Principal Trotter Observation Error with Truncated Commutators
Abstract
Hamiltonian simulation is one of the most promising applications of quantum computers, and the product formula is one of the most important methods for this purpose. Previous related work has mainly focused on the worstcase or averagecase scenarios. In this work, we consider the simulation error under a fixed observable. Under a fixed observable, errors that commute with this observable become less important. To illustrate this point, we define the observation error as the expectation under the observable and provide a commutativitybased upper bound using the BakerCampbellHausdorff formula. For highly commuting observables, the simulation error indicated by this upper bound can be significantly compressed. In the experiment with the Heisenberg model, the observation bound compresses the Trotter number by nearly half compared to recent commutator bounds. Additionally, we found that the evolution sequence significantly affects the observation error. By utilizing a simulated annealing algorithm, we designed a sequence optimization algorithm, achieving further compression of the Trotter number. The experiment on the hydrogen molecule Hamiltonian demonstrates that optimizing the sequence can lead to nearly half the reduction in the Trotter number.
Cite
@article{arxiv.2408.03891,
title = {Principal Trotter Observation Error with Truncated Commutators},
author = {Langyu Li},
journal= {arXiv preprint arXiv:2408.03891},
year = {2024}
}