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

Abstract

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, $H(p_{th})+H({\tilde{p}}_{th})\approx 1$, 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.

12 pages, 5 figures, 2 tables

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