Whether or not a physical system will thermalize from an initial state has been a key question in modern 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 determining thermalization is contained in PSPACE and is PSPACE-hard under quantum polynomial time reductions. Our results build off recent work by [Shiraishi & Matsumoto, Nature Communications, 12, 5084 (2021)] demonstrating undecidability of thermalization in the thermodynamic limit, and shows that the intractability of the problem is due to inherent difficulties in many-body physics rather than particularities of infinite systems.
Pizza and drinks will be served after the seminar in ATL 2117.
