Zu Chongzhi Research Seminar

Date and Time (China standard time): Tuesday, June 9, 10:30 am – 11:30 am

Location: WDR 1007

Zoom: 931 0023 4315, Passcode: dkumath

Title: Discrete Superconvergence Analysis for Quantum Magnus Algorithms of Unbounded Hamiltonian Simulation

Speaker: Jiaqi Zhang

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.

Bio: Jiaqi Zhang is a second-year Ph.D. student in the Department of Mathematics at Duke University. Her current research interests lie in applied and numerical analysis of differential equations, quantum algorithms, and the mathematical analysis of quantum systems.