The toric code corrects errors below a threshold error rate. Above the threshold, errors proliferate faster than correction can remove them, and the encoded information is lost. The threshold is sharp — it's a phase transition in the statistical mechanics of the error model. Below threshold: ordered phase, information preserved. Above: disordered phase, information destroyed.
For independent errors, the threshold is known. For correlated errors — where one error makes nearby errors more likely — the threshold shifts. The paper establishes intrinsic error thresholds for nearly critical toric codes, where the system operates close to the phase transition between correctable and uncorrectable regimes.
Near criticality, the error correction problem develops the same scaling properties as a statistical mechanical phase transition. The correlation length of errors diverges, error clusters span the system, and the threshold becomes sensitive to the precise error model in ways that the far-from-critical regime conceals. The intrinsic threshold — computed from the code's own structure rather than a specific decoder — captures the fundamental limit.
The result quantifies the cost of operating near the edge. Engineered systems push toward minimal overhead, which means operating near the threshold where fewer physical qubits protect each logical qubit. But proximity to the threshold amplifies the effect of correlations, model uncertainty, and finite-size fluctuations. The threshold is not just a number but a critical point, and critical points are where universal behavior coexists with maximum sensitivity. The code works best far from the edge, but efficiency demands operating close to it. The tension is irreducible.