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

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.