Computational complexity of non-equilibrium steady states of quantum spin chains

TL;DR

This paper investigates the computational complexity of non-equilibrium steady states in quantum spin chains, revealing that estimating local operator expectations in these states is as hard as, or harder than, quantum computation.

Contribution

It demonstrates that calculating local expectations in NESS of XX spin chains encoded as MPOs is computationally very hard, linking quantum many-body physics to complexity theory.

Findings

Estimations are as hard as quantum computational problems.

NESS of XX chains can encode complex quantum algorithms.

Hardness results suggest limitations for classical simulation of these states.

Abstract

We study non-equilibrium steady states (NESS) of spin chains with boundary Markovian dissipation from the computational complexity point of view. We focus on XX chains whose NESS are matrix product operators (MPO), i.e. with coefficients of a tensor operator basis described by transition amplitudes in an auxiliary space. Encoding quantum algorithms in the auxiliary space, we show that estimating expectations of operators, being local in the sense that each acts on disjoint sets of few spins covering all the system, provides the answers of problems at least as hard as, and believed by many computer scientists to be much harder than, those solved by quantum computers. We draw conclusions on the hardness of the above estimations.