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arXiv:2305.00137 · Quantum

Spatially-Coupled QLDPC Codes

Siyi Yang, Robert Calderbank

Published Apr 7, 2025• 9 citations
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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.

DOI
10.22331/q-2025-04-07-1693

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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.
In this chapter, we present an algebraic construction of a special type of LDPC code whose Tanner graph has a very specific structure. For an LDPC code of this type, its Tanner graph is locally connected . Every VN is only connected to the CNs that are confined to a (small) span of ρ col consecutive locations and every CN is only connected to VNs that are confined to a (small) span of ρ row consecutive locations . We call such constraints on the connections between VNs and CNs of a Tanner graph ( ρ col ,ρ row) -span-constraints . With this span-constraint, the Tanner graph of such an LDPC code is actually a chain of small Tanner graphs in which each graph is connected to its adjacent graphs on either side of it, except the first and the last ones. An LDPC code of this type is called a span-constrained LDPC code. The SC-LDPC code investigated in [59, 60, 22, 86] is a type of span-constrained LDPC code. An SC-LDPC code is an LDPC convolutional (LDPC-C) code viewed from a graphical point of view (or a spatial coupling point of view) [49, 104, 66, 67, 92]. An LDPC-C code [49, 104] is specified by a bi-infinite parity-check matrix whose nonzero entries are confined to a diagonal band of a certain width ρ row and a certain depth ρ col. The nonzero entries in every row are confined to a span of ρ row consecutive locations and the nonzero entries in every column are confined to a span of ρ col consecutive locations. With these constraints on the locations of the nonzero entries of the parity-check matrix of an LDPC-C code, every VN in its Tanner graph is only connected to the CNs that are confined to a span of ρ col consecutive locations and every CN is only connected to VNs that are confined to a span of ρ row consecutive locations. These constraints on the locations of the nonzero entries of the parity-check matrix of an LDPC-C code lead to the graphical ( ρ col ,ρ row)-span-constraint as mentioned above. Hence, an LDPC-C code is a span-constrained LDPC code, an SC-LDPC code.
We consider a special family of spatially coupled low-density parity-check (LDPC) codes known as LDPC convolutional codes and propose a design technique based on symbolic hyper-graphs aimed at achieving good codes with small syndrome former memory order. We define conditions on the minimum syndrome former memory order that must be imposed in order to avoid short local cycles in the codes Tanner graphs and to achieve good minimum distance. Code design examples are provided, accompanied by error rate performance assessments based on Montecarlo simulations.
In this paper, time-invariant spatially coupled low-density parity-check convolutional codes (SC-LDPC-CCs) are considered, and the connections existing between their low-weight codewords and cycles in their Tanner graphs are studied. Using the polynomial representation of these codes, we show that parity-check matrices having columns with weight ≥2 can be analyzed considering a certain number of parity-check sub-matrices having regular columns with weight 2. These sub-matrices are associated to cycles in the code Tanner graph and define as many codes we denote as component codes. Based on this observation, we find that codewords of the main code can be expressed as combinations of codewords of the component codes. The design of codes free of codewords up to a certain weight is also addressed. We show that low-weight codewords in the main code can be avoided by removing some types of cycles in its Tanner graph. Our design approach is applied to some well-known ensembles of SC-LDPC-CCs to prove its effectiveness.
Utility-scale quantum computing requires quantum error correction (QEC) to protect quantum information against noise. Currently, superconducting hardware is a promising candidate for achieving fault tolerance due to its fast gate times and feasible scalability. However, it is often restricted to two-dimensional nearest-neighbour connectivity, and therefore the variety of quantum low-density parity-check (qLDPC) codes that can be implemented on it without sacrificing QEC performance is believed to be greatly restricted. In this paper we construct a new family of qLDPC codes, which we call ``directional codes'', that outperforms the rotated toric code (RTC) while satisfying the connectivity requirements of the widely adopted square-grid, and some even the sparser hexagonal-grid, on a torus. The key idea is to utilise the iSWAP gate -- a native gate demonstrated on superconducting qubits -- to construct circuits that measure the stabilisers of these qLDPC codes without the need for additional connections. We numerically evaluate the performance of directional codes, encoding four, six, twelve and eighteen logical qubits, using a common superconducting-inspired circuit-level Pauli noise model. We also compare them to the RTC and to the bivariate bicycle (BB) codes, currently the two most popular quantum LDPC code families. As a concrete example, when evaluated with the Tesseract decoder with short beam setting, the best directional code family investigated achieves the same logical error rate as the RTC at physical error rate p=10−3p=10^{-3}p=10−3 but requires only a quarter to a third of the number of physical qubits to encode a given number of logical qubits. Our discovery opens a novel direction in QEC code design, suggesting that complex high-connectivity hardware may not be necessary for low-overhead fault-tolerant quantum computation.
Quantum low-density parity-check (QLDPC) codes are an important class of quantum error-correcting codes that have low-weight stabilizer generators and typically offer encoding rates much higher than popular topological quantum codes such as surface and toric codes. While recent constructions of QLDPC codes have been aimed at two-level quantum systems, several hardware platforms for quantum computing, including superconducting and photonic systems, support a much larger Hilbert space. This makes the design of qudit-based quantum LDPC codes an issue of paramount significance in harnessing the increased flexibility afforded by high -dimensional quantum systems. In this paper, we forge the first steps in addressing this gap and explore the generalization of QLDPC constructions to qudit systems. We review two methods of generalizing binary quantum codes to qudit codes, namely, the stack and merge constructions, and investigate the application of these two meth-ods to the well-known hypergraph product and lifted-product codes. Subsequently, we provide upper and lower bounds for the encoding rates of the resulting qudit codes. We also prove an interesting relationship between the qudit hypergraph product and qudit lifted-product codes: the merge construction applied to the lifted product code is equivalent to the stack construction applied to the hypergraph product code under certain conditions. We evaluate the performance of these qudit codes under non-binary belief-propagation decoding and observe that qudit codes offer us a greater degree of flexibility in optimizing their rate-performance trade-off. We also demonstrate that, under certain conditions, qudit codes can simultaneously achieve higher rates and higher frame error rate (FER) performance than their binary counterparts.
For designing low-density parity-check (LDPC) codes for quantum error-correction, we desire to satisfy the conflicting requirements below simultaneously. 1) The row weights of parity-check “should be large”: The minimum distances are bounded above by the minimum row weights of parity-check matrices of constituent classical codes. Small minimum distance tends to result in poor decoding performance at the error-floor region. 2) The row weights of parity-check matrices “should not be large”: The performance of the sum-product decoding algorithm at the water-fall region is degraded as the row weight increases. Recently, Kudekar et al. showed spatially-coupled (SC) LDPC codes exhibit capacity-achieving performance for classical channels. SC LDPC codes have both large row weight and capacity-achieving error-floor and water-fall performance. In this paper, we propose a new class of quantum LDPC codes based on spatially coupled quasi-cyclic LDPC codes. The performance outperforms that of quantum “non-coupled” quasi-cyclic LDPC codes.