The Complexity of Thermalization in Finite Quantum Systems

TL;DR

This work shows that predicting relaxation and thermalization in finite quantum systems is computationally PSPACE-hard in general, and that even for simple, translationally invariant 1D Hamiltonians the long-time behavior of local observables can encode PSPACE-complete problems. The authors establish PSPACE-hardness via a TM-to-Hamiltonian construction that links the halting of a universal reversible Turing machine to the time-averaged observable, and then prove PSPACE containment using block-encoding-based quantum algorithms and phase estimation. They further prove FSTherm(MC) lies in PSPACE and, under reductions, prove FSTherm(Gibbs) is PSPACE-hard and thus PSPACE-complete; a separate line shows FTFSRelax is BQP-complete, connecting finite-time relaxation to quantum computational power. Together, these results illustrate inherent intractability in the mechanisms of relaxation, suggesting that ETH-like sufficient conditions cannot be universally checked efficiently, and that finite-size quantum dynamics may resist general algorithmic prediction for relaxation and thermalization.

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.

Figures (2)

• Figure 1: The relaxation decision problem. For a given initial state $\ket{\psi}$, we plot the time-average of $\mathcal{A}_N \coloneqq \frac{1}{T}\int_0^Tdt \bra{\psi_0}e^{iHt}\mathcal{A}_N e^{-iHt}\ket{\psi_0}$. $A^*$ is indicated by the dashed green line. Given the promise that $\bar{\mathcal{A}}_N\coloneqq\lim_{T\rightarrow\infty} \langle \mathcal{A}_N \rangle_T$ is such that either $|\bar{\mathcal{A}}_N - A^*| \geq c\epsilon$ or $|\bar{\mathcal{A}}_N - A^*| \leq \epsilon, \mathsf{FSRelax}$ is the problem of deciding which of these two holds.
• Figure 2: The solid green and dashed-green lines represent the time-averaged expectation values in the halting and non-halting cases respectively, as $p$ is tuned. The red line represents the expectation values with respect to the microcanonical ensemble as $p$ is tuned.

Theorems & Definitions (71)

• Definition 1.1: (Informal) Finite-Size Relaxation, $\mathsf{FSRelax}$
• Definition 1.2: Microcanonical Ensemble
• Definition 1.3: (Informal) Finite-Size Thermalization to the microcanonical state, $\mathsf{FSTherm (MC)}$
• Definition 1.4: (Informal) Finite-Size Thermalization to the Gibbs state, $\mathsf{FSTherm (Gibbs)}$
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Definition 2.1: $\mathsf{PSPACE}$
• Definition 2.2: $\mathsf{BQPSPACE}$,watrous2008quantum
• Definition 2.3: $\mathsf{PrQSPACE}$ watrous2008quantum