Spatially-Coupled QLDPC Codes

TL;DR

This work develops spatially-coupled quantum LDPC codes as the quantum counterpart to 2D SC-LDPC constructions, using a two-variable polynomial representation F(U,V) to certify stabilizer commutativity via the condition ⟨F(U,V),F(U^{-1},V^{-1})⟩_s=0. It introduces 2D-SC-HGP and XYZ code families with small memories that achieve higher rates (around 1/3) and robust thresholds, and provides a rigorous cycle-analysis framework to minimize short cycles that impair BP decoding. The paper devises a gradient-based optimization (GRADE) and an alternating-optimization (AO) strategy to minimize cycles by tuning partitioning matrices, demonstrating substantial finite-length gains on depolarizing channels and revealing a rigid-cycle limit that bounds performance improvements. The findings indicate that high-performance SC-QLDPC codes can be realized with modest memory, and lay out pathways for windowed decoding and future code families with favorable trade-offs between rate, distance scaling, and decoding complexity. Overall, the algebraic 2D-SC framework enables systematic stabilizer construction, cycle-aware code optimization, and hardware-friendly implementations for quantum LDPC codes.

Abstract

Spatially-coupled (SC) codes is a class of convolutional LDPC codes that has been well investigated in classical coding theory thanks to their high performance and compatibility with low-latency decoders. We describe toric codes as quantum counterparts of classical two-dimensional spatially-coupled (2D-SC) codes, and introduce spatially-coupled quantum LDPC (SC-QLDPC) codes as a generalization. We use the convolutional structure to represent the parity check matrix of a 2D-SC code as a polynomial in two indeterminates, and derive an algebraic condition that is both necessary and sufficient for a 2D-SC code to be a stabilizer code. This algebraic framework facilitates the construction of new code families. While not the focus of this paper, we note that small memory facilitates physical connectivity of qubits, and it enables local encoding and low-latency windowed decoding. In this paper, we use the algebraic framework to optimize short cycles in the Tanner graph of 2D-SC hypergraph product (HGP) codes that arise from short cycles in either component code. While prior work focuses on QLDPC codes with rate less than 1/10, we construct 2D-SC HGP codes with small memories, higher rates (about 1/3), and superior thresholds.

Figures (10)

• Figure 1: A toric code defines on a $3\times 3$ grid. Vertices are indicated by circles and faces are indicated by squares. Vertices/edges with the same index are glued together to create a torus with $18$ edges, $9$ vertices, and $9$ faces. Vertex $5$ specifies the generator $Z_2\otimes Z_3\otimes Z_4\otimes Z_7$ and face $8$ specifies generator $X_{10}\otimes X_{13}\otimes X_{14}\otimes X_{15}$.
• Figure 2: Verifying pairwise orthogonality of blocks of rows in (\ref{['eqn quantum 2d sc codes 1']}).
• Figure 3: Layout of an SC code with a $(9,4)$ base code, memories $m_1 = m_2 = 2$, and coupling lengths $L_1 = L_2 = 5$. Blue dots represent data qubits in a fixed spatial light modulator (SLM) array, while red dots denote ancilla qubits in movable acousto-optic deflector (AOD) arrays. Qubit pairs within each black oval are positioned within the Rydberg distance, enabling four two-qubit gates in $\mathbf{H}_{0,0}$ of the corresponding replica via a Rydberg laser.
• Figure 4: Implementation of gates in $\mathbf{H}_{2,1}$ of the SC code shown in Fig. \ref{['fig: RAA1']}. Each AOD array is shifted two grids left and one grid up in a rotated layout. Qubit pairs enclosed within each black oval are positioned within the Rydberg distance, allowing the formation of four two-qubit gates in $\mathbf{H}_{2,1}$ of the corresponding replica via a Rydberg laser.
• Figure 5: Examples show how flexible cycle candidates of SC-HGP codes can be decomposed into two different components from $\mathbf{P}_a$ and $\mathbf{P}_b$. The matrices in the bottom panels of (a) and (b) are $\mathbf{P}_a$ and $\mathbf{P}_b$, and those in the top panels are $\mathbf{P}$. Nodes from copies of $\mathbf{A}$ and $\mathbf{B}$ are highlighted by red nodes and blue nodes, respectively. The number at each node is the value in its partitioning matrix $\mathbf{P}_a$ or $\mathbf{P}_b$. The red and blue cycles in (a) shows two flexible cycles-$8$ candidates where each of them is resulting from a pair of cycles-$4$ candidates from $\mathbf{P}_a$ and $\mathbf{P}_b$. Let $s_a=a_1-a_2+a_4-a_3$, $s_b=b_1-b_2+b_4-b_3$, the alternating sums associated with the blue and red cycle candidates are $s_a-s_b$ and $s_a+s_b$, respectively. The orange and green cycles in (a) shows two flexible cycles-$6$ candidates where each of them is resulting from a cycle-$4$ candidate from $\mathbf{P}_a$ and node $b_3$ from $\mathbf{P}_b$. The alternating sums of both of them are $s_a$. (b) shows three flexible cycles-$8$ candidates where each of them is resulting from a cycle-$6$ candidate from $\mathbf{P}_a$ and an entry $b$ from $\mathbf{P}_b$. The alternating sums of all three cases are $a_1-a_2+a_3-a_4+a_5-a_6$.

Theorems & Definitions (37)

• Example 1
• Example 2
• Remark 1
• Example 3
• Example 4
• Remark 2
• Theorem 1
• Remark 3
• proof
• Remark 4