Measuring Trotter error and its application to precision-guaranteed Hamiltonian simulations

TL;DR

This paper introduces a novel method for accurately measuring Trotter error in Hamiltonian simulations on quantum computers without ancillary qubits, enabling precision guarantees and adaptive step size selection for more efficient quantum simulations.

Contribution

The authors develop Trotter( m,n ), an algorithm that guarantees Trotter error within a preset tolerance and adaptively chooses step sizes, applicable to both time-independent and dependent Hamiltonians.

Findings

Trotter( m,n ) achieves error within specified tolerance.

Adaptive step size is about ten times larger than traditional bounds.

Method reduces circuit depth while maintaining accuracy.

Abstract

Trotterization is the most common and convenient approximation method for Hamiltonian simulations on digital quantum computers, but estimating its error accurately is computationally difficult for large quantum systems. Here, we develop a method for measuring the Trotter error without ancillary qubits on quantum circuits by combining the $m$th- and $n$th-order ($m<n$) Trotterizations rather than consulting with mathematical error bounds. Using this method, we make Trotterization precision-guaranteed, developing an algorithm named Trotter$(m,n)$, in which the Trotter error at each time step is within an error tolerance $\epsilon$ preset for our purpose. Trotter$(m,n)$ is applicable to both time- independent and dependent Hamiltonians, and it adaptively chooses almost the largest stepsize $\mathrm{d}t$, which keeps quantum circuits shallowest within the error tolerance. Benchmarking it in a quantum spin chain, we find the adaptively chosen $\mathrm{d}t$ to be about ten times larger than that inferred from known upper bounds of Trotter errors.