Slow convergence of Trotter decomposition for rotations

TL;DR

This paper investigates the convergence behavior of the Trotter approximation for orbital angular momentum operators, revealing that the error scales as 1/n for certain states but can be arbitrarily slow for others.

Contribution

It provides a detailed analysis of the state-dependent convergence rates of Trotter decomposition for angular momentum operators, highlighting conditions for slow convergence.

Findings

Trotter error scales as 1/n for states in the domains of the operators.

Convergence can be arbitrarily slow for states outside these domains.

The study clarifies the limitations of Trotter approximation in quantum simulations.

Abstract

We study the Trotter approximation for a pair of orbital angular momentum operators, $L_x$ and $L_y$. In particular, we investigate the scaling behavior of the state-dependent Trotter error. We show that for states in the domains of the orbital angular momentum operators the Trotter error scales as $n^{-1}$, where $n$ is the time discretization. Instead, the convergence rate can be arbitrarily slow for states that do not belong to the domains of all three angular momentum operators simultaneously.