Structural Analysis of Directional qLDPC Codes

TL;DR

This work develops a comprehensive word-first analysis framework for route-generated, translation-invariant CSS codes on rectangular tori, and provides explicit stabilizer dependencies, commuting-operator motifs, and an exact criterion for dimension collapse on thin rectangles.

Abstract

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 \emph{direction word} (route) on square- or hex-grid connectivity. In this work, we develop a comprehensive \emph{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 $W=\texttt{NE$^2$NE$^2$N}$ 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.

Figures (13)

• Figure 1: Checkerboard data/ancilla partition (shown as a patch) and a simple example layout with row alternation. Here, black dots are data qubits ($x+y$ even) and X/Z labels mark ancillas ($x+y$ odd).
• Figure 2: Toy instance illustrating the terminology. Gray sites are ancillas, black sites are data. For $W=\texttt{NE$^2$N}$ and anchor $a=(1,0)$ the four circled data qubits are the translated support $a+P(W)$.
• Figure 3: Support patterns $P(W)$ for five representative words, plotted as integer offsets relative to an ancilla anchor at the origin.
• Figure 4: For $W=\texttt{NE$^2$N}$, anchors separated by $\delta=(2,0)$ have supports that overlap on exactly one data qubit (star). Thus $\delta=(2,0)$ forbids assigning opposite types to those ancillas. This is the concrete meaning of an "odd-difference" displacement.
• Figure 5: Coset structure on ancillas for two lattice types that occur frequently in directional-code words. A layout is any $X/Z$ labeling constant on these cosets (Theorem \ref{['thm:layout']}).

Theorems & Definitions (53)

• Lemma 1: Route-to-offset formula
• Remark 1: Route interpretation as a local-gate schedule
• Definition 1: Support pattern
• Remark 2
• Example 1: word $\rightarrow$ check and a row of $H$
• Definition 2: Odd-multiplicity difference set and lattice
• Remark 3: Intuition for $L(W)$
• Example 2: computing $\Delta_{\mathrm{odd}}$ for NE$^2$N
• Remark 4: Choosing a basis for $L(W)$
• Theorem 1: Layout coset theorem