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:
-
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States). Theoretical Division
- Publication Date:
- Research Org.:
- Los Alamos National Laboratory (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. doi: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 = {Wed Jun 01 00:00:00 EDT 2016},
month = {Wed Jun 01 00:00:00 EDT 2016}
}
Free Publicly Available Full Text
Publisher's Version of Record
Other availability
Search WorldCat to find libraries that may hold this journal
Cited by: 13 works
Citation information provided by
Web of Science
Web of Science
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.
Works referenced in this record:
Simulated Quantum Computation of Molecular Energies
journal, September 2005
- Aspuru-Guzik, A.
- Science, Vol. 309, Issue 5741
Exponentially more precise quantum simulation of fermions in second quantization
journal, March 2016
- Babbush, Ryan; Berry, Dominic W.; Kivlichan, Ian D.
- New Journal of Physics, Vol. 18, Issue 3
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
Canonical representations of sp(2 n , R )
journal, April 1992
- Rangarajan, Govindan; Neri, Filippo
- Journal of Mathematical Physics, Vol. 33, Issue 4
Simulating Chemistry Using Quantum Computers
journal, May 2011
- Kassal, Ivan; Whitfield, James D.; Perdomo-Ortiz, Alejandro
- Annual Review of Physical Chemistry, Vol. 62, Issue 1
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
Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation
journal, August 1984
- Taha, Thiab R.; Ablowitz, Mark I.
- Journal of Computational Physics, Vol. 55, Issue 2
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
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
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
Simulating physics with computers
journal, June 1982
- Feynman, Richard P.
- International Journal of Theoretical Physics, Vol. 21, Issue 6-7
Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations
journal, June 1990
- Suzuki, Masuo
- Physics Letters A, Vol. 146, Issue 6
Simulating Hamiltonian Dynamics with a Truncated Taylor Series
journal, March 2015
- Berry, Dominic W.; Childs, Andrew M.; Cleve, Richard
- Physical Review Letters, Vol. 114, Issue 9
Quantum algorithms for fermionic simulations
journal, July 2001
- Ortiz, G.; Gubernatis, J. E.; Knill, E.
- Physical Review A, Vol. 64, Issue 2
Simulating physical phenomena by quantum networks
journal, April 2002
- Somma, R.; Ortiz, G.; Gubernatis, J. E.
- Physical Review A, Vol. 65, Issue 4
Gate-count estimates for performing quantum chemistry on small quantum computers
journal, August 2014
- Wecker, Dave; Bauer, Bela; Clark, Bryan K.
- Physical Review A, Vol. 90, Issue 2
Optimal Trotterization in universal quantum simulators under faulty control
journal, May 2015
- Knee, George C.; Munro, William J.
- Physical Review A, Vol. 91, Issue 5
Chemical basis of Trotter-Suzuki errors in quantum chemistry simulation
journal, February 2015
- Babbush, Ryan; McClean, Jarrod; Wecker, Dave
- Physical Review A, Vol. 91, Issue 2
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
Exponential improvement in precision for simulating sparse Hamiltonians
text, January 2013
- Berry, Dominic W.; Childs, Andrew M.; Cleve, Richard
- arXiv
Gate count estimates for performing quantum chemistry on small quantum computers
text, January 2013
- Wecker, Dave; Bauer, Bela; Clark, Bryan K.
- arXiv
Chemical Basis of Trotter-Suzuki Errors in Quantum Chemistry Simulation
text, January 2014
- Babbush, Ryan; McClean, Jarrod; Wecker, Dave
- arXiv
Simulating Hamiltonian dynamics with a truncated Taylor series
text, January 2014
- Berry, Dominic W.; Childs, Andrew M.; Cleve, Richard
- arXiv
Optimal Trotterization in universal quantum simulators under faulty control
text, January 2015
- Knee, George C.; Munro, William J.
- arXiv
Simulating Physical Phenomena by Quantum Networks
text, January 2001
- Somma, R.; Ortiz, G.; Gubernatis, J. E.
- arXiv
Efficient quantum algorithms for simulating sparse Hamiltonians
text, January 2005
- Berry, Dominic W.; Ahokas, Graeme; Cleve, Richard
- arXiv
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
Hamiltonian Simulation by Qubitization
journal, July 2019
- Low, Guang Hao; Chuang, Isaac L.
- Quantum, Vol. 3
Compilation by stochastic Hamiltonian sparsification
journal, February 2020
- Ouyang, Yingkai; White, David R.; Campbell, Earl T.
- Quantum, Vol. 4
Bounding the costs of quantum simulation of many-body physics in real space
text, January 2016
- Kivlichan, Ian D.; Wiebe, Nathan; Babbush, Ryan
- arXiv