Error Resilience of Fracton Codes and Near Saturation of Code-Capacity Threshold in Three Dimensions

Fracton codes have been intensively studied as novel topological states of matter, yet their fault-tolerant properties remain largely unexplored. Here, we investigate the optimal thresholds of self-dual fracton codes, in particular the checkerboard code, against stochastic Pauli noise. By utilizing a statistical-mechanical mapping combined with large-scale parallel tempering Monte Carlo simulations, we calculate the optimal code capacity of the checkerboard code to be p_th simeq 0.107(3). This value is the highest among known three-dimensional codes and nearly saturates the theoretical limit for topological codes. Our results further validate the generalized entropy relation for two mutually dual models, H\(p_th\) + H\({p}_th\) approx 1, and extend its applicability beyond standard topological codes. This verification indicates the Haah's code also possesses a code capacity near the theoretical limit p_th approx 0.11. These findings highlight fracton codes as highly resilient quantum memory and demonstrate the utility of duality techniques in analyzing intricate quantum error-correcting codes.