Optimal Convergence Rate of Lie-Trotter Approximation for Quantum Thermal Averages

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 challenge. This paper provides a quantitative error analysis for this approximation across two 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 $\mathcal{O}((\log N+1{)}^{1.5}/N^2)$. The derived error bounds provide a firm mathematical foundation for the second-order accuracy of path integral simulations in quantum statistical mechanics.

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