Complete classification of weight-3 logical operators in the Face Centered Cubic lattice CSS code

Abstract

The $$[[3L^3, 2L^3{+}2, 3]]$$ CSS code on the Face Centered Cubic (FCC) lattice reaches a 67.7% encoding rate with weight-12 stabilizers. Its distance is 3. We classify every weight-3 logical operator of the code. The X-type ones are the $$4L^3$$ triangular faces of the lattice. The Z-type ones are the $$4L^3$$ claws, each formed by the three edges of a tetrahedron meeting at one of its vertices. The two families are exact duals under the exchange of vertices and octahedral voids, and each spans $$2L^3 - 1$$ independent logical classes. We prove this count analytically from the homology of the 3-torus, not only numerically: the cycle space has dimension $$\tfrac{5}{2}L^3 + 1$$ , the boundary space has dimension $$\tfrac{5}{2}L^3 - 2$$ , and subtracting the stabilizer rank leaves $$2L^3 - 1$$ for every even L. So $$k = (2L^3 - 1) + 3$$ , and the excess rate is carried entirely by weight-3 operators. We then show that this sector cannot be protected. The three-body parity $$Z_{e_a}Z_{e_b}Z_{e_c}$$ at a face anticommutes with four octahedral stabilizers, so it is not measurable, and no measurement of any kind can distinguish $${|{\psi }\rangle }$$ from $$\hat{L}_T{|{\psi }\rangle }$$ for arbitrary encoded $${|{\psi }\rangle }$$ . Removing the sector requires adjoining all $$2L^3 - 1$$ independent directions, which returns $$k = 3$$ . Verified exactly over GF(2) at $$L = 4, 6, 8, 10$$ .