The Complexity of Thermalization in Finite Quantum Systems

TL;DR

This paper proves that determining whether a finite quantum system thermalizes or relaxes to a specific stationary value is computationally intractable, specifically PSPACE-complete, highlighting fundamental complexity barriers in many-body quantum physics.

Contribution

It establishes the PSPACE-completeness of thermalization decision problems for finite quantum systems, revealing inherent computational complexity in understanding quantum thermalization.

Findings

Thermalization decision problems are PSPACE-complete.

Determining relaxation to a stationary value is computationally intractable.

Intractability persists even for finite-sized quantum systems.

Abstract

Thermalization is the process through which a physical system evolves toward a state of thermal equilibrium. Determining whether or not a physical system will thermalize from an initial state has been a key question in condensed matter physics. Closely related questions are determining whether observables in these systems relax to stationary values, and what those values are. Using tools from computational complexity theory, we demonstrate that given a Hamiltonian on a finite-sized system, determining whether or not it thermalizes or relaxes to a given stationary value is computationally intractable, even for a quantum computer. In particular, we show that the problem of determining whether an observable of a finite-sized quantum system relaxes to a given value is PSPACE-complete, and so no efficient algorithm for determining the value is expected to exist. Further, we show the existence of Hamiltonians for which the problem of determining whether the system thermalizes to the Gibbs expectation value is PSPACE-complete. We also show that the related problem of determining whether the system thermalizes to the microcanonical expectation value is contained in PSPACE and is PSPACE-hard under quantum polynomial time reductions. In light of recent results demonstrating undecidability of thermalization in the thermodynamic limit, our work shows that the intractability of the problem is due to inherent difficulties in many-body physics rather than particularities of infinite systems.