi/​10.5555/​2481569.2481570.
https:/​/​dl.acm.org/​doi/​10.5555/​2481569.2481570

[17] Andrew M. Childs, Dmitri Maslov, Yunseong Nam, Neil J. Ross, and Yuan Su. Toward the first quantum simulation with quantum speedup. Proceedings of the National Academy of Sciences, 115 (38): 9456–9461, 2018. ISSN 0027-8424. https:/​/​doi.org/​10.1073/​pnas.1801723115. URL https:/​/​www.pnas.org/​content/​115/​38/​9456.
https:/​/​doi.org/​10.1073/​pnas.1801723115
https:/​/​www.pnas.org/​content/​115/​38/​9456

[18] Andrew M. Childs, Aaron Ostrander, and Yuan Su. Faster quantum simulation by randomization. Quantum, 3: 182, September 2019. ISSN 2521-327X. https:/​/​doi.org/​10.22331/​q-2019-09-02-182. URL https:/​/​doi.org/​10.22331/​q-2019-09-02-182.
https:/​/​doi.org/​10.22331/​q-2019-09-02-182

[19] Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu. Theory of trotter error with commutator scaling. Phys. Rev. X, 11: 011020, Feb 2021. https:/​/​doi.org/​10.1103/​PhysRevX.11.011020. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevX.11.011020.
https:/​/​doi.org/​10.1103/​PhysRevX.11.011020

[20] Richard P. Feynman. Simulating physics with computers. International Journal of Theoretical Physics, 21 (6): 467–488, Jun 1982. ISSN 1572-9575. https:/​/​doi.org/​10.1007/​BF02650179. URL https:/​/​doi.org/​10.1007/​BF02650179.
https:/​/​doi.org/​10.1007/​BF02650179

[21] Markus Heyl, Philipp Hauke, and Peter Zoller. Quantum localization bounds trotter errors in digital quantum simulation. Science advances, 5 (4): eaau8342, 2019. https:/​/​doi.org/​10.1126/​sciadv.aau8342. URL https:/​/​advances.sciencemag.org/​content/​5/​4/​eaau8342.
https:/​/​doi.org/​10.1126/​sciadv.aau8342
https:/​/​advances.sciencemag.org/​content/​5/​4/​eaau8342

[22] P. Jordan and E. Wigner. Über das Paulische Äquivalenzverbot. Zeitschrift für Physik, 47 (9): 631–651, September 1928. ISSN 0044-3328. https:/​/​doi.org/​10.1007/​BF01331938. URL https:/​/​doi.org/​10.1007/​BF01331938.
https:/​/​doi.org/​10.1007/​BF01331938

[23] Stephen P Jordan, Keith SM Lee, and John Preskill. Quantum algorithms for quantum field theories. Science, 336 (6085): 1130–1133, 2012. https:/​/​doi.org/​10.1126/​science.1217069. URL https:/​/​science.sciencemag.org/​content/​336/​6085/​1130.
https:/​/​doi.org/​10.1126/​science.1217069
https:/​/​science.sciencemag.org/​content/​336/​6085/​1130

[24] Seth Lloyd. Universal quantum simulators. Science, 273: 1073–1078, Aug 1996. https:/​/​doi.org/​10.1126/​science.273.5278.1073. URL http:/​/​science.sciencemag.org/​content/​273/​5278/​1073/​tab-article-info.
https:/​/​doi.org/​10.1126/​science.273.5278.1073
http:/​/​science.sciencemag.org/​content/​273/​5278/​1073/​tab-article-info

[25] Guang Hao Low and Isaac L. Chuang. Optimal hamiltonian simulation by quantum signal processing. Phys. Rev. Lett., 118: 010501, Jan 2017. https:/​/​doi.org/​10.1103/​PhysRevLett.118.010501. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.118.010501.
https:/​/​doi.org/​10.1103/​PhysRevLett.118.010501

[26] Guang Hao Low and Isaac L. Chuang. Hamiltonian Simulation by Qubitization. Quantum, 3: 163, July 2019. ISSN 2521-327X. https:/​/​doi.org/​10.22331/​q-2019-07-12-163. URL https:/​/​doi.org/​10.22331/​q-2019-07-12-163.
https:/​/​doi.org/​10.22331/​q-2019-07-12-163

[27] Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C. Benjamin, and Xiao Yuan. Quantum computational chemistry. Rev. Mod. Phys., 92: 015003, Mar 2020. https:/​/​doi.org/​10.1103/​RevModPhys.92.015003. URL https:/​/​link.aps.org/​doi/​10.1103/​RevModPhys.92.015003.
https:/​/​doi.org/​10.1103/​RevModPhys.92.015003

