Tong, Yu and Albert, Victor V. and McClean, Jarrod R. and Preskill, John and Su, Yuan
(2021)
*Provably accurate simulation of gauge theories and bosonic systems.*
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(Unpublished)
https://resolver.caltech.edu/CaltechAUTHORS:20220113-182219174

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## Abstract

Quantum many-body systems involving bosonic modes or gauge fields have infinite-dimensional local Hilbert spaces which must be truncated to perform simulations of real-time dynamics on classical or quantum computers. To analyze the truncation error, we develop methods for bounding the rate of growth of local quantum numbers such as the occupation number of a mode at a lattice site, or the electric field at a lattice link. Our approach applies to various models of bosons interacting with spins or fermions, and also to both abelian and non-abelian gauge theories. We show that if states in these models are truncated by imposing an upper limit Λ on each local quantum number, and if the initial state has low local quantum numbers, then an error at most ϵ can be achieved by choosing Λ to scale polylogarithmically with ϵ⁻¹, an exponential improvement over previous bounds based on energy conservation. For the Hubbard-Holstein model, we numerically compute a bound on Λ that achieves accuracy ϵ, obtaining significantly improved estimates in various parameter regimes. We also establish a criterion for truncating the Hamiltonian with a provable guarantee on the accuracy of time evolution. Building on that result, we formulate quantum algorithms for dynamical simulation of lattice gauge theories and of models with bosonic modes; the gate complexity depends almost linearly on spacetime volume in the former case, and almost quadratically on time in the latter case. We establish a lower bound showing that there are systems involving bosons for which this quadratic scaling with time cannot be improved. By applying our result on the truncation error in time evolution, we also prove that spectrally isolated energy eigenstates can be approximated with accuracy ϵ by truncating local quantum numbers at Λ = polylog(ϵ⁻¹).

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Additional Information: | We thank Kunal Sharma, Mark Wilde, Minh Cong Tran, Junyu Liu, Chi-Fang (Anthony) Chen, Ryan Babbush, Joonho Lee, Di Luo, Nathan Wiebe, Dominic Berry, and Lin Lin for helpful discussions. YT was partly supported by the NSF Quantum Leap Challenge Institute (QLCI) program through Grant No. OMA-2016245, and by the Department of Energy under Grant No. FWP-NQISCCAWL. JP was partly supported by the U.S. Department of Energy Office of Advanced Scientific Computing Research (DE-NA0003525, DE-SC0020290) and Office of High Energy Physics (DE-ACO2-07CH11359, DE-SC0018407), the Simons Foundation It from Qubit Collaboration, the Air Force Office of Scientific Research (FA9550-19-1-0360), and the National Science Foundation (PHY-1733907). YS was partly supported by the National Science Foundation RAISE-TAQS 1839204. The Institute for Quantum Information and Matter is an NSF Physics Frontiers Center. Contributions to this work by NIST, an agency of the US government, are not subject to US copyright. Any mention of commercial products does not indicate endorsement by NIST. | ||||||||||||||||||||||

Group: | AWS Center for Quantum Computing, Institute for Quantum Information and Matter | ||||||||||||||||||||||

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Record Number: | CaltechAUTHORS:20220113-182219174 | ||||||||||||||||||||||

Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20220113-182219174 | ||||||||||||||||||||||

Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||||||||||||||

ID Code: | 112871 | ||||||||||||||||||||||

Collection: | CaltechAUTHORS | ||||||||||||||||||||||

Deposited By: | George Porter | ||||||||||||||||||||||

Deposited On: | 13 Jan 2022 21:28 | ||||||||||||||||||||||

Last Modified: | 28 Oct 2022 16:46 |

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