Efficient decoding for the random matching code ensemble

Design an efficient decoder for the stabilizer codes generated by the shallow random matching circuits with locally twirled two-qubit Clifford gates, including the Clifford-twirled CNOT ensemble.

Background

The paper establishes that shallow random matching circuits can generate high-distance stabilizer codes attaining the quantum Gilbert–Varshamov rate–distance tradeoff, but its analysis concerns only encoding complexity and code distance. It does not provide an algorithm for decoding the resulting codes efficiently. Developing such a decoder is explicitly identified as an open problem.