More on Intractability of Thermalization: (almost) i.i.d. inputs and finite lattices

TL;DR

This paper investigates the computational complexity of predicting long-term local observables in one-dimensional lattice systems, showing that even simplified or finite cases remain computationally intractable or undecidable.

Contribution

It extends previous results by proving undecidability for restricted initial states and classifies the complexity for finite lattices as EXPSPACE or PSPACE-complete.

Findings

Undecidability persists for initial states with a single differing site.

Finite lattice problems are either EXPSPACE-complete or PSPACE-complete.

Long-term observable prediction remains computationally hard in simplified models.

Abstract

This work is an extention of Shiraishi and Matsumoto [10], and discusses the computational complexity of the long-term average of local observables in one-dimensional lattices with shift-invariant nearest-neighbor interactions for simple initial states. As shown in the previous paper, the problem is generally intractable. In this paper we refine the statement further. First, we consider restriction of the initial state, where the state of all the sites are the same except for a single site. We show this version of the problem is also undecidable (RE-complete). Then we turn to the case where the lattice size is finite: depening on the defitiniton of the input size, this version of problem is either EXPSPACE-complete or PSPACE-complete.