Does the optimal toric-code depolarizing threshold equal the hashing bound
In plain words
With a perfect decoder (the classical routine that infers which errors occurred), the toric code tolerates a certain depolarizing error rate, and numerics put it at about 18.9%. Whether this equals exactly the hashing bound, a limit from information theory with the same value to three digits, is unknown.
Precise statement
For the toric code under independent depolarizing noise of strength p with perfect syndrome measurement and maximum-likelihood decoding, the threshold $p_c$ equals the Nishimori-line critical point of a disordered eight-vertex (coupled Ising) model. Decide whether $p_c$ equals the hashing value $p_h \sim 0.1893$ solving $1 - H(p_h) - p_h \operatorname{log2} 3 = 0$ (H the binary entropy), or compute p_c to a precision that separates them; Bombin et al. found $p_c = 0.189(3)$ (Phys. Rev. X 2, 021004, 2012).
What would settle it
High-precision Monte Carlo or tensor-network evaluation of the Nishimori critical point with error below $10^{-4}$, or an exact duality argument fixing $p_{c}$.