Moderate-length lifted quantum Tanner codes

TL;DR

The article advances quantum LDPC code design by developing a lifting framework for quantum Tanner codes defined on bipartite square complexes. It generalizes Leverrier–Zémor’s approach to include square-complex covers, enabling odd-index lifts with $ ilde{n}=tn$, $ ilde{k}\ge k$, and $ ilde{d}\le td$, and shows that when $ ilde{k}=k$ the distance increases to $ ilde{d}\ge d$. The authors introduce transfer-homomorphism techniques to bound and characterize logical operators in lifted codes, and provide explicit moderate-length constructions using cyclic and double-circulant local codes, including a notable [[96,2,12]] code whose distance saturates the derived bound. They also demonstrate that lifting can reduce overlap between $X$- and $Z$-checks beyond previous methods, enabling distance advantages beyond $\,\sqrt{n}$ for certain codes, while preserving LDPC properties. The work presents a numerical exploration of lifts (up to index 30) and lays groundwork for future rigorous distance analyses and practical quantum LDPC code implementations.

Abstract

We introduce new families of quantum Tanner codes, a class of quantum codes that first appeared in the work of Leverrier and Zémor (FOCS 2022). These codes are built from two classical Tanner codes, for which the underlying graphs are extracted from coverings of 2D geometrical complexes, and the local linear codes are tensor-products of cyclic or double-circulant linear codes. The advantage of code lifting is that, for any lift of odd index $t$ of an $[[n,k,d]]$-code, we can adapt the study of the transfer homomorphism arising in cellular homology to describe symmetries of its logical operators and to establish that its dimension is lower bounded by $k$, and its distance is upper bounded by $t\cdot d$. Moreover, when the dimension of the lifted code is equal to $k$, its distance is lower bounded by $d$. These parameter bounds also apply to the previous methods of code lifting of Guémard (IEEE Trans. Inf. Theory, 2025). Finally, We present several explicit families, and identify instances of moderate length quantum codes which are degenerate, have low check weight, and whose distance surpasses the square root of the code length. Among them, we report the existence of a $[[96,2,12]]$-code whose distance growth saturates our bound, and for which half of the checks are of weight 8 and the other half of weight 4.

Figures (9)

• Figure 1: Bipartite square complex $\mathcal{S}=(V,E,F)$ in which one set of vertices is denoted $V_X$ and the other set $V_Z$. They give rise to two diagonal graphs, $\mathcal{G}_X^\Box=(V_X,F)$ represented in blue, and $\mathcal{G}_Z^\Box=(V_Z,F)$ in red.
• Figure 2: Subgraph of a bipartite square complex with a local product structure.
• Figure 3: Face neighborhood $F(v_x)$ and $F(v_z)$, identified with two product sets and represented as matrices. The gray column in the left array and in the right array corresponds to identical faces in the square complex. A $X$-check can be seen as a codeword of the product code $C\otimes C'$ in the left array, and a $Z$-check as a codeword of the product code $C^\perp \otimes C"^\perp$ in the right array. The restriction of the two codewords to the gray columns are orthogonal vectors, ensuring that the $X$-check and the $Z$-check commute.
• Figure 4: Face neighborhood of a vertex $v$ in the topological realization of a square complex. It is composed of four faces and its closure forms a subcomplex that is not simply connected: a cylinder.
• Figure 5: Left: space homotopy equivalent to the presentation complex of $\operatorname{L}(3)$. Right: its subdivision into a square complex and the indexing of the faces by $(i,j)\in [2]\times [\ell]$.

Theorems & Definitions (30)

• Theorem 2.1: Galois correspondence
• Definition 2.2: Classical Tanner code
• Definition 2.3: Lift of Tanner code
• Definition 3.1
• Lemma 3.2
• proof
• Proposition 3.3
• proof
• Lemma 3.4
• proof