Uniform observable error bounds of Trotter formulae for the semiclassical Schrödinger equation

by   Yonah Borns-Weil, et al.

By no fast-forwarding theorem, the simulation time for the Hamiltonian evolution needs to be O(H t), 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 Schrödinger 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 ≪ 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 Schrödinger 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 Schrödinger 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.


page 1

page 2

page 3

page 4


Observable Error Bounds of the Time-splitting Scheme for Quantum-Classical Molecular Dynamics

Quantum-classical molecular dynamics, as a partial classical limit of th...

Time-dependent unbounded Hamiltonian simulation with vector norm scaling

The accuracy of quantum dynamics simulation is usually measured by the e...

Time-dependent Hamiltonian simulation with L^1-norm scaling

The difficulty of simulating quantum dynamics depends on the norm of the...

Time-dependent Hamiltonian Simulation of Highly Oscillatory Dynamics

We propose a simple quantum algorithm for simulating highly oscillatory ...

Probability and consequences of living inside a computer simulation

It is shown that under reasonable assumptions a Drake-style equation can...

Convergence, error analysis and longtime behavior of the Scalar Auxiliary Variable method for the nonlinear Schrödinger equation

We carry out the convergence analysis of the Scalar Auxiliary Variable (...

Practical Black Box Hamiltonian Learning

We study the problem of learning the parameters for the Hamiltonian of a...

Please sign up or login with your details

Forgot password? Click here to reset