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Quantum Information Processing

Exploring quasi-geometric frameworks for quantum error-correcting codes: a systematic review

Valentine Nyirahafashimana, Nurisya Mohd Shah, Umair Abdul Halim, Мohamed Othman

Published Aug 28, 2025• 0 citations
Publisher

Abstract

This study investigates quasi-geometric strategies for improving quantum error correction in quantum computing, utilizing geometric principles to improve error detection and correction while maintaining computational efficiency. A comprehensive review of 20 studies, selected from 2988 publications spanning 2019 to 2024, reveals significant progress in quasi-cyclic codes, quasi-orthogonal codes, and quasi-structured geometric codes, highlighting their growing importance in quantum error correction and information theory. The findings indicate that quasi-orthogonal codes that employ coefficient vector differential quasi-orthogonal space-time frequency coding demonstrated a $$1.20$$ dB gain at a bit error rate of $$10^{-4}$$ , while reducing computational complexity. Quasi-structured geometric codes offered energy-efficient solutions, facilitating multi-state orthogonal signaling and reliable linear code construction. Furthermore, quasi-cyclic low-density parity-check codes with optimized information selection surpassed traditional forward error correction codes, achieving superior quantum error rates of $$10^{-5}$$ at $$10.00$$ dB and $$10^{-6}$$ at $$15.00$$ dB. Performance analysis showed that the effectiveness of error correction depends more on the frequency of six-length cycles than on girth, suggesting a new direction for optimization. The study emphasizes the transformative potential of quasi-geometric strategies in improving quantum communication by focusing on bit and quantum bit error rates within both stabilizer and classical frameworks. Future work focuses on integrating hybrid quantum-classical codes to raise error resilience and efficiency, addressing challenges like decoding instability, and limited orthogonality to enable reliable and computational quantum communication systems.

DOI
10.1007/s11128-025-04904-5

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Quantum error correction is an important building block for reliable quantum information processing. A challenging hurdle in the theory of quantum error correction is that it is significantly more difficult to design error-correcting codes with desirable properties for quantum information processing than for traditional digital communications and computation. A typical obstacle to constructing a variety of strong quantum error-correcting codes is the complicated restrictions imposed on the structure of a code. Recently, promising solutions to this problem have been proposed in quantum information science, where in principle any binary linear code can be turned into a quantum error-correcting code by assuming a small number of reliable quantum bits. This paper studies how best to take advantage of these latest ideas to construct desirable quantum error-correcting codes of very high information rate. Our methods exploit structured high-rate low-density parity-check codes available in the classical domain and provide quantum analogues that inherit their characteristic low decoding complexity and high error correction performance even at moderate code lengths. Our approach to designing high-rate quantum error-correcting codes also allows for making direct use of other major syndrome decoding methods for linear codes, making it possible to deal with a situation where promising quantum analogues of low-density parity-check codes are difficult to find.
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