Spacetime Markov length: a diagnostic for fault tolerance via mixed-state phases

TL;DR

This paper introduces the spacetime Markov length, a new diagnostic tool based on mixed-state phases, to assess fault-tolerance in quantum error correction codes, independent of decoding procedures.

Contribution

It establishes a novel connection between fault-tolerance in stabilizer codes and mixed-state phases, proposing a decoder-independent diagnostic called the spacetime Markov length.

Findings

The divergence of the spacetime Markov length signals fault-tolerance breakdown.

Decoherence can reveal phase transitions in symmetry-protected topological phases.

The diagnostic is applicable to both incoherent and coherent perturbations.

Abstract

We establish a correspondence between the fault-tolerance of local stabilizer codes experiencing measurement and physical errors and the mixed-state phases of decohered resource states in one higher dimension. Drawing from recent developments in mixed-state phases of matter, this motivates a diagnostic of fault-tolerance, which we refer to as the spacetime Markov length. This is a length scale determined by the decay of the (classical) conditional mutual information of repeated syndrome measurement outcomes in spacetime. The diagnostic is independent of the decoder, and its divergence signals the intrinsic breakdown of fault tolerance. As a byproduct, we find that decoherence may be useful for exposing transitions from higher-form symmetry-protected topological phases driven by both incoherent and coherent perturbations.