Suzuki type estimates for exponentiated sums and generalized Lie-Trotter formulas in JB-algebras

Chehade, Sarah; Wang, Shuzhou; Wang, Zhenhua

doi:10.1016/j.laa.2023.10.004

This content will become publicly available on Thu Oct 10 00:00:00 EDT 2024

Title: Suzuki type estimates for exponentiated sums and generalized Lie-Trotter formulas in JB-algebras

Abstract

Lie-Trotter-Suzuki product formulas are ubiquitous in quantum mechanics, computing, and simulations. Approximating exponentiated sums with such formulas are investigated in the JB-algebraic setting. We show that the Suzuki type approximation for exponentiated sums holds in JB-algebras, we give explicit estimation formulas, and we deduce three generalizations of Lie-Trotter formulas for arbitrary number elements in such algebras. In conclusion, we also extended the Lie-Trotter formulas in a Jordan Banach algebra from three elements to an arbitrary number of elements.

Authors:

Chehade, Sarah ^[1]; Wang, Shuzhou ^[2];

^[3]

Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
University of Georgia, Athens, GA (United States)
University of Georgia, Athens, GA (United States); Alabama A&M University, Normal, AL (United States)

Publication Date:: Tue Oct 10 00:00:00 EDT 2023

Research Org.:: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

OSTI Identifier:: 2204575

Grant/Contract Number:: AC05-00OR22725; ERKJ347

Resource Type:: Accepted Manuscript

Journal Name:: Linear Algebra and Its Applications

Additional Journal Information:: Journal Volume: 680; Journal Issue: 1; Journal ID: ISSN 0024-3795

Publisher:: Elsevier

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; Lie-Trotter formula; Suzuki approximation; JB-algebra; Jordan-Banach algebra

Citation Formats


                    Chehade, Sarah, Wang, Shuzhou, and Wang, Zhenhua. Suzuki type estimates for exponentiated sums and generalized Lie-Trotter formulas in JB-algebras.  United States: N. p., 2023. 
Web.  doi:10.1016/j.laa.2023.10.004.

Copy to clipboard


                    Chehade, Sarah, Wang, Shuzhou, & Wang, Zhenhua. Suzuki type estimates for exponentiated sums and generalized Lie-Trotter formulas in JB-algebras.  United States.  https://doi.org/10.1016/j.laa.2023.10.004

Copy to clipboard


                    Chehade, Sarah, Wang, Shuzhou, and Wang, Zhenhua. Tue .  
"Suzuki type estimates for exponentiated sums and generalized Lie-Trotter formulas in JB-algebras".  United States.  https://doi.org/10.1016/j.laa.2023.10.004.

Copy to clipboard


                    
@article{osti_2204575,

  title        = {Suzuki type estimates for exponentiated sums and generalized Lie-Trotter formulas in JB-algebras},

  author       = {Chehade, Sarah and Wang, Shuzhou and Wang, Zhenhua},

  abstractNote = {Lie-Trotter-Suzuki product formulas are ubiquitous in quantum mechanics, computing, and simulations. Approximating exponentiated sums with such formulas are investigated in the JB-algebraic setting. We show that the Suzuki type approximation for exponentiated sums holds in JB-algebras, we give explicit estimation formulas, and we deduce three generalizations of Lie-Trotter formulas for arbitrary number elements in such algebras. In conclusion, we also extended the Lie-Trotter formulas in a Jordan Banach algebra from three elements to an arbitrary number of elements.},

  doi          = {10.1016/j.laa.2023.10.004},

  journal      = {Linear Algebra and Its Applications},

  number       = 1,

  volume       = 680,

  place        = {United States},

  year         = {Tue Oct 10 00:00:00 EDT 2023},

  month        = {Tue Oct 10 00:00:00 EDT 2023}

}

Copy to clipboard

Journal Article:

Free Publicly Available Full Text

This content will become publicly available on October 10, 2024

Publisher's Version of Record

https://doi.org/10.1016/j.laa.2023.10.004

Other availability

Search WorldCat to find libraries that may hold this journal

Save / Share:

Export Metadata

Save to My Library

Works referenced in this record:

Generalized Trotter's formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems
journal, June 1976

Suzuki, Masuo
Communications in Mathematical Physics, Vol. 51, Issue 2
DOI: 10.1007/BF01609348

Transfer-matrix method and Monte Carlo simulation in quantum spin systems
journal, March 1985

Suzuki, Masuo
Physical Review B, Vol. 31, Issue 5
DOI: 10.1103/PhysRevB.31.2957

Jordan $C^*$-algebras.
journal, January 1977

Wright, J. D. Maitland
Michigan Mathematical Journal, Vol. 24, Issue 3
DOI: 10.1307/mmj/1029001946

Similar Records in DOE PAGES and OSTI.GOV collections:

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

Journal Article Barthel, Thomas ; Zhang, Yikang - Annals of Physics

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. 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).more »« less
Cited by 17
https://doi.org/10.1016/j.aop.2020.168165

Full Text Available
A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation

Journal Article Somma, Rolando D., E-mail: somma@lanl.gov - Journal of Mathematical Physics

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 ofmore »« less
https://doi.org/10.1063/1.4952761
A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation

Journal Article Somma, Rolando D. - Journal of Mathematical Physics

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 themore »« less
Cited by 13
https://doi.org/10.1063/1.4952761

Full Text Available
Analogues of Chernoff's theorem and the Lie-Trotter theorem

Journal Article Neklyudov, Alexander Yu - Sbornik. Mathematics

This paper is concerned with the abstract Cauchy problem .x=Ax, x(0)=x{sub 0} element of D(A), where A is a densely defined linear operator on a Banach space X. It is proved that a solution x( {center_dot} ) of this problem can be represented as the weak limit lim {sub n{yields}}{sub {infinity}}{l_brace}F(t/n){sup n}x{sub 0}{r_brace}, where the function F:[0,{infinity}){yields}L(X) satisfies the equality F'(0)y=Ay, y element of D(A), for a natural class of operators. As distinct from Chernoff's theorem, the existence of a global solution to the Cauchy problem is not assumed. Based on this result, necessary and sufficient conditions are found formore »« less
https://doi.org/10.1070/SM2009V200N10ABEH004047; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)
Exponentiation and deformations of Lie-admissible algebras

Conference Myung, H C - Hadronic J.; (United States)

The exponential function is defined for a finite-dimensional real power-associative algebra with unit element. The application of the exponential function is focused on the power-associative (p,q)-mutation of a real or complex associative algebra. Explicit formulas are computed for the (p,q)-mutation of the real envelope of the spin 1 algebra and the Lie algebra so(3) of the rotation group, in light of earlier investigations of the spin 1/2. A slight variant of the mutated exponential is interpreted as a continuous function of the Lie algebra into some isotope of the corresponding linear Lie group. The second part of this paper ismore »« less

Similar Records

This content will become publicly available on Thu Oct 10 00:00:00 EDT 2024

Title: Suzuki type estimates for exponentiated sums and generalized Lie-Trotter formulas in JB-algebras

Abstract

Citation Formats

Generalized Trotter's formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems journal, June 1976

Transfer-matrix method and Monte Carlo simulation in quantum spin systems journal, March 1985

Jordan $C^*$-algebras. journal, January 1977

Generalized Trotter's formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems
journal, June 1976

Transfer-matrix method and Monte Carlo simulation in quantum spin systems
journal, March 1985

Jordan $C^*$-algebras.
journal, January 1977