Time-dependent Hamiltonian Simulation via Magnus Expansion: Algorithm and Superconvergence

TL;DR

This paper presents a new time-dependent Hamiltonian simulation algorithm using Magnus expansion that achieves superconvergence, improving accuracy and efficiency especially for oscillatory systems, with theoretical proof and practical implications.

Contribution

Introduces a Magnus expansion-based Hamiltonian simulation algorithm with superconvergence properties, extending previous methods and providing rigorous semiclassical analysis.

Findings

Second-order algorithm exhibits fourth-order superconvergence.

Error preconstant is independent of spatial grid size.

Algorithm has commutator scaling with weak dependence on derivatives.

Abstract

Hamiltonian simulation becomes more challenging as the underlying unitary becomes more oscillatory. In such cases, an algorithm with commutator scaling and a weak dependence, such as logarithmic, on the derivatives of the Hamiltonian is desired. We introduce a new time-dependent Hamiltonian simulation algorithm based on the Magnus series expansion that exhibits both features. Importantly, when applied to unbounded Hamiltonian simulation in the interaction picture, we prove that the commutator in the second-order algorithm leads to a surprising fourth-order superconvergence, with an error preconstant independent of the number of spatial grids. This extends the qHOP algorithm [An, Fang, Lin, Quantum 2022] based on first-order Magnus expansion, and the proof of superconvergence is based on semiclassical analysis that is of independent interest.