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Optimized Lie–Trotter–Suzuki decompositions for two and three noncommuting terms
Abstract
Lie–Trotter–Suzuki decompositions are an efficient way to approximate operator exponentials $exp\left(tH\right)$ when $H$ is a sum of $n$ (noncommuting) terms which, individually, can be exponentiated easily. They are employed in timeevolution 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 1norm 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 twodimensional lattice model with finiterange interactions. This follows by solving a partitioning problem for the interaction graph.
 Authors:

 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:
 Accepted Manuscript
 Journal Name:
 Annals of Physics
 Additional Journal Information:
 Journal Volume: 418; Journal Issue: C; Journal ID: ISSN 00034916
 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; LieTrotter product formula; operator exponential; tensor network states; quantum Monte Carlo; BakerCampbellHausdorff formula; Gröbner basis
Citation Formats
Barthel, Thomas, and Zhang, Yikang. Optimized Lie–Trotter–Suzuki decompositions for two and three noncommuting 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 noncommuting 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 noncommuting terms". United States. doi:10.1016/j.aop.2020.168165.
@article{osti_1646280,
title = {Optimized Lie–Trotter–Suzuki decompositions for two and three noncommuting 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 (noncommuting) terms which, individually, can be exponentiated easily. They are employed in timeevolution 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 1norm 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 twodimensional lattice model with finiterange interactions. This follows by solving a partitioning problem for the interaction graph.},
doi = {10.1016/j.aop.2020.168165},
journal = {Annals of Physics},
number = C,
volume = 418,
place = {United States},
year = {2020},
month = {4}
}
Web of Science