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.

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• Exotic phases of matter
• Measurement-based quantum computing
• Open quantum systems & decoherence
• Quantum algorithms & computation
• Quantum computation
• Quantum error correction
• Quantum information theory
• Quantum phase transitions
• Quasiparticles & collective excitations
• Topological quantum computing