Two-dimensional Entanglement-assisted Quantum Quasi-cyclic Low-density Parity-check Codes

TL;DR

The paper addresses robust coding for two-dimensional burst errors in storage and quantum channels by deriving a general condition for the existence of $2g$-cycles in 2-D Tanner graphs of QC-LDPC codes and then constructing several 2-D classical QC-LDPC families with controlled girth and burst-erasure correction capabilities. It introduces a stacking framework based on permutation tensors to realize girth-$>4$ codes for odd prime $p$ (and composite-$p$ extensions), achieving at least $p^2$ burst-erasure correction, and further extends to 4- and 6-cycle-free designs using Behrend sequences. From these classical codes, it builds two families of 2-D entanglement-assisted QLDPC codes: one using a pair of 2-D codes with a 4-cycle-free unassisted graph requiring only a single ebit, and another from a single 2-D code free of $4$-cycles, both preserving the $p^2$ erasure-correction property. The results offer structured, scalable 2-D classical and EA-QLDPC codes suitable for reliable quantum storage and communication under 2-D error models.

Abstract

For any positive integer $g \ge 2$, we derive general conditions for the existence of a $2g$-cycle in the Tanner graph of two-dimensional ($2$-D) classical quasi-cyclic (QC) low-density parity-check (LDPC) codes. Based on these conditions, we construct a family of $2$-D classical QC-LDPC codes with girth greater than $4$ by stacking $p \times p \times p$ tensors, where $p$ is an odd prime. Furthermore, for composite values of $p$, we propose two additional families of $2$-D classical LDPC codes obtained via similar tensor stacking. In this case, one family achieves girth greater than $4$, while the other attains girth greater than $6$. All the proposed $2$-D classical QC-LDPC codes exhibit an erasure correction capability of at least $p \times p$. Based on the constructed classical $2$-D QC-LDPC codes, we derive two families of $2$-D entanglement-assisted (EA) quantum low-density parity-check (QLDPC) codes. The first family of $2$-D EA-QLDPC codes is obtained from a pair of binary $2$-D classical LDPC codes and is designed such that the unassisted part of the Tanner graph of the resulting EA-QLDPC code is free of cycles of length four, while requiring only a single ebit to be shared across the quantum transceiver. The second family is constructed from a single $2$-D classical LDPC code whose Tanner graph is free from $4$-cycles. Moreover, the constructed EA-QLDPC codes inherit an erasure correction capability of $p \times p$, as the underlying classical codes possess the same erasure correction property.

Figures (4)

• Figure 1: Identity tensor and its various shifted versions, where the shifts are applied in the positive $i$- and $j$-directions.
• Figure 2: Parity-check tensor $H_{2\text{-}D}$ for $p=3$, where the shifts used are the same as those defined in \ref{['shifts']} with $\phi(i)=i$, and $\psi(j)=j$ and $\eta(k)=k$.
• Figure 3: First layer in parity-check tensor $H_{2\text{-}D}$ for $p=3$, constructed using the specific shifts with $\phi(i)=i$, $\psi(j)=j$, and $\eta(k)=k \pmod{p}$ for all $i,j,k$.
• Figure 4: Unfolding of the $3$-D parity-check tensor $H_{2\text{-}D}$ into a $2$-D parity-check matrix $H_{1\text{-}D}$, where different colors represent a vertical layer in $H_{2\text{-}D}$.

Theorems & Definitions (23)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1: Condition for a $2g$-cycle
• proof
• Lemma 3
• proof
• Lemma 4
• proof