No $((n,K,d<127))$ Code Can Violate the Quantum Hamming Bound
It is well-known that nondegenerate quantum error correcting codes (QECCs) are constrained by a quantum version of the Hamming bound. Whether degenerate codes also obey such a bound, however, remains a long-standing question with practical implications for the efficacy of QECCs. We employ a combination of previously derived bounds on QECCs to demonstrate that a subset of all codes must obey the quantum Hamming bound. Specifically, we combine an analytical bound due to Rains with a numerical bound due to Li and Xing to show that no $((n,K,d))((n,K,d))$ code with $d<127d<127$ can violate the quantum Hamming bound.
Related Articles
QKD Based on Time-Entangled Photons and Its Key-Rate Promise Publisher: Â鶹´«Ã½AV
For secure practical systems, quantum key distribution (QKD) must provide high key rates over long distances. Time-entanglement-based QKD promises to increas...
Special Issue on Quantum Computation and Quantum Information
Quantum information was once a niche domain focused on foundational questions. It has grown into a broad research area with industrial, academic, and national research efforts around the world, but many foundational questions remain. The special issue papers consider cryptography and error correction from the viewpoint of quantum information and computation.
Shaping Postquantum Cryptography: The Hidden Subgroup and Shift Problems
The security of popular public key-cryptographic protocols, such as RSA, Diffie–Hellman key exchange and the digital signature algorithm (DSA), is endangered...