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arXiv:2512.22116 · Physical review. B./Physical review. B

General Construction of Quantum Error-Correcting Codes from Multiple Classical Codes

Yue Wu, Meng-Yuan Li, Chengshu Li, Hui Zhai

quant-phcond-mat.quant-gascond-mat.str-elSubmitted Dec 26, 2025• Announced Dec 29, 2025• 0 citations
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Abstract

The hypergraph product (HGP) construction of quantum error-correcting codes (QECC) offers a general and explicit method for building a QECC from two classical codes, thereby paving the way for the discovery of good quantum low-density parity-check codes. In this letter, we propose a general and explicit construction recipe for QECCs from a total of D classical codes for arbitrary D. Following this recipe guarantees the obtainment of a QECC within the stabilizer formalism and nearly exhausts all possible constructions. As examples, we demonstrate that our construction recovers the HGP construction when D = 2 and leads to four distinct types of constructions for D = 3, including a previously studied case as one of them. When the input classical codes are repetition codes, our D = 3 constructions unify various three-dimensional lattice models into a single framework, encompassing the three-dimensional toric code model, a fracton model, and two other intriguing models not previously investigated. Among these, two types of constructions exhibit a trade-off between code distance and code dimension for a fixed number of qubits by adjusting the lengths of the different classical codes, and the optimal choice can simultaneously achieve relatively large values for both code distance and code dimension. Our general construction protocol provides another perspective for enriching the structure of QECCs and enables the exploration of richer possibilities for good codes.

