Effective dynamics from minimising dissipation

TL;DR

This paper develops a method to find the best unitary approximation for large-scale quantum dynamics by minimizing dissipation, especially when only partial system information is accessible, with applications to quantum walks and Dirac equation limits.

Contribution

It introduces a principle of minimal dissipation to derive effective IR dynamics from discretely evolving quantum systems, including a mean-field approximation for weak IR-UV coupling.

Findings

Optimal unitary is a mean-field dynamics when IR and UV are weakly coupled.

Error in approximation is quantified by energy variances.

Application to quantum walk shows IR dynamics as a mass redefinition.

Abstract

It is known that the same physical system can be described by different effective theories depending on the scale at which it is observed. In this work, we formulate a prescription for finding the unitary that best approximates the large scale dynamics of a quantum system evolving discretely in time, as it is the case for digital quantum simulators. We consider the situation in which the degrees of freedom of the system can be divided between an IR part that we can observe, and a UV part that we cannot observe. Following a principle of minimal dissipation, our goal is to find the unitary dynamics that best approximates the (generally non unitary) time evolution of the IR degrees of freedom. We first prove that when the IR and UV degrees of freedom are weakly coupled, the unitary that maximises the fidelity is given by a mean-field dynamics and the error is given by a sum of energy variances. We then apply our results to a one dimensional quantum walk, which is known to reproduce the Dirac equation in the small mass and momenta limit. We find that in this limit the effective IR dynamics is obtained by a mass redefinition.