Discrete Superconvergence Analysis for Quantum Magnus Algorithms of Unbounded Hamiltonian Simulation

TL;DR

The paper tackles the challenge of simulating unbounded Hamiltonians on discrete spatial grids using quantum Magnus algorithms. It introduces a two-parameter semiclassical framework and discrete microlocal analysis to achieve uniform-in-$N$ fourth-order (superconvergence) error bounds for the second-order Magnus expansion in the interaction picture, even under finite-difference discretization. A pivotal advance is the two-parameter symbol calculus, which overcomes the breakdown of Egorov’s theorem in the discrete setting and yields $N$-uniform commutator bounds, leading to polylogarithmic quantum algorithm complexity in the discretization size. The results significantly extend the applicability of Magnus-based quantum simulation to fully discrete settings and propose a framework potentially extensible to higher-order Magnus schemes and other discretizations.

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.

Theorems & Definitions (16)

• Lemma 1: Derivative of Matrix Exponential without time ordering
• Theorem 2: Local Truncation Error via Exact Error Representation
• Theorem 3: Long-time Error for Hamiltonian Simulation
• proof
• Theorem 4
• Lemma 5: Lemma 9 in BornsWeilFang2022
• Theorem 6
• Lemma 7: Long-time complexity of the second-order quantum Magnus algorithm
• Lemma 8
• proof