Statistical Mechanics of Quantum Error-Correcting Codes

We study stabilizer quantum error-correcting codes (QECC) generated under hybrid dynamics of local Clifford unitaries and local Pauli measurements in one dimension. Building upon a general formula relating the error-susceptibility of a subregion to its entanglement properties, and a previously established mapping between entanglement entropies and domain wall free energies of an underlying spin model, we propose a statistical mechanical description of the QECC in terms of "entanglement domain walls". Such domain walls are most easily accounted for by capillary-wave theory of liquid-gas interfaces, which we use as an illustrative tool. We show that the information-theoretic decoupling criterion corresponds to a geometric decoupling of domain walls when transverse fluctuations dominate over the surface tension. It follows that the "contiguous code distance" diverges with the system size, and a finite code rate is protected against local undetectable errors. We support these correspondences with numerical evidences, where we find capillary-wave theory describes many qualitative features of the QECC; we also discuss when and why it fails to do so.