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Title: Optimized Lie–Trotter–Suzuki decompositions for two and three non-commuting terms

Abstract

Lie–Trotter–Suzuki decompositions are an efficient way to approximate operator exponentials exp ( t H ) when H is a sum of n (non-commuting) terms which, individually, can be exponentiated easily. They are employed in time-evolution algorithms for tensor network states, digital quantum simulation protocols, path integral methods like quantum Monte Carlo, and splitting methods for symplectic integrators in classical Hamiltonian systems. Here, we provide optimized decompositions up to order t 6 . The leading error term is expanded in nested commutators (Hall bases) and we minimize the 1-norm of the coefficients. For n = 2 terms, several of the optima we find are close to those in McLachlan (1995). Generally, our results substantially improve over unoptimized decompositions by Forest, Ruth, Yoshida, and Suzuki. We explain why these decompositions are sufficient to efficiently simulate any one- or two-dimensional lattice model with finite-range interactions. This follows by solving a partitioning problem for the interaction graph.

Authors:
 [1];  [1]
  1. Duke Univ., Durham, NC (United States). Dept. of Physics
Publication Date:
Research Org.:
Duke Univ., Durham, NC (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Basic Energy Sciences (BES)
OSTI Identifier:
1646280
Alternate Identifier(s):
OSTI ID: 1617108
Grant/Contract Number:  
SC0019449
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Annals of Physics
Additional Journal Information:
Journal Volume: 418; Journal Issue: C; Journal ID: ISSN 0003-4916
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
Lie–Trotter–Suzuki decomposition; quantum Hamiltonian simulation; symplectic integrators; splitting methods; Hall basis; graph partitioning problem; Lie-Trotter product formula; operator exponential; tensor network states; quantum Monte Carlo; Baker-Campbell-Hausdorff formula; Gröbner basis

Citation Formats

Barthel, Thomas, and Zhang, Yikang. Optimized Lie–Trotter–Suzuki decompositions for two and three non-commuting terms. United States: N. p., 2020. Web. doi:10.1016/j.aop.2020.168165.
Barthel, Thomas, & Zhang, Yikang. Optimized Lie–Trotter–Suzuki decompositions for two and three non-commuting terms. United States. doi:10.1016/j.aop.2020.168165.
Barthel, Thomas, and Zhang, Yikang. Thu . "Optimized Lie–Trotter–Suzuki decompositions for two and three non-commuting terms". United States. doi:10.1016/j.aop.2020.168165.
@article{osti_1646280,
title = {Optimized Lie–Trotter–Suzuki decompositions for two and three non-commuting terms},
author = {Barthel, Thomas and Zhang, Yikang},
abstractNote = {Lie–Trotter–Suzuki decompositions are an efficient way to approximate operator exponentials exp(tH) when H is a sum of n (non-commuting) terms which, individually, can be exponentiated easily. They are employed in time-evolution algorithms for tensor network states, digital quantum simulation protocols, path integral methods like quantum Monte Carlo, and splitting methods for symplectic integrators in classical Hamiltonian systems. Here, we provide optimized decompositions up to order t6. The leading error term is expanded in nested commutators (Hall bases) and we minimize the 1-norm of the coefficients. For n=2 terms, several of the optima we find are close to those in McLachlan (1995). Generally, our results substantially improve over unoptimized decompositions by Forest, Ruth, Yoshida, and Suzuki. We explain why these decompositions are sufficient to efficiently simulate any one- or two-dimensional lattice model with finite-range interactions. This follows by solving a partitioning problem for the interaction graph.},
doi = {10.1016/j.aop.2020.168165},
journal = {Annals of Physics},
issn = {0003-4916},
number = C,
volume = 418,
place = {United States},
year = {2020},
month = {4}
}

Journal Article:
Free Publicly Available Full Text
This content will become publicly available on April 16, 2021
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