Optimal Convergence Rate of Lie-Trotter Approximation for Quantum Thermal Averages
Xuda Ye, Zhennan Zhou
TL;DR
This work provides a rigorous trace-error analysis for the Lie–Trotter approximation of quantum thermal quantities with unbounded Hamiltonians, using a path-integral framework anchored by the Araki–Lieb–Thirring inequality. It proves an optimal $\mathcal{O}(1/N^2)$ convergence rate for the partition function $\mathcal{Z}$ and thermal averages in the periodic-potential setting, and a nearly optimal $\mathcal{O}((\log N+1)^{3/2}/N^2)$ rate for confining potentials on $\mathbb{R}$, via a domain-truncation strategy and Brownian-bridge second-moment bounds. The analysis translates into quantitative guarantees for common quantum simulation techniques such as PIMC and PIMD, strengthening their mathematical foundation. An appendix discusses an alternative splitting that can achieve $\mathcal{O}(1/N^2)$ in the confining case under suitable conditions.
Abstract
The Lie--Trotter product formula is a foundational approximation for the quantum partition function, yet obtaining rigorous error bounds for the unbounded Hamiltonians common in physics remains a significant challenge. This paper provides a quantitative error analysis for this approximation across two key systems. For a particle in a smooth, periodic potential, we establish an optimal convergence rate of $\mathcal O(1/N^2)$ for both the partition function and thermal averages, where $N$ is the number of imaginary time steps. We then extend this analysis to the more challenging case of a confining potential on $\mathbb R$, proving a nearly optimal rate of $\mathcal O((\log N+1)^{\frac32}/N^2)$. The derived error bounds provide a firm mathematical foundation for the high-order accuracy of path integral simulations in quantum statistical mechanics.