An exact Error Threshold of Surface Code under Correlated Nearest-Neighbor Errors: A Statistical Mechanical Analysis

The surface code represents a promising candidate for fault-tolerant quantum computation due to its high error threshold and experimental accessibility with nearest-neighbor interactions. However, current exact surface code threshold analyses are based on the assumption of independent and identically distributed (i.i.d.) errors. Though there are numerical studieds for threshold with correlated error, they are only the lower bond ranther than exact value, this offers potential for higher error thresholds.Here, we establish an error-edge map, which allows for the mapping of quantum error correction to a square-octagonal random bond Ising model. We then present the exact threshold under a realistic noise model that combines independent single-qubit errors with correlated errors between nearest-neighbor data qubits. Our method is applicable for any ratio of nearest-neighbor correlated errors to i.i.d. errors. We investigate the error correction threshold of surface codes and we present analytical constraints giving exact value of error threshold. This means that our error threshold is both upper bound and achievable and hence on the one hand the existing numerical threshold values can all be improved to our threshold value, on the other hand, our threshold value is highest achievable value in principle.