Understanding Stabilizer Codes Under Local Decoherence Through a General Statistical Mechanics Mapping

TL;DR

This paper maps the effects of local decoherence on stabilizer codes to classical statistical mechanics models, providing new insights into their information capacity and decoding thresholds.

Contribution

It introduces a novel mapping from stabilizer code decoherence to statistical mechanics, enabling analysis of phase transitions and thresholds in quantum error correction.

Findings

Mapping relates quantum information measures to thermodynamic quantities

Bounds on decoding thresholds for 3D toric and X-cube models

Classical models can be gauged to recover the original quantum codes

Abstract

We consider the problem of a generic stabilizer Hamiltonian under local, incoherent Pauli errors. Using two different approaches -- (i) Haah's polynomial formalism arXiv:1204.1063 and (ii) the homological perspective on CSS codes -- we construct a mapping from the $n$th moment of the decohered ground state density matrix to a classical statistical mechanics model. We demonstrate that various measures of information capacity -- (i) quantum relative entropy, (ii) coherent information, and (iii) entanglement negativity -- map to thermodynamic quantities in the statistical mechanics model and can be used to characterize the decoding phase transition. As examples, we analyze the 3D toric code and X-cube model, deriving bounds on their optimal decoding thresholds and gaining insight into their information properties under decoherence. Additionally, we demonstrate that the SM mapping acts an an "ungauging" map; the classical models that describe a given code under decoherence also can be gauged to obtain the same code. Finally, we comment on correlated errors and non-CSS stabilizer codes.