Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

IEEE Transactions on Information Theory

A Construction of Quantum LDPC Codes From Cayley Graphs

Alain Couvreur, Nicolas Delfosse, Gilles Zémor

Published May 21, 2013• 30 citations
Publisher

Abstract

We study a construction of quantum LDPC codes proposed by MacKay, Mitchison, and Shokrollahi. It is based on the Cayley graph of \BBF2ntogether with a set of generators regarded as the columns of the parity-check matrix of a classical code. We give a general lower bound on the minimum distance of the quantum code in O(dn2) where d is the minimum distance of the classical code. This bound is logarithmic in the blocklength 2nof the quantum code. When the classical code is the [n,1,n] repetition code, we are able to compute the exact parameters of the associated quantum code which are [[2n, 2[(n+1)/2], 2[(n-1)/2]]].

DOI
10.1109/tit.2013.2261116

Related papers

A construction of quantum LDPC codes from Cayley graphs

Alain Couvreur, Nicolas Delfosse, Gilles Zémor

We study a construction of Quantum LDPC codes proposed by MacKay, Mitchison and Shokrollahi in the draft [6]. It is based on the Cayley graph of F2ntogether with a set of generators regarded as the columns of the parity-check matrix of a classical code. We give a general lower bound on the minimum distance of the quantum code in O(dn2) where d is the minimum distance of the classical code. When the classical code is the [n, 1, n] repetition code, we are able to compute the exact parameters of the associated quantum code which are [[2n-1, 2 n/2, 2 n/2-1]].

arXiv:1206.2656•IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2011•8 citations

A Class of Quantum LDPC Codes Constructed From Finite Geometries

Salah A. Aly

Low-density parity check (LDPC) codes are a significant class of classical codes with many applications. Several good LDPC codes have been constructed using random, algebraic, and finite geometries approaches, with containing cycles of length at least six in their Tanner graphs. However, it is impossible to design a self-orthogonal parity check matrix of an LDPC code without introducing cycles of length four. In this paper, a new class of quantum LDPC codes based on lines and points of finite geometries is constructed. The parity check matrices of these codes are adapted to be self- orthogonal with containing only one cycle of length four in each pair of two rows. Also, the column and row weights, and bounds on the minimum distance of these codes are given. As a consequence, these codes can be encoded using shift-register encoding algorithms and can be decoded using iterative decoding algorithms over various quantum depolarizing channels.

arXiv:0712.4115•Globecom•Jan 1, 2008•47 citations

Quantum LDPC Codes With Positive Rate and Minimum Distance Proportional to the Square Root of the Blocklength

Jean–Pierre Tillich, Gilles Zémor

The current best asymptotic lower bound on the minimum distance of quantum LDPC codes with a fixed non-zero rate is logarithmic in the blocklength. We propose a construction of quantum LDPC codes with fixed non-zero rate and prove that the minimum distance grows proportionally to the square root of the blocklength.

arXiv:0903.0566•IEEE Transactions on Information Theory•Jan 24, 2014•345 citations

Design of Quasi-Cyclic Cycle LDPC Codes over GF(q)

HU Shu-kai, Chao Chen, Rong Xia Sun +1 more

Quasi-cyclic (QC) low-density parity-check (LDPC) codes have several appealing properties regarding decoding, storage requirements and encoding aspects. In this paper, we focus on the QC LDPC codes over GF(q) whose parity-check matrices have fixed column weight j=2. By investigating two subgraphs in the Tanner graphs of the corresponding base matrices, we derive two upper bounds on the minimum Hamming distance for this class of codes. In addition, a method is proposed to construct QC LDPC codes over GF(q), which have good Hamming distance distributions. Simulations show that our designed codes have good performance.

IEICE Transactions on Communications•Jan 1, 2012•0 citations

Quantum Tanner codes

Anthony Leverrier, Gilles Zémor

Tanner codes are long error correcting codes obtained from short codes and a graph, with bits on the edges and parity-check constraints from the short codes enforced at the vertices of the graph. Combining good short codes together with a spectral expander graph yields the celebrated expander codes of Sipser and Spielman, which are asymptotically good classical LDPC codes. In this work we apply this prescription to the left-right Cayley complex that lies at the heart of the recent construction of a c3locally testable code by Dinur et at. Specifically, we view this complex as two graphs that share the same set of edges. By defining a Tanner code on each of those graphs we obtain two classical codes that together define a quantum code. This construction can be seen as a simplified variant of the Panteleev and Kalachev asymptotically good quantum LDPC code, with improved estimates for its minimum distance. This quantum code is closely related to the Dinur et at. code in more than one sense: indeed, we prove a theorem that simultaneously gives a linearly growing minimum distance for the quantum code and recovers the local testability of the Dinur et at. code.

arXiv:2202.13641•Annual Symposium on Foundations of Computer Science•Oct 1, 2022•137 citations

