Error-Correction Transitions in Finite-Depth Quantum Channels

We study error-correction protocols in which a quantum channel encodes logical information into an enlarged Hilbert space through a one-dimensional random circuit with local gates. We consider both noise acting after the encoding step and noise affecting the encoding circuit itself. Using the coherent information, we show that in both settings the infinite-depth limit is governed by a universal random-matrix transition at a critical noise rate, separating a recoverable phase from one where information is lost. We then characterize the leading finite-depth deviations from this universal regime. When noise acts only after a unitary encoder, convergence to perfect encoding is exponential in circuit depth, although boundary effects can delay saturation beyond the design time. When the encoder itself is noisy, the relevant control parameter becomes the circuit fidelity, which replaces the Hashing bound, and convergence is only polynomial in depth.