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arXiv:2609.01305

Lifted-Product QLDPC Codes in the Polynomial Domain

Vahid Nourozi, Mohsen Moradi, Roxana Smarandache, David G. M. Mitchell

Published Sep 1, 2026• 0 citations
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Abstract

This paper presents a finite-length polynomial-domain formulation of lifted-product quantum low-density parity-check (QLDPC) codes. We formulate the code construction over the quotient ring F 2[D]/(D L + 1), where polynomial base matrices are lifted entrywise to binary circulant blocks. This representation gives a compact algebraic description of the lifted-product parity-check matrices and allows CSS orthogonality to be analyzed before binary expansion. We show that the standard circulant lifting map is compatible with polynomial conjugation, which implies that the resulting binary matrices satisfy the CSS commutation constraint. The construction is illustrated with a constraint length 7, rate 1/2 NASA convolutional code example, and numerical examples are provided from a 3 x 4 polynomial parity-check matrix. The finite-length performance of selected constructed codes is then evaluated over the depolarizing channel using various benchmark decoders. The resulting framework gives a structured method to construct finite-length lifted-product QLDPC codes from small polynomial base matrices.

DOI
10.48550/arxiv.2609.01305

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This paper presents a finite-length polynomial-domain formulation of lifted-product quantum low-density parity-check (QLDPC) codes. We formulate the code construction over the quotient ring F 2[D]/(D L + 1), where polynomial base matrices are lifted entrywise to binary circulant blocks. This representation gives a compact algebraic description of the lifted-product parity-check matrices and allows CSS orthogonality to be analyzed before binary expansion. We show that the standard circulant lifting map is compatible with polynomial conjugation, which implies that the resulting binary matrices satisfy the CSS commutation constraint. The construction is illustrated with a constraint length 7, rate 1/2 NASA convolutional code example, and numerical examples are provided from a 3 x 4 polynomial parity-check matrix. The finite-length performance of selected constructed codes is then evaluated over the depolarizing channel using various benchmark decoders. The resulting framework gives a structured method to construct finite-length lifted-product QLDPC codes from small polynomial base matrices.
Lifted product codes are an important family of quantum low-density parity-check (QLDPC) codes, as they were the first QLDPC code family shown to be asymptotically good. Understanding the structure of their parity-check matrices HXH_{\mathsf{X}}HX​ and HZH_{\mathsf{Z}}HZ​, as well as the associated Tanner graphs, is essential for analyzing their decoding behavior and error-floor performance. In this work, we show that the Tanner graphs of HXH_{\mathsf{X}}HX​ and HZH_{\mathsf{Z}}HZ​ are indeed isomorphic, and investigate their graph-theoretical structure. We establish conditions ensuring the connectivity of these graphs and provide bounds on their minimal absorbing sets, providing new insight into the combinatorial structures influencing decoding performance.
Quantum low-density parity-check (QLDPC) codes offer a promising route to scalable fault-tolerant quantum computation, but their performance under iterative decoding is strongly influenced by short-cycle structure and other harmful subgraphs in the associated Tanner graphs. This paper develops an algebraic framework for constructing and analyzing (Q)LDPC codes from dyadic and quasi-dyadic matrices-translation-invariant 2ℓ×2ℓ2^\ell \times 2^\ell2ℓ×2ℓ binary matrices specified compactly by a signature row and forming a commutative ring with recursive block structure. Leveraging this structure, we relate cycles in lifted Tanner graphs to tailless backtrackless closed walks in the protograph and derive efficient, implementable methods to enumerate and control short cycles (notably 444-, 666-, and 888-cycles). We introduce dyadic-aware PEG-style construction algorithms that use forbidden sets of shifts to maximize attainable girth when possible and otherwise minimize the multiplicity of the shortest cycles at the target girth. Motivated by error-floor phenomena, we further characterize and explicitly enumerate absorbing sets in key dyadic layout boundary cases, identifying configurations that induce abundant (a,0)(a,0)(a,0)-absorbing sets. Finally, we propose CSS-oriented dyadic constructions that satisfy commutation constraints by design and demonstrate via belief-propagation simulations that reducing short-cycle multiplicity can yield substantial decoding gains even when girth cannot be increased.
