Maximum-likelihood decoding for the honeycomb code under symmetric noise

The honeycomb code is an example of a Floquet code, a quantum error-correcting code that is dynamically generated by a periodic sequence of repeated measurements. This approach to error correction yields surprising phenomena, such as logical operators that have no counterpart in the associated subsystem code and evolve in time with subsequent measurement rounds. We are thus motivated to find a general theoretical understanding of these codes. In particular, we consider Floquet codes that can be exactly mapped to systems of free fermions through a graphical framework. These codes are defined by Pauli measurement observables that can be associated to the edges of a graph such that anticommuting Paulis correspond to incident edges. We additionally assume a noise model whose action on the logical subspace can be faithfully inferred from the free-fermion description. By combining our graphical framework with a description of the Floquet code by Majorana worldlines, we express the maximum-likelihood decoding problem in-terms of a partition function defined with respect to a particular graph. We expect this mapping to yield insight into tailoring codes for highly biased noise and the more general setting of random measurements.