A matching decoder for bivariate bicycle codes

The discovery of new quantum error-correcting codes that encode several logical qubits into relatively few physical qubits motivates the development of efficient and accurate methods of decoding these systems. Here, we adopt the minimum-weight perfect matching algorithm, a subroutine invaluable to decoding topological codes, to decode bivariate bicycle codes. Using the equivalence of bivariate bicycle codes to copies of the toric code, we propose a method we call the `cylinder trick' to rapidly find a correction using matching on code symmetries. We benchmark our decoder on the gross code family, cyclic hypergraph-product codes, generalized toric codes, and recently proposed directional codes under code capacity and phenomenological noise models, demonstrating the general applicability of our protocol. For a subset of these codes, we find that our decoder can be significantly improved by augmenting matching with strategies including belief propagation and `over-matching', thus achieving performance competitive with state-of-the-art approaches.