Constructions and analysis of linear codes for quantum error correction and secure information transmission

Jincheng Zhang

This paper investigates a novel approach to quantum error correction (QEC) leveraging topological codes. The core idea centers around dynamically adapting the topology of a topological code in real-time to counteract the effects of fluctuating environmental noise, exceeding the error correction capabilities of traditional, static codes. The proposed mechanism utilizes the non-Abelian exchange statistics inherent in quantum topological networks and integrates machine learning algorithms for dynamic optimization of topological parameters, minimizing correction overhead. The research addresses the limitations of existing QEC schemes, which primarily rely on static encoding, by introducing a dynamic framework capable of effectively handling the non-linear disturbances inherent in quantum systems. The theoretical framework presented here outlines a pathway toward more robust and efficient quantum information transmission.

Markus Grassl

Jincheng Zhang

This paper investigates the construction and properties of topological entanglement-based codes for quantum information transmission. Traditional quantum error correction codes face significant challenges in mitigating channel losses and decoherence, particularly over long distances. We propose a novel approach utilizing the inherent topological properties of entangled states to enhance error correction efficiency. The core concept involves leveraging entanglement as the fundamental resource for the code, employing carefully designed entanglement networks and encoding schemes to achieve robust error detection and correction. Our framework offers a potential pathway to overcoming limitations inherent in conventional quantum error correction, promising improved performance in long-distance quantum communication. The theoretical analysis presented herein establishes a foundation for future research and development in this area.

Mitsuru Hamada

Pavel Bažant

M. N. Isaeva

When transmitting information over channels with grouping errors, the traditional approach is channel decorrelation and use of codes correcting independent errors. The decorrelation procedure lowers achievable rates of reliable transmission, therefore the problem of using special codes for channels with memory and construction of computationally effective decoding methods for correction of grouping errors is actual. For the class of random codes, an approach is known using information sets of limited diameter to correct error bursts. The size of the set of information sets grows linearly with increasing code length, and the construction of the set is described by a probabilistic procedure. This article considers the construction of a set of information sets for a special class of codes that correct error bursts called Gilbert codes. The sets of code positions of the smallest possible diameter are considered. Based on the calculation of the ranks of the submatrices of the parity-check matrix of the Gilbert code, the probability that the set of positions is an information set is estimated. For a given location of the information set, the positions of the corrected bursts are analyzed. Based on the analysis, a method for constructing a set of dense information sets for Gilbert codes for correcting all error bursts within the code correcting capacity is proposed. Using the features of setting the parameters of Gilbert codes, an estimate of the size of the resulting set of dense information sets is carried out. For a simple block size of the paritycheck matrix of a quasi-cyclic code, it is shown that for Gilbert codes a dense information set is located at any position. In the case of extended Gilbert codes, it is shown that sets of minimum diameter exist only at the last position of each block. A procedure for constructing a set of dense information sets of minimum diameter for Gilbert codes and their extensions is proposed. A comparison is made of the set size of information sets and the probability of obtaining it for Gilbert codes and random codes. It is shown that the number of information sets obtained by the proposed procedure does not increase with the length of the code. The results obtained in the paper demonstrate the possibility of developing computationally efficient decoders based on information sets when correcting single error bursts. Unlike random linear codes, for which the methods of constructing information sets including dense ones, are probabilistic, a procedure for guaranteed construction of a set of information sets of minimal diameter is specified for Gilbert codes. The quasi-cyclic structure of Gilbert codes allows constructing sets of dense information sets of smaller dimension than for random codes. The obtained results allow us to guarantee the correction of error bursts within the correcting capacity of Gilbert codes and their extensions with low computational complexity. The use of computationally efficient procedures for encoding and decoding error bursts will improve the reliability of message delivery in channels with memory.