Exponentially Reduced Circuit Depths Using Trotter Error Mitigation

TL;DR

This paper presents an improved analysis of Trotter error mitigation techniques, demonstrating exponential reductions in circuit depth for quantum simulations using Richardson extrapolation and polynomial interpolation.

Contribution

It provides a rigorous analysis showing that error mitigation techniques can exponentially reduce circuit depth in quantum simulation, improving upon previous bounds.

Findings

Circuit depths of O(T^{1+1/p} polylog(1/ε)) are sufficient for error ε.

Achieves commutator scaling and better complexity with T.

Does not require fractional Trotter step implementations.

Abstract

Product formulae are a popular class of digital quantum simulation algorithms due to their conceptual simplicity, low overhead, and performance which often exceeds theoretical expectations. Recently, Richardson extrapolation and polynomial interpolation have been proposed to mitigate the Trotter error incurred by use of these formulae. This work provides an improved, rigorous analysis of these techniques for the task of calculating time-evolved expectation values. We demonstrate that, to achieve error $\epsilon$ in a simulation of time $T$ using a $p^\text{th}$-order product formula with extrapolation, circuits depths of $O\left(T^{1+1/p} \textrm{polylog}(1/\epsilon)\right)$ are sufficient -- an exponential improvement in the precision over product formulae alone. Furthermore, we achieve commutator scaling, improve the complexity with $T$, and do not require fractional implementations of Trotter steps. Our results provide a more accurate characterisation of the algorithmic error mitigation techniques currently proposed to reduce Trotter error.