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Heade kvantveaparanduskoodide otsing

Aitsen, Daglas

Published Jan 1, 2025• 0 citations
OpenAlex

Abstract

Quantum error correction codes are developed to correct errors that occur in quantum data. However, their design is complicated by the fundamental constraints of quantum mechanics. This thesis explores how to adapt quasi-cyclic low-density parity-check (QC-LDPC) codes, used in classical communications, for quantum error correction. Since a small girth in an LDPC code is linked to poor decoding performance, this thesis focuses on constructing quantum QC-LDPC codes with large girth. To achieve this, we present two methods: one based on minimal-weight codewords and another based on multiplicity constraints to avoid short cycles. Using these approaches, we construct quantum QC-LDPC codes with girth 8. The construction methods offer a more flexible alternative to existing techniques, enabling the design of large-girth codes with a wider range of structures and sizes.

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Quantum error correction codes are developed to correct errors that occur in quantum data. However, their design is complicated by the fundamental constraints of quantum mechanics. This thesis explores how to adapt quasi-cyclic low-density parity-check (QC-LDPC) codes, used in classical communications, for quantum error correction. Since a small girth in an LDPC code is linked to poor decoding performance, this thesis focuses on constructing quantum QC-LDPC codes with large girth. To achieve this, we present two methods: one based on minimal-weight codewords and another based on multiplicity constraints to avoid short cycles. Using these approaches, we construct quantum QC-LDPC codes with girth 8. The construction methods offer a more flexible alternative to existing techniques, enabling the design of large-girth codes with a wider range of structures and sizes.
This study proposes an explicit construction method for quantum quasi-cyclic low-density parity-check (QC-LDPC) codes with a girth of 12. The proposed method designs parity-check matrices that maximize the girth while maintaining an orthogonal structure suitable for quantum error correction. By utilizing algebraic techniques, short cycles are eliminated, which improves error correction performance. Additionally, this method is extended to non-binary LDPC codes and spatially-coupled LDPC codes, demonstrating that both the girth and orthogonality can be preserved. The results of this study enable the design of high-performance quantum error-correcting codes without the need for random search.
In this paper, we propose a new approach to construct quasi-cyclic low-density parity-check (QC-LDPC) codes using Hamming codewords of weight three, namely the HM-QC-LDPC codes. Thanks to the distance property of Hamming codes, length four cycles are avoided from the Tanner graph of the constructed codes, resulting in codes of girth six. This approach can be further extended to construct QC-LDPC codes of girth eight. Our numerical results show that the proposal has a better cycle profile than the progressive-edge-growth (PEG) constructed codes, and consequently prevails in performance. Moreover, the constructed QC-LDPC code prevails the LDPC code of the IEEE802.22 standard by 0.2 dB at the bit error rate (BER) of 10-8when the codeword length is within hundreds of bits, making it a promising candidate for scenarios where strict decoding latency is imposed.
Short cycles in the Tanner graph of a given LDPC code has negative effect on the performance of iterative decoding. To improve the performance, we presented an approach for constructing Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes whose associated Tanner graph has large girth and a small amount of short cycles. Each column block of the parity check matrix was designed by adding circulant permutation sub-matrices one after another, provided the current parity check matrix has minimal number of short cycles. The condition for the proposed QC-LDPC codes to have girth at least 6 was also analyzed. Regular and irregular QC-LDPC codes with various lengths and rates can be constructed. Simulations show that the proposed QC-LDPC codes have less short cycles and better performance than existing progressive edge growth based QC-LDPC codes.
Classical low-density parity-check (LDPC) codes are a widely deployed and well-established technology, forming the backbone of modern communication and storage systems. It is well known that, in this classical setting, increasing the girth of the Tanner graph while maintaining regular degree distributions leads simultaneously to good belief-propagation (BP) decoding performance and large minimum distance. In the quantum setting, however, this principle does not directly apply because quantum LDPC codes must satisfy additional orthogonality constraints between their parity-check matrices. When one enforces both orthogonality and regularity in a straightforward manner, the girth is typically reduced and the minimum distance becomes structurally upper bounded. In this work, we overcome this limitation by using permutation matrices with controlled commutativity and by restricting the orthogonality constraints to only the active part of the construction, while preserving regular check-matrix structures. This design circumvents conventional structural distance limitations induced by parent-matrix orthogonality, and enables the construction of quantum LDPC codes with large girth while avoiding latent low-weight logical operators. As a concrete demonstration, we construct a girth-8, (3,12)-regular [[9216,4612,≤48]][[9216,4612, \leq 48]][[9216,4612,≤48]] quantum LDPC code and show that, under BP decoding combined with a low-complexity post-processing algorithm, it achieves a frame error rate as low as 10−810^{-8}10−8 on the depolarizing channel with error probability 4%4 \%4%.
The objective of this thesis is to develop schemes to construct algebraically structured low-density parity-check (LDPC) codes for improving their performance and memory efficiency. It is well-known that the sum-product algorithm provides optimum decoding for LDPC codes with no cycles. However, short cycles in the Tanner graph of LDPC codes may induce a serious performance degradation. Several researchers made an effort to obtain the LDPC codes with no short cycles. Quasi-cyclic LDPC (QC-LDPC) code is a type of LDPC codes, based on circulant permutation matrices. It required a smaller size of memory for encoding and/or decoding and are easier to analyze their algebraic properties than random LDPC codes. This motivates us to take an intensive interest in constructing nonbinary QC-LDPC codes without short cycles. In this thesis, we propose a construction method of nonbinary QC-LDPC codes without short cycles. We present the mathematical proof for girth and cycle properties of proposed design method. As a result, we present the constructing method for nonbinary QC-LDPC codes without short cycles and show it performs well.
We present a novel construction of quasi-cyclic (QC) low-density parity-check (LDPC) codes with arbitrary column weight and girth at least eight. By carefully designing the parity-check matrix, our approach ensures both structural flexibility and improved error-correction capabilities. We evaluate the performance of these codes under the additive white Gaussian noise (AWGN) channel. Simulation results demonstrate that the proposed QC-LDPC codes outperform the protograph LDPC codes included in the Consultative Committee for Space Data Systems (CCSDS) telecommand standards. Additionally, we benchmark our proposed codes against existing short-length codes with girth 8, observing improvements in the decoding performance while ensuring suitability for hardware implementation. The findings of this work contribute to the ongoing development of efficient LDPC codes suitable for ultra-reliable low-latency communications and area-efficient architecture designs for the Internet of Things (IoT), machine-to-machine (M2M), and satellite communications.