Dynamical error bounds for continuum discretisation via Gauss quadrature rules, -- a Lieb-Robinson bound approach

TL;DR

This paper develops analytical error bounds for discretizing continuum quantum systems using Gauss quadrature, leveraging Lieb-Robinson bounds to quantify approximation accuracy over time.

Contribution

It introduces a novel approach combining Lieb-Robinson bounds and orthogonal polynomial theory to derive error estimates for continuum discretization.

Findings

Error bounds depend on time and number of modes

Gauss quadrature rules optimize mode selection

Bounds are applicable to a wide class of quantum systems

Abstract

Instances of discrete quantum systems coupled to a continuum of oscillators are ubiquitous in physics. Often the continua are approximated by a discrete set of modes. We derive analytical error bounds on expectation values of system observables that have been time evolved under such discretised Hamiltonians. These bounds take on the form of a function of time and the number of discrete modes, where the discrete modes are chosen according to Gauss quadrature rules. The derivation makes use of tools from the field of Lieb-Robinson bounds and the theory of orthonormal polynominals.