Statistical mechanical model for surface code with partially heralded errors

The surface code is one of the leading quantum error correction codes for experimental implementation. Its connection with the random bond Ising model (RBIM) has long been known, and the phase transition conditions in the RBIM provide insights into the surface code's error threshold. Heralded errors of known locations are easier to correct compared to Pauli errors of unknown locations. A statistical mechanical model for surface codes with a portion of heralded errors enhances our understanding of such systems and their potential for fault-tolerant quantum computing. In this work, we map the surface code with a portion of heralded errors to a diluted RBIM. The effect of erasure becomes a dilution in the RBIM, while parameters still adhere to the Nishimori condition. Furthermore, the time-evolving block decimation method is applied to compute the partition function of the diluted RBIM, allowing efficient computation of the logical error rate for large systems when the erasure portion is close to 100%. Notably, techniques in neutral atom systems that can convert a large portion of Pauli errors into heralded erasure errors offer promising applications for this research.