SYK thermal expectations are classically easy at any temperature

TL;DR

This paper presents a classical algorithm for approximating thermal expectations in local Hamiltonians, including the SYK model at any temperature, highlighting a classically easy high-temperature phase.

Contribution

It introduces a simple classical algorithm with quasi-polynomial complexity for thermal expectations, applicable to models like SYK with no phase transition.

Findings

The algorithm works efficiently above a phase transition in free energy.

It applies to the SYK model at all constant temperatures.

A universal high-temperature phase is established for local Hamiltonians.

Abstract

Estimating thermal expectations of local observables is a natural target for quantum advantage. We give a simple classical algorithm that approximates thermal expectations for Gibbs states of local Hamiltonians, and we show it has quasi-polynomial cost $n^{O(\log (n/\epsilon))}$ for all temperatures above a phase transition in the free energy. For many natural models, this coincides with the entire fast-mixing, quantumly easy phase. Our results apply to the Sachdev-Ye-Kitaev (SYK) model at any constant temperature due to its absence of a phase transition -- despite its entanglement, sign problem, and polynomial quantum circuit lower bounds. Beyond SYK, we rigorously establish a universal classically easy high-temperature phase for all local, bounded-degree Hamiltonians and show that it extends to temperatures strictly colder than the death of entanglement transition.