Trotter error time scaling separation via commutant decomposition

TL;DR

This paper introduces a commutant decomposition framework to better estimate Trotter errors in quantum simulation, revealing two error components with distinct time-scaling behaviors, and improves error bounds for higher-order formulas.

Contribution

The paper presents a novel commutant decomposition method that separates Trotter error components with different scaling, enhancing error estimates for quantum simulation.

Findings

Identifies two error components scaling as $O( au^pt)$ and $O( au^p)$

Provides improved bounds for higher-order product formulas

Demonstrates analytical and numerical validation of the new estimates

Abstract

Suppressing the Trotter error in dynamical quantum simulation typically requires running deeper circuits, posing a great challenge for noisy near-term quantum devices. Studies have shown that the empirical error is usually much smaller than the one suggested by existing bounds, implying the actual circuit cost required is much less than the ones based on those bounds. Here, we improve the estimate of the Trotter error over existing bounds, by introducing a general framework of commutant decomposition that separates disjoint error components that have fundamentally different scaling with time. In particular we identify two error components that each scale as $O(\tau^pt)$ and $O(\tau^p)$ for a $p$th-order product formula evolving to time $t$ using a fixed step size $\tau$, it implies one would scale linearly with time $t$ and the other would be constant of $t$. We show that this formalism not only straightforwardly reproduces previous results but also provides a better error estimate for higher-order product formulas. We demonstrate the improvement both analytically and numerically. We also apply the analysis to observable error relating to the heating in Floquet dynamics and thermalization, which is of independent interest.