Decoding Quasi-Cyclic Quantum LDPC Codes

Louis Golowich, Venkatesan Guruswami

Quantum low-density parity-check (qLDPC) codes are an important component in the quest for quantum fault tolerance. Dramatic recent progress on qLDPC codes has led to constructions which are asymptotically good, and which admit linear-time decoders to correct errors affecting a constant fraction of codeword qubits. These constructions, while theoretically explicit, rely on inner codes with strong properties only shown to exist by probabilistic arguments, resulting in lengths that are too large to be practically relevant. In practice, the surface/toric codes, which are the product of two repetition codes, are still often the qLDPC codes of choice. A previous construction of qLDPC codes based on the lifted product of an expander-based classical LDPC code with a repetition code (Panteleev and Kalachev, 2020) achieved a near-linear distance, and avoids the need for such intractable inner codes. Our main result is an efficient decoding algorithm for these codes that corrects a near-linear number of adversarial errors. En route, we give a similar algorithm for the hypergraph product version these codes, which are simpler but have distance growing only as the square root of the block length. Our decoding algorithms leverage the fact that the codes we consider are quasi-cyclic, meaning that they respect a cyclic group symmetry. Since the repetition code is not based on expanders, previous approaches to decoding expander-based qLDPC codes, which typically worked by greedily flipping code bits to reduce some potential function, do not apply in our setting. Instead, we reduce our decoding problem (in a black-box manner) to that of decoding classical expander-based LDPC codes under noisy parity-check syndromes. For completeness, we also include a treatment of such classical noisy-syndrome decoding that is sufficient for our application to the quantum setting.

arXiv:2411.04464•Annual Symposium on Foundations of Computer Science•Oct 27, 2024•1 citation
We study a construction of quantum LDPC codes proposed by MacKay, Mitchison, and Shokrollahi. It is based on the Cayley graph of \BBF2ntogether with a set of generators regarded as the columns of the parity-check matrix of a classical code. We give a general lower bound on the minimum distance of the quantum code in O(dn2) where d is the minimum distance of the classical code. This bound is logarithmic in the blocklength 2nof the quantum code. When the classical code is the [n,1,n] repetition code, we are able to compute the exact parameters of the associated quantum code which are [[2n, 2[(n+1)/2], 2[(n-1)/2]]].
We study a construction of Quantum LDPC codes proposed by MacKay, Mitchison and Shokrollahi in the draft [6]. It is based on the Cayley graph of F2ntogether with a set of generators regarded as the columns of the parity-check matrix of a classical code. We give a general lower bound on the minimum distance of the quantum code in O(dn2) where d is the minimum distance of the classical code. When the classical code is the [n, 1, n] repetition code, we are able to compute the exact parameters of the associated quantum code which are [[2n-1, 2 n/2, 2 n/2-1]].
Low-density parity check (LDPC) codes are a significant class of classical codes with many applications. Several good LDPC codes have been constructed using random, algebraic, and finite geometries approaches, with containing cycles of length at least six in their Tanner graphs. However, it is impossible to design a self-orthogonal parity check matrix of an LDPC code without introducing cycles of length four. In this paper, a new class of quantum LDPC codes based on lines and points of finite geometries is constructed. The parity check matrices of these codes are adapted to be self- orthogonal with containing only one cycle of length four in each pair of two rows. Also, the column and row weights, and bounds on the minimum distance of these codes are given. As a consequence, these codes can be encoded using shift-register encoding algorithms and can be decoded using iterative decoding algorithms over various quantum depolarizing channels.
Quasi-cyclic (QC) low-density parity-check (LDPC) codes have several appealing properties regarding decoding, storage requirements and encoding aspects. In this paper, we focus on the QC LDPC codes over GF(q) whose parity-check matrices have fixed column weight j=2. By investigating two subgraphs in the Tanner graphs of the corresponding base matrices, we derive two upper bounds on the minimum Hamming distance for this class of codes. In addition, a method is proposed to construct QC LDPC codes over GF(q), which have good Hamming distance distributions. Simulations show that our designed codes have good performance.
Tanner codes are long error correcting codes obtained from short codes and a graph, with bits on the edges and parity-check constraints from the short codes enforced at the vertices of the graph. Combining good short codes together with a spectral expander graph yields the celebrated expander codes of Sipser and Spielman, which are asymptotically good classical LDPC codes. In this work we apply this prescription to the left-right Cayley complex that lies at the heart of the recent construction of a c3locally testable code by Dinur et at. Specifically, we view this complex as two graphs that share the same set of edges. By defining a Tanner code on each of those graphs we obtain two classical codes that together define a quantum code. This construction can be seen as a simplified variant of the Panteleev and Kalachev asymptotically good quantum LDPC code, with improved estimates for its minimum distance. This quantum code is closely related to the Dinur et at. code in more than one sense: indeed, we prove a theorem that simultaneously gives a linearly growing minimum distance for the quantum code and recovers the local testability of the Dinur et at. code.
Quantum low-density parity-check (qLDPC) codes are an important component in the quest for quantum fault tolerance. Dramatic recent progress on qLDPC codes has led to constructions which are asymptotically good, and which admit linear-time decoders to correct errors affecting a constant fraction of codeword qubits. These constructions, while theoretically explicit, rely on inner codes with strong properties only shown to exist by probabilistic arguments, resulting in lengths that are too large to be practically relevant. In practice, the surface/toric codes, which are the product of two repetition codes, are still often the qLDPC codes of choice. A previous construction of qLDPC codes based on the lifted product of an expander-based classical LDPC code with a repetition code (Panteleev and Kalachev, 2020) achieved a near-linear distance, and avoids the need for such intractable inner codes. Our main result is an efficient decoding algorithm for these codes that corrects a near-linear number of adversarial errors. En route, we give a similar algorithm for the hypergraph product version these codes, which are simpler but have distance growing only as the square root of the block length. Our decoding algorithms leverage the fact that the codes we consider are quasi-cyclic, meaning that they respect a cyclic group symmetry. Since the repetition code is not based on expanders, previous approaches to decoding expander-based qLDPC codes, which typically worked by greedily flipping code bits to reduce some potential function, do not apply in our setting. Instead, we reduce our decoding problem (in a black-box manner) to that of decoding classical expander-based LDPC codes under noisy parity-check syndromes. For completeness, we also include a treatment of such classical noisy-syndrome decoding that is sufficient for our application to the quantum setting.