Quantum Cross Subspace Alignment Codes via the $N$-sum Box Abstraction (2304.14676v1)
Abstract: Cross-subspace alignment (CSA) codes are used in various private information retrieval (PIR) schemes (e.g., with secure storage) and in secure distributed batch matrix multiplication (SDBMM). Using a recently developed $N$-sum box abstraction of a quantum multiple-access channel (QMAC), we translate CSA schemes over classical multiple-access channels into efficient quantum CSA schemes over a QMAC, achieving maximal superdense coding gain. Because of the $N$-sum box abstraction, the underlying problem of coding to exploit quantum entanglements for CSA schemes, becomes conceptually equivalent to that of designing a channel matrix for a MIMO MAC subject to given structural constraints imposed by the $N$-sum box abstraction, such that the resulting MIMO MAC is able to implement the functionality of a CSA scheme (encoding/decoding) over-the-air. Applications include Quantum PIR with secure and MDS-coded storage, as well as Quantum SDBMM.
- S. Song and M. Hayashi, “Capacity of quantum private information retrieval with multiple servers,” IEEE Transactions on Information Theory, vol. 67, no. 1, pp. 452–463, 2020.
- ——, “Capacity of quantum private information retrieval with colluding servers,” IEEE Transactions on Information Theory, vol. 67, no. 8, pp. 5491–5508, 2021.
- M. Allaix, S. Song, L. Holzbaur, T. Pllaha, M. Hayashi, and C. Hollanti, “On the capacity of quantum private information retrieval from MDS-coded and colluding servers,” IEEE Journal on Selected Areas in Communications, vol. 40, no. 3, pp. 885–898, 2022.
- B. Chor, E. Kushilevitz, O. Goldreich, and M. Sudan, “Private information retrieval,” Journal of the ACM, vol. 45, no. 6, pp. 965–981, 1998.
- H. Sun and S. A. Jafar, “The capacity of private information retrieval,” IEEE Transactions on Information Theory, vol. 63, no. 7, pp. 4075–4088, 2017.
- A. S. Holevo, “Bounds for the quantity of information transmitted by a quantum communication channel,” Problemy Peredachi Informatsii, vol. 9, no. 3, pp. 3–11, 1973.
- H. Sun and S. A. Jafar, “The capacity of robust private information retrieval with colluding databases,” IEEE Trans. on Information Theory, vol. 64, no. 4, pp. 2361–2370, 2017.
- K. Banawan and S. Ulukus, “The capacity of private information retrieval from coded databases,” IEEE Transactions on Information Theory, vol. 64, no. 3, pp. 1945–1956, 2018.
- R. Tajeddine, O. W. Gnilke, D. Karpuk, R. Freij-Hollanti, and C. Hollanti, “Private information retrieval from coded storage systems with colluding, Byzantine, and unresponsive servers,” IEEE Transactions on information theory, vol. 65, no. 6, pp. 3898–3906, 2019.
- S. Ulukus, S. Avestimehr, M. Gastpar, S. A. Jafar, R. Tandon, and C. Tian, “Private retrieval, computing, and learning: Recent progress and future challenges,” IEEE Journal on Selected Areas in Communications, vol. 40, no. 3, pp. 729–748, 2022.
- Z. Jia, H. Sun, and S. A. Jafar, “Cross subspace alignment and the asymptotic capacity of X𝑋Xitalic_X-secure T𝑇Titalic_T-private information retrieval,” IEEE Trans. on Information Theory, vol. 65, no. 9, pp. 5783–5798, Sep. 2019.
- Z. Jia and S. A. Jafar, “X𝑋Xitalic_X-secure T𝑇Titalic_T-private information retrieval from MDS coded storage with Byzantine and unresponsive servers,” IEEE Transactions on Information Theory, vol. 66, no. 12, pp. 7427–7438, 2020.
- ——, “On the capacity of secure distributed batch matrix multiplication,” IEEE Transactions on Information Theory, vol. 67, no. 11, pp. 7420–7437, 2021.
- Z. Chen, Z. Jia, Z. Wang, and S. A. Jafar, “GCSA codes with noise alignment for secure coded multi-party batch matrix multiplication,” IEEE Journal on Selected Areas in Information Theory, vol. 2, no. 1, pp. 306–316, 2021.
- M. Allaix, Y. Lu, Y. Yao, T. Pllaha, C. Hollanti, and S. Jafar, “N𝑁Nitalic_N-sum box: An abstraction for linear computation over many-to-one quantum networks,” CPCC Technical Report, 2023. [Online]. Available: http://escholarship.org/uc/item/44p655jp
- A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codes exist,” Physical Review A, vol. 54, no. 2, p. 1098, 1996.
- A. Steane, “Multiple-particle interference and quantum error correction,” Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, vol. 452, no. 1954, pp. 2551–2577, 1996.