Statistical mechanics of quantum error correcting codes
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Abstract
We study stabilizer quantum error correcting codes (QECC) generated under\nhybrid dynamics of local Clifford unitaries and local Pauli measurements in one\ndimension. Building upon 1) a general formula relating the error-susceptibility\nof a subregion to its entanglement properties, and 2) a previously established\nmapping between entanglement entropies and domain wall free energies of an\nunderlying spin model, we propose a statistical mechanical description of the\nQECC in terms of "entanglement domain walls". Free energies of such domain\nwalls generically feature a leading volume law term coming from its "surface\nenergy", and a sub-volume law correction coming from thermodynamic entropies of\nits transverse fluctuations. These are most easily accounted for by\ncapillary-wave theory of liquid-gas interfaces, which we use as an illustrative\ntool. We show that the information-theoretic decoupling criterion corresponds\nto a geometric decoupling of domain walls, which further leads to the\nidentification of the "contiguous code distance" of the QECC as the crossover\nlength scale at which the energy and entropy of the domain wall are comparable.\nThe contiguous code distance thus diverges with the system size as the\nsubleading entropic term of the free energy, protecting a finite code rate\nagainst local undetectable errors. We support these correspondences with\nnumerical evidence, where we find capillary-wave theory describes many\nqualitative features of the QECC; we also discuss when and why it fails to do\nso.\n