Bulletin of the American Physical Society
APS March Meeting 2023
Volume 68, Number 3
Las Vegas, Nevada (March 5-10)
Virtual (March 20-22); Time Zone: Pacific Time
Session A64: Algorithms for Hamiltionian Simulations
8:00 AM–11:00 AM,
Monday, March 6, 2023
Room: Room 415
Sponsoring
Unit:
DQI
Chair: William Munizzi, Arizona State University
Abstract: A64.00002 : Improved Trotter formula for time-dependent Hamiltonians
8:12 AM–8:24 AM
Presenter:
Asir Abrar
(NTT Research, Inc.)
Authors:
SHO SUGIURA
(NTT Research, Inc.)
Asir Abrar
(NTT Research, Inc.)
Isaac L Chuang
(Massachusetts Institute of Technology)
Tatsuhiko N Ikeda
(Institute for Solid State Physics, University of Tokyo)
What is lacking is an algorithm that incorporates the time variant of the Hamiltonian and a theory to quantify the error of the algorithm. The most common time-dependent simulation algorithm is based on the Lie-Trotter formula. However, it is inefficient because it discretizes the time dependence until each step can be approximated as time-independent. A time-dependent version of the Trotter expansion has also been proposed, but it is not considered an independent algorithm because it requires time-ordered integration even after the expansion. The Magnus and Dyson series expansions have been studied as algorithms that incorporate time variation, but they are inefficient at higher orders because the number of terms required combinatorially increases. Also, there is currently no error formula to compare these algorithms. Although error analysis for Trotter's time step has been done, there is no general formula for the error for each of the terms that make up the Hamiltonian.
In this project, we propose an extended Trotter formula for Hamiltonian simulation in the time-dependent case. This algorithm uses the Taylor expansion of the Hamiltonian so that the time derivative of the Hamiltonian appears in the formula. Then, by constructing a general theory to analyze the Trotter error in the time-dependent case, we show that this algorithm has less gate complexity for the same error than conventional algorithms. Our expansion and error theory provide a construction that allows us to derive the extended Trotter formulas with the smallest error in a given ansatz for arbitrary higher order.
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