DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

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:
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:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 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. https://doi.org/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. https://doi.org/10.1016/j.aop.2020.168165. https://www.osti.gov/servlets/purl/1646280.
@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},
number = C,
volume = 418,
place = {United States},
year = {2020},
month = {4}
}

Journal Article:

Citation Metrics:
Cited by: 2 works
Citation information provided by
Web of Science

Save / Share:

Works referenced in this record:

On the product of semi-groups of operators
journal, April 1959


Efficient Simulation of One-Dimensional Quantum Many-Body Systems
journal, July 2004


Real-Time Evolution Using the Density Matrix Renormalization Group
journal, August 2004


Infinite time-evolving block decimation algorithm beyond unitary evolution
journal, October 2008


Criticality, the Area Law, and the Computational Power of Projected Entangled Pair States
journal, June 2006


Quantum phase transition in spin-3/2 systems on the hexagonal lattice — optimum ground state approach
journal, January 1997

  • Niggemann, H.; Klümper, A.; Zittartz, J.
  • Zeitschrift für Physik B Condensed Matter, Vol. 104, Issue 1
  • DOI: 10.1007/s002570050425

Self-consistent tensor product variational approximation for 3D classical models
journal, June 2000


Stripe ansätze from exactly solved models
journal, July 2001


Universal Quantum Simulators
journal, August 1996


Efficient Quantum Algorithms for Simulating Sparse Hamiltonians
journal, December 2006

  • Berry, Dominic W.; Ahokas, Graeme; Cleve, Richard
  • Communications in Mathematical Physics, Vol. 270, Issue 2
  • DOI: 10.1007/s00220-006-0150-x

Universal Digital Quantum Simulation with Trapped Ions
journal, September 2011


Dissipative Quantum Church-Turing Theorem
journal, September 2011


An open-system quantum simulator with trapped ions
journal, February 2011

  • Barreiro, Julio T.; Müller, Markus; Schindler, Philipp
  • Nature, Vol. 470, Issue 7335
  • DOI: 10.1038/nature09801

Toward the first quantum simulation with quantum speedup
journal, September 2018

  • Childs, Andrew M.; Maslov, Dmitri; Nam, Yunseong
  • Proceedings of the National Academy of Sciences, Vol. 115, Issue 38
  • DOI: 10.1073/pnas.1801723115

Monte Carlo Simulation of Quantum Spin Systems. I
journal, November 1977

  • Suzuki, M.; Miyashita, S.; Kuroda, A.
  • Progress of Theoretical Physics, Vol. 58, Issue 5
  • DOI: 10.1143/PTP.58.1377

Applications of quantum Monte Carlo methods in condensed systems
journal, January 2011


A Can0nical Integrati0n Technique
journal, August 1983


Construction of higher order symplectic integrators
journal, November 1990


On the Numerical Integration of Ordinary Differential Equations by Symmetric Composition Methods
journal, January 1995

  • McLachlan, Robert I.
  • SIAM Journal on Scientific Computing, Vol. 16, Issue 1
  • DOI: 10.1137/0916010

Fourth-order symplectic integration
journal, May 1990


General theory of fractal path integrals with applications to many‐body theories and statistical physics
journal, February 1991

  • Suzuki, Masuo
  • Journal of Mathematical Physics, Vol. 32, Issue 2
  • DOI: 10.1063/1.529425

Composition constants for raising the orders of unconventional schemes for ordinary differential equations
journal, July 1997


Optimized Forest–Ruth- and Suzuki-like algorithms for integration of motion in many-body systems
journal, July 2002


On a Law of Combination of Operators (Second Paper) *
journal, November 1897


Alternants and Continuous Groups
journal, January 1905


A basis for free Lie rings and higher commutators in free groups
journal, May 1950


Die Unterringe der freien Lieschen Ringe
journal, December 1956


A new efficient algorithm for computing Gröbner bases (F4)
journal, June 1999


The finite group velocity of quantum spin systems
journal, September 1972

  • Lieb, Elliott H.; Robinson, Derek W.
  • Communications in Mathematical Physics, Vol. 28, Issue 3
  • DOI: 10.1007/BF01645779

Lieb-Robinson Bound and Locality for General Markovian Quantum Dynamics
journal, May 2010


The accuracy of symplectic integrators
journal, March 1992


Multispinon Continua at Zero and Finite Temperature in a Near-Ideal Heisenberg Chain
journal, September 2013


Matrix product purifications for canonical ensembles and quantum number distributions
journal, September 2016


Higher-order methods for simulations on quantum computers
journal, September 1999


Nonequilibrium electron transport using the density matrix renormalization group method
journal, September 2004


Time evolution of Matrix Product States
journal, December 2006


Lanczos algorithm with matrix product states for dynamical correlation functions
journal, May 2012


Out-of-equilibrium dynamics with matrix product states
journal, December 2012


Time-Dependent Variational Principle for Quantum Lattices
journal, August 2011


Unifying time evolution and optimization with matrix product states
journal, October 2016


The density-matrix renormalization group in the age of matrix product states
journal, January 2011


Time-evolution methods for matrix-product states
journal, December 2019


Simulating Hamiltonian Dynamics with a Truncated Taylor Series
journal, March 2015


Optimal Hamiltonian Simulation by Quantum Signal Processing
journal, January 2017