  Efficient high-performance decoding of topological stabilizer codes has the
potential to crucially improve the balance between logical failure rates and
the number and individual error rates of the constituent qubits. High-threshold
maximum-likelihood decoders require an explicit error model for Pauli errors to
decode a specific syndrome, whereas lower-threshold heuristic approaches such
as minimum weight matching are "error agnostic". Here we consider an
intermediate approach, formulating a decoder that depends on the bias, i.e.,
the relative probability of phase-flip to bit-flip errors, but is agnostic to
error rate. Our decoder is based on counting the number and effective weight of
the most likely error chains in each equivalence class of a given syndrome. We
use Metropolis-based Monte Carlo sampling to explore the space of error chains
and find unique chains, that are efficiently identified using a hash table.
Using the error-rate invariance the decoder can sample chains effectively at an
error rate which is higher than the physical error rate and without the need
for "thermalization" between chains in different equivalence classes. Applied
to the surface code and the XZZX code, the decoder matches maximum-likelihood
decoders for moderate code sizes or low error rates. We anticipate that,
because of the compressed information content per syndrome, it can be taken
full advantage of in combination with machine-learning methods to extrapolate
Monte Carlo-generated data.
