A parity obstruction to degree-optimal syndrome extraction in bivariate bicycle codes

arXiv:2509.26494v5 Announce Type: replace-cross Abstract: Syndrome extraction for a CSS code has depth at least the Tanner-graph degree $\Delta$, attained by edge colouring if gate order is ignored; correctness adds a parity constraint. We prove three general obstructions to depth-$\Delta$ extraction and show that, for every two-block group-algebra code with weight-three polynomials whose checks overlap in at most two qubits, no correct single-round circuit with one ancilla per check and one CNOT per Tanner edge has depth six: the minimum is seven, so the seven-layer cycle of Bravyi \emph{et al.} is optimal for their seven bivariate bicycle codes. This is the first analytic, family-wide impossibility result, extending prior instance-specific solver-based infeasibility results.