Principal Trotter Observation Error with Truncated Commutators

TL;DR

This paper introduces a commutativity-based error bound for Hamiltonian simulation under a fixed observable, demonstrating significant Trotter number reductions through theoretical bounds and sequence optimization in quantum simulations.

Contribution

It develops a new observation error bound based on commutativity and introduces a sequence optimization method to reduce Trotter errors in quantum simulations.

Findings

Observation error bound compresses Trotter number by nearly half in Heisenberg model.

Sequence optimization further reduces Trotter number, nearly halving it in hydrogen molecule simulation.

The approach improves simulation efficiency by leveraging observable-specific error analysis.

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 worst$-$case or average$-$case 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 commutativity$-$based upper bound using the Baker$-$Campbell$-$Hausdorff 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.