Theory of Trotter Error with Commutator Scaling

The Lie-Trotter formula, together with its higher-order generalizations, provides a direct approach to decomposing the exponential of a sum of operators. Despite significant effort, the error scaling of such product formulas remains poorly understood. We develop a theory of Trotter error that overco...

Descripción completa

Guardado en:

Detalles Bibliográficos
Autores principales:	Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, Shuchen Zhu
Formato:	article
Lenguaje:	EN
Publicado:	American Physical Society 2021
Materias:	Physics QC1-999
Acceso en línea:	https://doaj.org/article/0f379eec97814cbbbfe28873fb53ab68
Etiquetas:	Agregar Etiqueta Sin Etiquetas, Sea el primero en etiquetar este registro!

id	oai:doaj.org-article:0f379eec97814cbbbfe28873fb53ab68
record_format	dspace
spelling	oai:doaj.org-article:0f379eec97814cbbbfe28873fb53ab682021-12-02T14:09:11ZTheory of Trotter Error with Commutator Scaling10.1103/PhysRevX.11.0110202160-3308https://doaj.org/article/0f379eec97814cbbbfe28873fb53ab682021-02-01T00:00:00Zhttp://doi.org/10.1103/PhysRevX.11.011020http://doi.org/10.1103/PhysRevX.11.011020https://doaj.org/toc/2160-3308The Lie-Trotter formula, together with its higher-order generalizations, provides a direct approach to decomposing the exponential of a sum of operators. Despite significant effort, the error scaling of such product formulas remains poorly understood. We develop a theory of Trotter error that overcomes the limitations of prior approaches based on truncating the Baker-Campbell-Hausdorff expansion. Our analysis directly exploits the commutativity of operator summands, producing tighter error bounds for both real- and imaginary-time evolutions. Whereas previous work achieves similar goals for systems with geometric locality or Lie-algebraic structure, our approach holds, in general. We give a host of improved algorithms for digital quantum simulation and quantum Monte Carlo methods, including simulations of second-quantized plane-wave electronic structure, k-local Hamiltonians, rapidly decaying power-law interactions, clustered Hamiltonians, the transverse field Ising model, and quantum ferromagnets, nearly matching or even outperforming the best previous results. We obtain further speedups using the fact that product formulas can preserve the locality of the simulated system. Specifically, we show that local observables can be simulated with complexity independent of the system size for power-law interacting systems, which implies a Lieb-Robinson bound as a by-product. Our analysis reproduces known tight bounds for first- and second-order formulas. Our higher-order bound overestimates the complexity of simulating a one-dimensional Heisenberg model with an even-odd ordering of terms by only a factor of 5, and it is close to tight for power-law interactions and other orderings of terms. This result suggests that our theory can accurately characterize Trotter error in terms of both asymptotic scaling and constant prefactor.Andrew M. ChildsYuan SuMinh C. TranNathan WiebeShuchen ZhuAmerican Physical SocietyarticlePhysicsQC1-999ENPhysical Review X, Vol 11, Iss 1, p 011020 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	Physics QC1-999
spellingShingle	Physics QC1-999 Andrew M. Childs Yuan Su Minh C. Tran Nathan Wiebe Shuchen Zhu Theory of Trotter Error with Commutator Scaling
description	The Lie-Trotter formula, together with its higher-order generalizations, provides a direct approach to decomposing the exponential of a sum of operators. Despite significant effort, the error scaling of such product formulas remains poorly understood. We develop a theory of Trotter error that overcomes the limitations of prior approaches based on truncating the Baker-Campbell-Hausdorff expansion. Our analysis directly exploits the commutativity of operator summands, producing tighter error bounds for both real- and imaginary-time evolutions. Whereas previous work achieves similar goals for systems with geometric locality or Lie-algebraic structure, our approach holds, in general. We give a host of improved algorithms for digital quantum simulation and quantum Monte Carlo methods, including simulations of second-quantized plane-wave electronic structure, k-local Hamiltonians, rapidly decaying power-law interactions, clustered Hamiltonians, the transverse field Ising model, and quantum ferromagnets, nearly matching or even outperforming the best previous results. We obtain further speedups using the fact that product formulas can preserve the locality of the simulated system. Specifically, we show that local observables can be simulated with complexity independent of the system size for power-law interacting systems, which implies a Lieb-Robinson bound as a by-product. Our analysis reproduces known tight bounds for first- and second-order formulas. Our higher-order bound overestimates the complexity of simulating a one-dimensional Heisenberg model with an even-odd ordering of terms by only a factor of 5, and it is close to tight for power-law interactions and other orderings of terms. This result suggests that our theory can accurately characterize Trotter error in terms of both asymptotic scaling and constant prefactor.
format	article
author	Andrew M. Childs Yuan Su Minh C. Tran Nathan Wiebe Shuchen Zhu
author_facet	Andrew M. Childs Yuan Su Minh C. Tran Nathan Wiebe Shuchen Zhu
author_sort	Andrew M. Childs
title	Theory of Trotter Error with Commutator Scaling
title_short	Theory of Trotter Error with Commutator Scaling
title_full	Theory of Trotter Error with Commutator Scaling
title_fullStr	Theory of Trotter Error with Commutator Scaling
title_full_unstemmed	Theory of Trotter Error with Commutator Scaling
title_sort	theory of trotter error with commutator scaling
publisher	American Physical Society
publishDate	2021
url	https://doaj.org/article/0f379eec97814cbbbfe28873fb53ab68
work_keys_str_mv	AT andrewmchilds theoryoftrottererrorwithcommutatorscaling AT yuansu theoryoftrottererrorwithcommutatorscaling AT minhctran theoryoftrottererrorwithcommutatorscaling AT nathanwiebe theoryoftrottererrorwithcommutatorscaling AT shuchenzhu theoryoftrottererrorwithcommutatorscaling
_version_	1718391887213101056

Theory of Trotter Error with Commutator Scaling

Ejemplares similares