Discrete Superconvergence Analysis for Quantum Magnus Algorithms of Unbounded Hamiltonian Simulation

TL;DR

This paper establishes a novel superconvergence estimate for quantum Magnus algorithms applied to unbounded Hamiltonian simulation in a fully discrete setting, using a semiclassical framework and two-parameter symbol class analysis.

Contribution

It provides the first discrete superconvergence analysis for quantum Magnus algorithms with uniform error bounds, extending previous continuous setting results.

Findings

First superconvergence estimate in fully discrete setting

Error bounds are uniform in the number of spatial discretization points

Introduces a semiclassical framework with two-parameter symbol class analysis

Abstract

Motivated by various applications, unbounded Hamiltonian simulation has recently garnered great attention. Quantum Magnus algorithms, designed to achieve commutator scaling for time-dependent Hamiltonian simulation, have been found to be particularly efficient for such applications. When applied to unbounded Hamiltonian simulation in the interaction picture, they exhibit an unexpected superconvergence phenomenon. However, existing proofs are limited to the spatially continuous setting and do not extend to discrete spatial discretizations. In this work, we provide the first superconvergence estimate in the fully discrete setting with a finite number of spatial discretization points $N$, and show that it holds with an error constant uniform in $N$. The proof is based on the two-parameter symbol class, which, to our knowledge, is applied for the first time in algorithm analysis. The key idea is to establish a semiclassical framework by identifying two parameters through the discretization number and the time step size rescaled by the operator norm, such that the semiclassical uniformity guarantees the uniformity of both. This approach may have broader applications in numerical analysis beyond the specific context of this work.