Learning conservation laws in unknown quantum dynamics

TL;DR

This paper introduces a learning algorithm that identifies conservation laws in unknown quantum systems by analyzing local observables, applicable to both closed and open quantum dynamics, with practical experimental implementation.

Contribution

The paper presents a novel algorithm combining classical shadows and data analysis to discover all conservation laws in unknown quantum dynamics with rigorous guarantees.

Findings

Successfully identifies conservation laws in simulated quantum systems.

Applicable to both closed and open quantum systems.

Demonstrated in models like $\

Abstract

We present a learning algorithm for discovering conservation laws given as sums of geometrically local observables in quantum dynamics. This includes conserved quantities that arise from local and global symmetries in closed and open quantum many-body systems. The algorithm combines the classical shadow formalism for estimating expectation values of observable and data analysis techniques based on singular value decompositions and robust polynomial interpolation to discover all such conservation laws in unknown quantum dynamics with rigorous performance guarantees. Our method can be directly realized in quantum experiments, which we illustrate with numerical simulations, using closed and open quantum system dynamics in a $\mathbb{Z}_2$-gauge theory and in many-body localized spin-chains.