Topological Order and Memory Time in Marginally Self-Correcting Quantum Memory

TL;DR

This paper investigates two quantum memory proposals, proving they lack topological order at non-zero temperatures and establishing that the welded code's memory time grows doubly exponentially with inverse temperature.

Contribution

It provides a rigorous proof of the absence of topological order in these codes at finite temperature and introduces a framework linking thermal order and self-correction.

Findings

No topological order above zero temperature in the studied codes

Memory time in the welded code is doubly exponential in inverse temperature

Framework for reducing quantum memory problems to classical ones

Abstract

We examine two proposals for marginally self-correcting quantum memory, the cubic code by Haah and the welded code by Michnicki. In particular, we prove explicitly that they are absent of topological order above zero temperature, as their Gibbs ensembles can be prepared via a short-depth quantum circuit from classical ensembles. Our proof technique naturally gives rise to the notion of free energy associated with excitations. Further, we develop a framework for an ergodic decomposition of Davies generators in CSS codes which enables formal reduction to simpler classical memory problems. We then show that memory time in the welded code is doubly exponential in inverse temperature via the Peierls argument. These results introduce further connections between thermal topological order and self-correction from the viewpoint of free energy and quantum circuit depth.