Small quantum Tanner codes from left--right Cayley complexes

TL;DR

This work treats quantum Tanner codes built from left–right Cayley complexes as lifts of a base CSS code on an $n_A\times n_B$ grid, extended by a finite-group lift with multisets $A,B$ to produce qubits indexed by $(i,j,g)$. It provides an explicit base-code parameterization, including a dimension formula $k = k_{01}k'_{01} + k_{01}^\perp k'_{01}^\perp$ and a distance bound $d = \min(d_{01},d_{01}',d_{01}^\perp,d'_{01}^\perp)$, and, in the special case $|B|=2$ with $B$-side repetition, shows $k = k_{01} + k_{01}^\perp$ (Theorem). Through extensive numerical searches over small groups and local codes, the authors identify moderate-length codes with parameters such as $[[144,12,11]]$, $[[432,20,\le 22]]$, and $[[576,28,\le 24]]$ for generator weight $9$, demonstrating practical viability at hundreds of qubits. The work also discusses decoding challenges and open questions about the structure of logical operators and how to push performance toward larger, decodable quantum LDPC codes.

Abstract

Quantum Tanner codes are a class of quantum low-density parity-check codes that provably display a linear minimum distance and a constant encoding rate in the asymptotic limit. When built from left--right Cayley complexes, they can be described through a lifting procedure and a base code, which we characterize. We also compute the dimension of quantum Tanner codes when the right degree of the complex is 2. Finally, we perform an extensive search over small groups and identify instances of quantum Tanner codes with parameters $[[144,12,11]]$, $[[432,20,\leq 22]]$ and $[[576,28,\leq 24]]$ for generators of weight 9.

Figures (1)

• Figure 1: Three-dimensional representation of the quantum Tanner codes, with $\mathcal{G} = C_{46}, n_A=n_B=6$ and $A = (0,1,2,3,4,5), B=(0,2,4,6,8,10)$, where we use additive notation for cyclic groups. Qubits are associated with tiles indexed by $(i,j,g)$ with $i \in[n_A], j \in [n_B], g\in \mathcal{G}$. Each generator is supported within the translate of a horizontal slice. The figure depicts 4 generators, each associated with specific choices of local codewords (marked with dark disks) and a group element. The whole set of generators is obtained by considering a basis of the local product codes, and all $\mathcal{G}$-translates. From bottom to top, an $X$-generator defined on the (red) set $\{ (i,j, g) : i \in [n_A], j\in [n_B]\}$ corresponding to a row of $H_0 \otimes G_0' \otimes I_{|\mathcal{G}|}$, a $Z$-generator defined on the (blue) set $\{ (i,j, gb_j^{-1}) : i \in [n_A], j\in [n_B]\}$ corresponding to a row of $(G_0 \otimes H_1' \otimes I_{|\mathcal{G}|})R_B$, an $X$-generator defined on the set $\{ (i,j, a_i gb_j^{-1}) : i \in [n_A], j\in [n_B]\}$ corresponding to a row of $(H_1 \otimes G_1' \otimes I_{|\mathcal{G}|})L_A R_B$, and a $Z$-generator defined on the set $\{ (i,j, a_i g) : i \in [n_A], j\in [n_B]\}$ corresponding to a row of $(G_1 \otimes H_0' \otimes I_{|\mathcal{G}|})L_A$. The support of an $X$-generator can only overlap that of a $Z$-generator on a single row or column of such slices (depicted with a darker color on the figure): in that case, the supports correspond to orthogonal codewords of the local codes, ensuring the commutation of the generators.

Theorems & Definitions (2)

• Lemma 1: Base-code parameters
• Theorem 1