Finite-Basis Duality Estimate for the Surface-Code Threshold under Correlated Bit-Flip Errors

Abstract: We apply finite-basis duality to a statistical-mechanical model introduced by Chubb and Flammia for the surface code under spatially correlated bit-flip noise. Their mapping gives a random-bond Ising model with both two-body edge interactions and four-body face interactions. The single-equation estimate based on the duality analysis is slightly deviated from their Monte Carlo estimate (p_c=0.1004(6)). Following the finite-basis and graph-polynomial strategy, we instead use periodic and twisted-periodic sectors on toroidal bases. We have obtained an improved estimate of the critical point, $p_c = 0.10348$.