Work

Uniform Semiclassical Observable Error Bound of Trotter–Suzuki Splitting: A Simple Algebraic Proof

Year: 2026

Type: article

Abstract: . Efficient simulation of the semiclassical Schrödinger equation has garnered significant attention in the numerical analysis community. While controlling the error in the unitary evolution or the wavefunction typically requires the time step size to shrink as the semiclassical parameter [Formula: see text] decreases, it has been observed–and proved for first- and second-order Trotterization schemes–that the error in certain classes of observables admits a time step size independent of [Formula: see text]. In this work, we explicitly characterize this class of observables and present a new, simple algebraic proof of uniform-in-[Formula: see text] error bounds for arbitrarily high-order Trotterization schemes. Our proof relies solely on the algebraic structure of the underlying operators in both the continuous and discrete settings. Unlike previous analyses, it avoids Egorov-type theorems and bypasses heavy semiclassical machinery. To the best of our knowledge, this is the first proof of uniform-in-[Formula: see text] observable error bounds for Trotterization in the semiclassical regime that relies only on algebraic structure, without invoking the semiclassical limit.

Source: SIAM Journal on Numerical Analysis

DOI: 10.1137/25m1777098

Authors: Di Fang, Conrad Qu

Institution: Duke University

Language: English

Topic: Numerical methods for differential equations

Subfield: Numerical Analysis

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, U.S. Department of Energy, Office of Science

Awards: Towards Quantum Speedup for Solving High-Dimensional Partial Differential Equations, QUANTUM UTILITY THROUGH ADVANCED COMPUTATIONAL QUANTUM ALGORITHM (QUACQ), CAREER: Quantum Simulation with Unbounded Operators: Algorithms, Analysis, and Applications, G697302361