Time-dependent Hamiltonian Simulation of Highly Oscillatory Dynamics and Superconvergence for Schr\"odinger Equation

TL;DR

This paper introduces a simple, robust quantum algorithm for simulating highly oscillatory, time-dependent quantum dynamics that achieves superconvergence, with potential for efficient Schrödinger equation simulation.

Contribution

It presents the first quantum algorithm insensitive to rapid Hamiltonian changes, exhibiting commutator scaling and superconvergence for Schrödinger equation simulation.

Findings

Achieves second order convergence rate for Schrödinger equation

Demonstrates superconvergence through numerical results

Operates without complex quantum control logic

Abstract

We propose a simple quantum algorithm for simulating highly oscillatory quantum dynamics, which does not require complicated quantum control logic for handling time-ordering operators. To our knowledge, this is the first quantum algorithm that is both insensitive to the rapid changes of the time-dependent Hamiltonian and exhibits commutator scaling. Our method can be used for efficient Hamiltonian simulation in the interaction picture. In particular, we demonstrate that for the simulation of the Schr\"odinger equation, our method exhibits superconvergence and achieves a surprising second order convergence rate, of which the proof rests on a careful application of pseudo-differential calculus. Numerical results verify the effectiveness and the superconvergence property of our method.