Structural Analysis of Directional qLDPC Codes

Directional codes, recently introduced by Gehér--Byfield--Ruban \cite{Geher2025Directional}, constitute a hardware-motivated family of quantum low-density parity-check (qLDPC) codes. These codes are defined by stabilizers measured by ancilla qubits executing a fixed direction word (route) on square- or hex-grid connectivity. In this work, we develop a comprehensive word-first analysis framework for route-generated, translation-invariant CSS codes on rectangular tori. Under this framework, a direction word W deterministically induces a finite support pattern P(W), from which we analytically derive: (i) a closed-form route-to-support map; (ii) the odd-multiplicity difference lattice L(W) that classifies commutation-compatible X/Z layouts; and (iii) conservative finite-torus admissibility criteria. Furthermore, we provide: (iv) a rigorous word equivalence and canonicalization theory (incorporating dihedral lattice symmetries, reversal/inversion, and cyclic shifts) to enable symmetry-quotiented searches; (v) an "inverse problem" criterion to determine when a translation-invariant support pattern is realizable by a single route, including reconstruction and non-realizability certificates; and (vi) a quasi-cyclic (group-algebra) reduction for row-periodic layouts that explains the sensitivity of code dimension k to boundary conditions. As a case study, we analyze the word W=texttt{NE^2NE^2N} end-to-end. We provide explicit stabilizer dependencies, commuting-operator motifs, and an exact criterion for dimension collapse on thin rectangles: for \(L_x, L_y\) = (2d, d) with row alternation, we find k=4 if 6 mid d, and k=0 otherwise.