[28] Jarrod R McClean, Nicholas C Rubin, Kevin J Sung, Ian D Kivlichan, Xavier Bonet-Monroig, Yudong Cao, Chengyu Dai, E Schuyler Fried, Craig Gidney, Brendan Gimby, Pranav Gokhale, Thomas Häner, Tarini Hardikar, Vojtěch Havlíček, Oscar Higgott, Cupjin Huang, Josh Izaac, Zhang Jiang, Xinle Liu, Sam McArdle, Matthew Neeley, Thomas O'Brien, Bryan O'Gorman, Isil Ozfidan, Maxwell D Radin, Jhonathan Romero, Nicolas P D Sawaya, Bruno Senjean, Kanav Setia, Sukin Sim, Damian S Steiger, Mark Steudtner, Qiming Sun, Wei Sun, Daochen Wang, Fang Zhang, and Ryan Babbush. OpenFermion: the electronic structure package for quantum computers. Quantum Science and Technology, 5 (3): 034014, jun 2020. https:/​/​doi.org/​10.1088/​2058-9565/​ab8ebc. URL https:/​/​doi.org/​10.1088/​2058-9565/​ab8ebc.
https:/​/​doi.org/​10.1088/​2058-9565/​ab8ebc

[29] Richard Meister, Simon C Benjamin, and Earl T Campbell. Tailoring term truncations for electronic structure calculations using a linear combination of unitaries. arXiv preprint arXiv:2007.11624, 2020.
arXiv:2007.11624

[30] R. Somma, G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme. Simulating physical phenomena by quantum networks. Phys. Rev. A, 65: 042323, Apr 2002. https:/​/​doi.org/​10.1103/​PhysRevA.65.042323. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.65.042323.
https:/​/​doi.org/​10.1103/​PhysRevA.65.042323

[31] Yuan Su, Hsin-Yuan Huang, and Earl T. Campbell. Nearly tight Trotterization of interacting electrons. Quantum, 5: 495, July 2021. ISSN 2521-327X. https:/​/​doi.org/​10.22331/​q-2021-07-05-495. URL https:/​/​doi.org/​10.22331/​q-2021-07-05-495.
https:/​/​doi.org/​10.22331/​q-2021-07-05-495

[32] Masuo Suzuki. Generalized trotter's formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems. Comm. Math. Phys., 51: 183–190, 1976. https:/​/​doi.org/​10.1007/​BF01609348. URL https:/​/​rdcu.be/​cv10X.
https:/​/​doi.org/​10.1007/​BF01609348
https:/​/​rdcu.be/​cv10X

[33] Masuo Suzuki. General theory of fractal path integrals with applications to many-body theories and statistical physics. Journal of Mathematical Physics, 32 (2): 400–407, 1991. https:/​/​doi.org/​10.1063/​1.529425.
https:/​/​doi.org/​10.1063/​1.529425

[34] Minh C. Tran, Su-Kuan Chu, Yuan Su, Andrew M. Childs, and Alexey V. Gorshkov. Destructive error interference in product-formula lattice simulation. Physical Review Letters, 124 (22), Jun 2020. ISSN 1079-7114. https:/​/​doi.org/​10.1103/​physrevlett.124.220502. URL http:/​/​dx.doi.org/​10.1103/​PhysRevLett.124.220502.
https:/​/​doi.org/​10.1103/​physrevlett.124.220502

[35] Dave Wecker, Bela Bauer, Bryan K. Clark, Matthew B. Hastings, and Matthias Troyer. Gate-count estimates for performing quantum chemistry on small quantum computers. Phys. Rev. A, 90: 022305, Aug 2014. https:/​/​doi.org/​10.1103/​PhysRevA.90.022305. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.90.022305.
https:/​/​doi.org/​10.1103/​PhysRevA.90.022305

[36] Dave Wecker, Matthew B. Hastings, Nathan Wiebe, Bryan K. Clark, Chetan Nayak, and Matthias Troyer. Solving strongly correlated electron models on a quantum computer. Phys. Rev. A, 92: 062318, Dec 2015. https:/​/​doi.org/​10.1103/​PhysRevA.92.062318. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.92.062318.
https:/​/​doi.org/​10.1103/​PhysRevA.92.062318

Cited by

On Crossref's cited-by service no data on citing works was found (last attempt 2021-09-23 07:03:03). On SAO/NASA ADS no data on citing works was found (last attempt 2021-09-23 07:03:03).