qector-two-stage-csslisted

# QECTOR Two-Stage CSS Decoding Source of authority: v1.0.0 reference manual, chapter 12. ## The problem with independent X/Z decoders A CSS code splits stabilizers into X-type and Z-type. Independent X and Z decoders assume P(X, Z) = P(X) P(Z) Depolarising noise violates this: a `Y` error flips both sectors, so the X and Z syndromes share information. The standard single-stage correction is a residual logical failure that a two-stage pass can avoid. ## The feedforward construction (manual 12.1) ``` c_X <- DecodeX(s_X) s'_Z = s_Z xor H_{Z,X} c_X (mod 2) c_Z <- DecodeZ(s'_Z) c = c_X xor c_Z ``` The X correction `c_X` induces a syndrome on the Z sector through the cross-coupling `H_{Z,X}`; subtracting it before the Z decode removes the cross-talk. Both stages may use any faithful decoder (Blossom, Sparse Blossom, Union-Find, BP-OSD). ## Theorem 13 (joint faithfulness, manual 12.1) > If `DecodeX` and `DecodeZ` are syndrome-faithful on their > respective inputs, then the combined correction satisfies > `H c = s` for the joint CSS code. The proof uses the fact that the full parity-check matrix of the CSS code applies `H_X` to the X sector and `H_Z` to the Z sector with the cross-coupling accounted by the update: H c = (H_X c_X, H_Z (c_X xor c_Z)) = (s_X, H_{Z,X} c_X xor s_Z xor H_{Z,X} c_X) = (s_X, s_Z) = s ## Worked example (manual 12, appendix E.4) ``` H_X = [[1, 1, 0], [0, 1, 1]] H_Z = [[0, 1, 1], [1,