Comments
6 pages, 4 figures + Supplementary Material 5 pages, 4 figures

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The hypergraph product (HGP) construction of quantum error-correcting codes (QECC) offers a general and explicit method for building a QECC from two classical codes, thereby paving the way for the discovery of good quantum low-density parity-check codes. In this letter, we propose a general and explicit construction recipe for QECCs from a total of D classical codes for arbitrary D. Following this recipe guarantees the obtainment of a QECC within the stabilizer formalism and nearly exhausts all possible constructions. As examples, we demonstrate that our construction recovers the HGP construction when D = 2 and leads to four distinct types of constructions for D = 3, including a previously studied case as one of them. When the input classical codes are repetition codes, our D = 3 constructions unify various three-dimensional lattice models into a single framework, encompassing the three-dimensional toric code model, a fracton model, and two other intriguing models not previously investigated. Among these, two types of constructions exhibit a trade-off between code distance and code dimension for a fixed number of qubits by adjusting the lengths of the different classical codes, and the optimal choice can simultaneously achieve relatively large values for both code distance and code dimension. Our general construction protocol provides another perspective for enriching the structure of QECCs and enables the exploration of richer possibilities for good codes.
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.
Product constructions constitute a powerful method for generating quantum CSS codes, yielding celebrated examples such as toric codes and asymptotically good low-density parity check (LDPC) codes. Since a CSS code is fully described by a chain complex, existing product formalisms are predominantly homological, defined via the tensor product of the underlying chain complexes of input codes, thereby establishing a natural connection between quantum codes and topology. In this Letter, we introduce the \textit{quantum bootstrap product} (QBP), an approach that extends beyond this standard homological paradigm. Specifically, a QBP code is determined by solving a consistency condition termed the ``bootstrap equation''. We find that the QBP paradigm unifies a wide range of important codes, including general hypergraph product (HGP) codes of arbitrary dimensions and fracton codes typically represented by the X-cube code. Crucially, the solutions to the bootstrap equation yield chain complexes where the chain groups and associated boundary maps consist of multiple components. We term such structures \textit{fork complexes}. This structure elucidates the underlying topological structures of fracton codes, akin to foliated fracton order theories. Beyond conceptual insights, we demonstrate that the QBP paradigm can generate self-correcting quantum codes from input codes with constant energy barriers and surpass the code-rate upper bounds inherent to HGP codes. Our work thus substantially extends the scope of quantum product codes and provides a versatile framework for designing fault-tolerant quantum memories.
We continue the study of classical and quantum low-density parity check (LDPC) codes from a physical perspective. We focus on constructive approaches and formulate a general framework for systematically constructing codes with various features on generic Euclidean and non-Euclidean graphs. These codes can serve as fixed-point limits for phases of matter. To build our machinery, we unpack various product constructions from the coding literature in terms of physical principles such as symmetries and redundancies, introduce a new cubic product, and combine these products with the ideas of gauging and Higgsing introduced in Part I. We illustrate the usefulness of this approach in finite Euclidean dimensions by showing that using the one-dimensional Ising model as a starting point, we can systematically produce a very large zoo of classical and quantum phases of matter, including type I and type II fractons and SPT phases with generalized symmetries. We also use the balanced product to construct new Euclidean models, including one with topological order enriched by translation symmetry, and another exotic fracton model whose excitations are formed by combining those of a fractal spin liquid with those of a toric code, resulting in exotic mobility constraints. Moving beyond Euclidean models, we give a review of existing constructions of good qLDPC codes and classical locally testable codes and elaborate on the relationship between quantum code distance and classical energy barriers, discussed in Part I, from the perspective of product constructions.
Hypergraph product (HGP) codes are one of the most popular family of quantum low-density parity-check (LDPC) codes. Circuit-level simulations show that they can achieve the same logical error rate as surface codes with a reduced qubit overhead. They have been extensively optimized by importing classical techniques such as the progressive edge growth, or through random search, simulated annealing or reinforcement learning techniques. In this work, instead of machine learning (ML) algorithms that improve the code performance through local transformations, we impose additional global symmetries, that are hard to discover through ML, and we perform an exhaustive search. Precisely, we focus on the hypergraph product of two cyclic codes, which we call CxC codes and we study C2 codes which are the product a cyclic code with itself and CxR codes which are the product of a cyclic codes with a repetition code. We discover C2 codes and CxR codes that significantly outperform previously optimized HGP codes, achieving better parameters and a logical error rate per logical qubit that is up to three orders of magnitude better. Moreover, some C2 codes achieve simultaneously a lower logical error rate and a smaller qubit overhead than state-of-the-art LDPC codes such as the bivariate bicycle codes, at the price of a larger block length. Finally, leveraging the cyclic symmetry imposed on the codes, we design an efficient planar layout for the QCCD architecture, allowing for a trapped ion implementation of the syndrome extraction circuit in constant depth.
We study a three-fold variant of the hypergraph product code construction, differing from the standard homological product of three classical codes. When instantiated with 3 classical LDPC codes, this "XYZ product" yields a non CSS quantum LDPC code which might display a large minimum distance. The simplest instance of this construction, corresponding to the product of 3 repetition codes, is a non CSS variant of the 3-dimensional toric code known as the Chamon code. The general construction was introduced in Denise Maurice's PhD thesis, but has remained poorly understood so far. The reason is that while hypergraph product codes can be analyzed with combinatorial tools, the XYZ product codes also depend crucially on the algebraic properties of the parity-check matrices of the three classical codes, making their analysis much more involved.Our main motivation for studying XYZ product codes is that the natural representatives of logical operators are two-dimensional objects. This contrasts with standard hypergraph product codes in 3 dimensions which always admit one-dimensional logical operators. In particular, specific instances of XYZ product codes with constant rate might display a minimum distance as large as Θ(N2/3) . While we do not prove this result here, we obtain the dimension of a large class of XYZ product codes, and when restricting to codes with dimension 1, we reduce the problem of computing the minimum distance to a more elementary combinatorial problem involving binary 3-tensors. We also discuss in detail some families of XYZ product codes that can be embedded in three dimensions with local interaction. Some of these codes seem to share properties with Haah's cubic codes and might be interesting candidates for self-correcting quantum memories with a logarithmic energy barrier.