Uniform observable error bounds of Trotter formulae for the semiclassical Schr\"odinger equation

TL;DR

This paper establishes uniform observable error bounds for Trotter formulae in semiclassical Schrödinger equations, demonstrating that certain observable simulations can be performed efficiently at classical timescales in the semiclassical regime.

Contribution

It provides the first uniform-in-h observable error bounds for semiclassical Schrödinger equations without reducing the convergence order, leveraging multiscale analysis techniques.

Findings

Number of Trotter steps can be O(1) for certain observables in semiclassical regime

Uniform observable error bounds are achieved without sacrificing convergence order

Simulation time for some observables is comparable to classical scale

Abstract

Known as no fast-forwarding theorem in quantum computing, the simulation time for the Hamiltonian evolution needs to be $O(\|H\| t)$ in the worst case, which essentially states that one can not go across the multiple scales as the simulation time for the Hamiltonian evolution needs to be strictly greater than the physical time. We demonstrated in the context of the semiclassical Schr\"odinger equation that the computational cost for a class of observables can be much lower than the state-of-the-art bounds. In the semiclassical regime (the effective Planck constant $h \ll 1$), the operator norm of the Hamiltonian is $O(h^{-1})$. We show that the number of Trotter steps used for the observable evolution can be $O(1)$, that is, to simulate some observables of the Schr\"odinger equation on a quantum scale only takes the simulation time comparable to the classical scale. In terms of error analysis, we improve the additive observable error bounds [Lasser-Lubich 2020] to uniform-in-$h$ observable error bounds. This is, to our knowledge, the first uniform observable error bound for semiclassical Schr\"odinger equation without sacrificing the convergence order of the numerical method. Based on semiclassical calculus and discrete microlocal analysis, our result showcases the potential improvements taking advantage of multiscale properties, such as the smallness of the effective Planck constant, of the underlying dynamics and sheds light on going across the scale for quantum dynamics simulation.