Quantifying finite-size corrections for surface codes under biased noise

We study the XY and XZZX surface codes under code-capacity phase-biased noise by mapping to random-bond Ising models. By appropriately formulating the boundary conditions, in the rotated code geometry, we find exact solutions at a special disordered point, where the codes reduce to a one-dimensional Ising model [1]. In the thermodynamic limit the special point is in the disordered phase of the Ising model, i.e. above the code-capacity threshold, except for pure phase biased noise where it coincides with the threshold. From the exact solutions we find that the logical bit-flip failure rate is strongly suppressed when the code distance is small relative to the bias. This implies a significant finite size effect that influences the reliability of threshold estimates based on finite-size scaling fits to numerically calculated failure rates for moderate code distances. We demonstrate this by studying the convergence with code distance of threshold fits for failures with respect to the three logical Paulis separately, compared to only fitting to the total failure rate. For the XZZX model this confirms that the thresholds for moderate bias are above the hashing bound, whereas for larger bias, the finite size corrections are too large to determine precise thresholds.

[1] Y. Xiao, B. Srivastava, M. Granath, Quantum, 8, 1468 (2024)