Trotter product formulae for $*$-automorphisms of quantum lattice systems

TL;DR

This paper develops efficient product formula approximations for the dynamics of infinite quantum lattice systems generated by local interactions, with error bounds that improve with the number of product terms.

Contribution

It introduces a method to approximate quantum lattice dynamics using Trotter-like product formulas with controllable error bounds.

Findings

Error scales as n^{-m} for any integer m

Approximations hold in norm for well-behaved algebra elements

Efficient decomposition of dynamics into local automorphisms

Abstract

We consider the dynamics $t\mapsto\tau_t$ of an infinite quantum lattice system that is generated by a local interaction. If the interaction decomposes into a finite number of terms that are themselves local interactions, we show that $\tau_t$ can be efficiently approximated by a product of $n$ automorphisms, each of them being an alternating product generated by the individual terms. For any integer $m$, we construct a product formula (in the spirit of Trotter) such that the approximation error scales as $n^{-m}$. Our bounds hold in norm, pointwise for algebra elements that are sufficiently well approximated by finite volume observables.