Threshold saturation for Haah's cubic code

Establish whether the code-capacity threshold of self-dual Haah's cubic code lies close to the optimal value of approximately 0.11, equivalently whether its Fractal Ising disorder model satisfies the principal-factor condition underlying the generalized Kramers–Wannier threshold relation.

Background

The thesis numerically estimates the Checkerboard code threshold as 0.107(3), finding near-saturation of the quantum threshold bound predicted by the generalized Kramers–Wannier duality framework. The same mechanism is argued to apply heuristically to Haah's code because the associated Fractal Ising model is self-dual and exhibits a single transition in the disorder-free limit.

A controlled finite-size scaling study of the disordered Fractal Ising model is currently obstructed by its system-size-dependent symmetries, strong finite-size effects, and computational limitations. Consequently, the threshold prediction for Haah's code remains unresolved rather than numerically established.