Error-Correction Transitions in Finite-Depth Quantum Channels

TL;DR

This paper investigates phase transitions in error correction for quantum channels modeled by local random circuits, revealing universal behavior governed by random matrix theory and characterizing finite-depth deviations.

Contribution

It introduces a universal phase transition framework for quantum error correction in local random circuits and analyzes finite-depth effects beyond the infinite limit.

Findings

Universal phase transition at a critical noise rate predicted by random matrix theory

Error correction phase preserves information, irretrievable phase leads to information loss

Finite-depth deviations from universality depend on noise location and circuit depth

Abstract

We study error correction type protocols in which a quantum channel encodes logical information into an enlarged Hilbert space. Specifically, we consider channels realized by one dimensional random noisy quantum circuits with spatially local interaction gates. We analyze both noise acting after the encoding and noise affecting the encoding circuit itself. Using the coherent information as a metric, we show that in both cases the infinite depth limit is governed by random matrix theory, which predicts a universal phase transition at a critical noise rate. This critical point separates an error correcting phase, in which encoded information is preserved, from a phase in which it is irretrievably lost. Going beyond the infinite depth limit, we characterize the systematic finite depth deviations from random matrix universality. In particular, we show that these deviations behave parametrically differently depending on whether the noise acts after the encoding or also affects the encoding itself. For noiseless encoders, the approach is exponential in circuit depth, although boundary effects can delay perfect encoding relative to the circuit design time. For noisy encoders, we find that the circuit fidelity effectively replaces the Hashing bound, and perfect encoding is approached polynomially with depth.