MATHEMATICAL FOUNDATIONS OF QUANTUM COMPUTING AND QUANTUM ERROR CORRECTION: FROM QUANTUM STATES TO CSS CODES
Nội dung chính của bài viết
Tóm tắt
Quantum computing exploits quantum-mechanical principles, including superposition, entanglement, and interference, to process information in ways that differ fundamentally from classical computation. However, quantum systems are highly vulnerable to environmental noise and hardware imperfections, making quantum error correction (QEC) essential for reliable and fault-tolerant quantum computing. This paper provides a systematic introduction to the mathematical and physical foundations of quantum computing and QEC.
Key concepts, including complex vector spaces, tensor products, unitary transformations, quantum measurements, projective measurements, positive operator valued measures (POVMs), and density operators, are reviewed. The roles of state orthogonality and the no-cloning theorem in quantum information processing are also discussed.
Furthermore, the relationship between classical and quantum error-correcting codes is examined through the Calderbank-Shor-Steane (CSS) construction. CSS codes are developed from nested classical linear codes using cosets and superpositions of computational basis states, with emphasis on the relationship between classical and quantum code parameters. This framework establishes a foundation for studying advanced stabilizer, cyclic, constacyclic, entanglement-assisted, and asymmetric quantum error-correcting codes.
Từ khóa
Quantum computing, quantum information, quantum error correction, quantum states, quantum measurement, density operator, no-cloning theorem, Calderbank–Shor–Steane codes, classical error- correcting codes, fault-tolerant quantum computing.
Chi tiết bài viết
Tài liệu tham khảo
pp. 7551–7558, 2025. https://doi.org/10.1109/TCE.2025.3594368.
[2] U. Jayasinghe, P. Samarathunga, N. Pollwaththage, Y. Ganearachchi, T. Fernando, and A. Fernando, "Quantum Communication for Video Transmission Over Error-Prone Channels," IEEE Transactions on Consumer Electronics, vol. 71,
pp. 1148–1155, 2025. https://doi.org/10.1109/TCE.2025.3552930.
[3] V. Karthick, M. R. Prabhu, P. Nancy, A. Devipriya, and R. A. A. Rosaline, "Advancements in Fault-Tolerant Quantum Error Correction for Current Progress and Future Directions," in Proceedings of the Second International Conference on Advances in Information Technology (ICAIT), Chikkamagaluru, Karnataka, India,
pp. 1–5, 2024. https://doi.org/10.1109/ICAIT61638.2024.10690433.
[4] S. Kundu, T. Gupta, A. Sardar, A. Bandyopadhyay, S. Swain, and S. Mallik, "A survey on quantum computing: Transforming cryptography, AI/ML, blockchain, and network communication," Franklin Open, vol. 12, 2025. https://doi.org/
10.1016/j.fraope.2025.100371.
[5] A. Chatterjee, K. Phalak, and S. Ghosh, "Quantum Error Correction For Dummies," in Proceedings of the IEEE International Conference on Quantum Computing and Engineering (QCE), Bellevue, WA, USA, pp. 70–81, 2023. https://doi.org/
10.1109/QCE57702.2023.00017.
[6] S. Ling and C. Xing, Coding Theory: A First Course. Cambridge University Press, 2004.
[7] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes. Cambridge University Press, 2010. [8] F. Gaitan, Quantum Error Correction and Fault Tolerant Quantum Computing.
Southern Illinois University, Carbondale, Illinois, 2008. [9] J. Roffe, "Quantum Error Correction: An Introductory Guide," Contemporary Physics, vol. 60, no. 3, pp. 226–245, 2019. [10] A. C. Quillen, PHY265 Notes: Introducing Quantum Error Correction, 2023. [11] R. D Wolf, Quantum Computing: Lecture Notes, QuSoft, CWI and University of
Amsterdam, 2023. Available: https://arxiv.org/abs/1907.09415.
[12] H. Q. Dinh, H. T. Le, B. T. Nguyen, and P. Maneejuk, “Some classes of new quantum MDS and synchronizable codes constructed from repeated-root cyclic codes of length 6푝푠,” IEEE Access, vol. 9, pp. 138543–138552, 2021. DOI: 10.1109/ACCESS.2021.3117595.
[13] T. Zhang and G. Ge, “Some new classes of quantum MDS codes from constacyclic codes,” IEEE Transactions on Information Theory, vol. 61, no. 9, pp. 5224–5228, 2015. DOI: 10.1109/TIT.2015.2444417.
[14] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information.
Cambridge University Press, 2010. [15] W. Fedus, Entropy in Classical and Quantum Information Theory, Physics Department, University of California, San Diego. [16] J. Watrous, Lecture 1: Overview of Quantum Information, CPSC 519/619: Quantum Computation, University of Calgary, January 10, 2006. [17] R. Wan and S. Zhu, "New Quantum MDS codes from Hermitian self-orthogonal generalized Reed-Solomon codes," Journal of Mathematical Physics, vol. 68, no. 4, pp. 1234-1248, 2023. https://doi.org/xxxx. [18] D. Gottesman, "A theory of fault-tolerant quantum computation," California Institute of Technology, Pasadena, CA, 1997. [Online]. Available: arXiv:quant-ph/9702029v2. [Accessed: 14 Oct. 2026]. [19] T. A. Brun, I. Devetak, and M.-H. Hsieh, "Correcting quantum errors with entanglement," Science, vol. 314, no. 5798, pp. 436–439, Oct. 2006, doi: 10.1126/science.1131563. [20] D. Elkouss, A. Leverrier, R. Alléaume, and J. J. Boutros, “Efficient reconciliation protocol for discrete-variable quantum key distribution,” in Proc. 2009 IEEE International Symposium on Information Theory (ISIT), Seoul, Korea, 2009, pp. 1879–1883.
[21] C. I. Kaya, A Review on Quantum Error Correction, M.S. thesis, Dept. of Physics, Middle East Technical University, Ankara, Turkey, Dec. 2023. [Online]. Available: https://open.metu.edu.tr/handle/11511/107782