skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation

Abstract

In this paper, we present a product formula to approximate the exponential of a skew-Hermitian operator that is a sum of generators of a Lie algebra. The number of terms in the product depends on the structure factors. When the generators have large norm with respect to the dimension of the Lie algebra, or when the norm of the effective operator resulting from nested commutators is less than the product of the norms, the number of terms in the product is significantly less than that obtained from well-known results. We apply our results to construct product formulas useful for the quantum simulation of some continuous-variable and bosonic physical systems, including systems whose potential is not quadratic. For many of these systems, we show that the number of terms in the product can be sublinear or even subpolynomial in the dimension of the relevant local Hilbert spaces, where such a dimension is usually determined by the energy scale of the problem. Our results emphasize the power of quantum computers for the simulation of various quantum systems.

Authors:
 [1]
  1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States). Theoretical Division
Publication Date:
Research Org.:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
USDOE Laboratory Directed Research and Development (LDRD) Program
OSTI Identifier:
1337097
Alternate Identifier(s):
OSTI ID: 1421146
Report Number(s):
LA-UR-15-29416
Journal ID: ISSN 0022-2488
Grant/Contract Number:  
AC52-06NA25396
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Mathematical Physics
Additional Journal Information:
Journal Volume: 57; Journal Issue: 6; Journal ID: ISSN 0022-2488
Publisher:
American Institute of Physics (AIP)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICS AND COMPUTING; Quantum Computing; Quantum Algorithms

Citation Formats

Somma, Rolando D. A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation. United States: N. p., 2016. Web. https://doi.org/10.1063/1.4952761.
Somma, Rolando D. A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation. United States. https://doi.org/10.1063/1.4952761
Somma, Rolando D. Wed . "A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation". United States. https://doi.org/10.1063/1.4952761. https://www.osti.gov/servlets/purl/1337097.
@article{osti_1337097,
title = {A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation},
author = {Somma, Rolando D.},
abstractNote = {In this paper, we present a product formula to approximate the exponential of a skew-Hermitian operator that is a sum of generators of a Lie algebra. The number of terms in the product depends on the structure factors. When the generators have large norm with respect to the dimension of the Lie algebra, or when the norm of the effective operator resulting from nested commutators is less than the product of the norms, the number of terms in the product is significantly less than that obtained from well-known results. We apply our results to construct product formulas useful for the quantum simulation of some continuous-variable and bosonic physical systems, including systems whose potential is not quadratic. For many of these systems, we show that the number of terms in the product can be sublinear or even subpolynomial in the dimension of the relevant local Hilbert spaces, where such a dimension is usually determined by the energy scale of the problem. Our results emphasize the power of quantum computers for the simulation of various quantum systems.},
doi = {10.1063/1.4952761},
journal = {Journal of Mathematical Physics},
number = 6,
volume = 57,
place = {United States},
year = {2016},
month = {6}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Citation Metrics:
Cited by: 1 work
Citation information provided by
Web of Science

Save / Share:

Works referenced in this record:

Quantum algorithms for fermionic simulations
journal, July 2001


Simulated Quantum Computation of Molecular Energies
journal, September 2005


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


Gate-count estimates for performing quantum chemistry on small quantum computers
journal, August 2014


Exponentially more precise quantum simulation of fermions in second quantization
journal, March 2016


Universal Quantum Simulators
journal, August 1996


Adiabatic quantum state generation and statistical zero knowledge
conference, January 2003

  • Aharonov, Dorit; Ta-Shma, Amnon
  • Proceedings of the thirty-fifth ACM symposium on Theory of computing - STOC '03
  • DOI: 10.1145/780542.780546

Simulating physics with computers
journal, June 1982

  • Feynman, Richard P.
  • International Journal of Theoretical Physics, Vol. 21, Issue 6-7
  • DOI: 10.1007/bf02650179

Simulating physical phenomena by quantum networks
journal, April 2002


Canonical representations of sp(2 n , R )
journal, April 1992

  • Rangarajan, Govindan; Neri, Filippo
  • Journal of Mathematical Physics, Vol. 33, Issue 4
  • DOI: 10.1063/1.529702

Simulating Chemistry Using Quantum Computers
journal, May 2011


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

Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation
journal, August 1984


Optimal Trotterization in universal quantum simulators under faulty control
journal, May 2015


Exponential improvement in precision for simulating sparse Hamiltonians
conference, January 2014

  • Berry, Dominic W.; Childs, Andrew M.; Cleve, Richard
  • Proceedings of the 46th Annual ACM Symposium on Theory of Computing - STOC '14
  • DOI: 10.1145/2591796.2591854

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

Higher order decompositions of ordered operator exponentials
journal, January 2010

  • Wiebe, Nathan; Berry, Dominic; Høyer, Peter
  • Journal of Physics A: Mathematical and Theoretical, Vol. 43, Issue 6
  • DOI: 10.1088/1751-8113/43/6/065203

Chemical basis of Trotter-Suzuki errors in quantum chemistry simulation
journal, February 2015


Simulating physics with computers
journal, June 1982

  • Feynman, Richard P.
  • International Journal of Theoretical Physics, Vol. 21, Issue 6-7
  • DOI: 10.1007/BF02650179

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


Quantum algorithms for fermionic simulations
journal, July 2001


Simulating physical phenomena by quantum networks
journal, April 2002


Gate-count estimates for performing quantum chemistry on small quantum computers
journal, August 2014


Optimal Trotterization in universal quantum simulators under faulty control
journal, May 2015


Chemical basis of Trotter-Suzuki errors in quantum chemistry simulation
journal, February 2015


Exponential Improvement in Precision for Simulating Sparse Hamiltonians
journal, January 2017

  • Berry, Dominic W.; Childs, Andrew M.; Cleve, Richard
  • Forum of Mathematics, Sigma, Vol. 5
  • DOI: 10.1017/fms.2017.2

    Works referencing / citing this record:

    Bounding the costs of quantum simulation of many-body physics in real space
    journal, June 2017

    • Kivlichan, Ian D.; Wiebe, Nathan; Babbush, Ryan
    • Journal of Physics A: Mathematical and Theoretical, Vol. 50, Issue 30
    • DOI: 10.1088/1751-8121/aa77b8

    Nearly Optimal Lattice Simulation by Product Formulas
    journal, August 2019


    Hamiltonian Simulation by Qubitization
    journal, July 2019


    Compilation by stochastic Hamiltonian sparsification
    journal, February 2020