We study two explicit constructions of binary Calderbank--Shor--Steane quantum low-density parity-check (QLDPC) codes with check weight six. The first uses a fixed circulant shift pattern and yields parameters $\code{6N}{2N+2}{6}$ with Tanner graph girth twelve for N=31N=31N=31 and every N≥37N\ge37N≥37. A graph incidence interpretation gives the ranks and girths, and a uniform weight six logical operator establishes the exact distance. The second construction uses fixed matrix polynomials with circulant and affine permutation coefficients. Here, a polynomial commutation identity guarantees orthogonality for every cyclic lift. Four realizations give [[192,16,8]][[192,16,8]][[192,16,8]], [[288,16,12]][[288,16,12]][[288,16,12]], [[384,16,14]][[384,16,14]][[384,16,14]], and [[576,32,20]][[576,32,20]][[576,32,20]], with Tanner girth eight and computationally certified distances. We give reproducible construction data, distinguish classical cycle distance from quantum distance, and identify the limitations of both families.
Quantum low-density parity-check (QLDPC) codes provide a practical balance between error-correction capability and implementation complexity in quantum error correction (QEC). In this paper, we propose an algebraic construction based on dyadic matrices for designing both classical and quantum LDPC codes. The method first generates classical binary quasi-dyadic LDPC codes whose Tanner graphs have girth 6. It is then extended to the Calderbank-Shor-Steane (CSS) framework, where the two component parity-check matrices are built to satisfy the compatibility condition required by the recently introduced CAMEL-ensemble quaternary belief propagation decoder. This compatibility condition ensures that all unavoidable cycles of length 4 are assembled in a single variable node, allowing the mitigation of their detrimental effects by decimating that variable node.
Generalized low-density parity-check (GLDPC) codes, where single parity-check constraints on the code bits are replaced with generalized constraints (an arbitrary linear code), are a promising class of codes for low-latency communication. The block error rate performance of the GLDPC codes, combined with a complementary outer code, has been shown to outperform a variety of state-of-the-art code and decoder designs with suitable lengths and rates for the 5G ultra-reliable low-latency communication (URLLC) regime. A major drawback of these codes is that it is not known how to construct appropriate polynomial matrices to encode them efficiently. In this paper, we analyze practical constructions of quasi-cyclic GLDPC (QC-GLDPC) codes and show how to construct polynomial generator matrices in various forms using minors of the polynomial matrix. The approach can be applied to fully generalized matrices or partially generalized (with mixed constraint node types) to find better performance/rate trade-offs. The resulting encoding matrices are presented in useful forms that facilitate efficient implementation. The rich substructure displayed also provides us with new methods of determining low weight codewords, providing lower and upper bounds on the minimum distance and often giving those of weight equal to the minimum distance. Based on the minors of the polynomial parity-check matrix, we also give a formula for the rank of any parity-check matrix representing a QC-LDPC or QC-GLDPC code, and hence, the dimension of the code. Finally, we show that by applying double graph-liftings, the code parameters can be improved without affecting the ability to obtain a polynomial generator matrix.
Quantum low-density parity-check (qLDPC) codes offer a promising route to scalable fault-tolerant quantum computing, but their practical implementation requires efficient circuits for syndrome extraction. Many qLDPC families are assembled from a small set of components through explicit operations that imprint edge symmetries on their Tanner graphs. We show that disassembling the code by quotienting its symmetries one at a time, reduces parity-check circuit design to a much smaller problem acting only on the underlying components. Solutions to the reduced problem can then be lifted to the full code, yielding circuit constructions that apply to entire families of codes built from the same components. For lifted-product codes, our approach provides an optimal analytical construction for parity-check circuits achieving the lower bound on CNOT depth. We obtain analogous constructions for balanced-product codes and quantum--classical homological product codes. For quantum Tanner codes, the reduced problem is small enough to solve numerically, reaching the CNOT-depth lower bound for nearly all instance tested, including codes with up to 576 data qubits.