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]].
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]].
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]]].
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.
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.
Quasi-cyclic (QC) low-density parity-check (LDPC) codes are an important instance of proto-graph-based LDPC codes. In this paper we present upper bounds on the minimum Hamming distance of QC LDPC codes and study how these upper bounds depend on graph structure parameters (like variable degrees, check node degrees, girth) of the Tanner graph and of the underlying proto-graph. Moreover, for several classes of proto-graphs we present explicit QC LDPC code constructions that achieve (or come close) to the respective minimum Hamming distance upper bounds.
Quantum Low-Density Parity-Check (LDPC) codes provide a promising solution to protect quantum information from errors and are therefore considered an important step in the realisation of a fullscale fault-tolerant quantum computer. In this thesis, we study quantum LDPC codes under three different approaches, providing insights into renormalisation decoding for Kitaev’s toric code, establishing bounds on the minimum distance of geometrically-local codes, and constructing small instances of quantum Tanner codes. Kitaev’s toric code is a prominent quantum LDPC code that is currently the most widely pursued for experimental implementation. We focus on the probabilistic renormalisation decoders introduced by Duclos- Cianci and Poulin, a family of decoders that exhibit one of the best trade-offs between accuracy and efficiency. We study how they handle adversarial errors by introducing a deterministic renormalisation decoder that does not use a priori probabilities of the error model. We find that this renormalisation decoder allows for fractal-like uncorrectable error patterns, and we obtain a lower bound on the weight of such errors. Due to physical constraints, a lot of attention has gone into quantum LDPC codes whose stabilisers act only on a limited number of nearby qubits. We extend the well-known Bravyi-Terhal bound to quantum codes defined by local constraints on a lattice quotient, providing a crucial insight into the limitations of geometrically-local stabiliser codes. As an application, we provide upper bounds on the minimum distance of Abelian Two-Block Group Algebra codes, a family of quantum LDPC codes for which the minimum distance is currently unknown. In recent years, asymptotically good quantum LDPC codes were constructed, but provable results on their minimum distances only apply to impractically long codes. Following the effort in the design of near-term implementable codes, we construct short instances of quantum Tanner codes using square complexes obtained from lifted products of regular graphs. Several explicit instances of these codes exhibit minimum distances surpassing the square root of the code length.