The random coupled-plaquette gauge model and the surface code under circuit-level noise.

We introduce the random coupled-plaquette gauge model (RCPGM), which enables optimal accounting for Y-errors in decoding a surface code with noisy syndrome measurements. Using Parallel Tempering Monte Carlo simulations, we determine the code's fundamental error thresholds. For phenomenological depolarizing data and bit-flip syndrome noise we determine a threshold of 6%, to be compared to the "uncoupled" random plaquette gauge model (RPGM) with a mere 4.3%. We then tackle the circuit-level noise scenario, where an approximate reduction technique allows us to exploit the RCPGM. Within the assumptions of the reduction technique, we find a threshold of 1.4%, to be compared to 0.7% when marginalizing Y-errors for the "uncoupled" RPGM. These results crucially enlarge the landscape of statistical mechanical mappings for quantum error correction. In particular they show further room for improvement of the surface code for fault-tolerant quantum computation and should be highly encouraging for practical decoder development.