Entanglement-assisted Reed–Solomon codes over qudits: theory and architecture.

We develop a systematic theory for the construction of quantum codes from classical Reed–Solomon over F p k , where p is prime and k ∈ Z + . Based on two classical [ n , n - d m + 1 , d m ] Reed–Solomon codes over F p k , we provide the construction of an [ [ n , n + n e - 2 (d m - 1) , d m ] ] entanglement-assisted Reed–Solomon code over qudits of dimension d = p k that saturates the quantum Singleton bound and needs n e entangled qudits, which involves obtaining the explicit form of the stabilizers for the code. Our contributions towards the entanglement-assisted Reed–Solomon code are multi-fold as follows: (a) Based on the parity check matrices of the classical Reed–Solomon codes, the explicit form of the stabilizers of the entanglement-assisted Reed–Solomon code are provided and the rate-optimal code is obtained. (b) Based on the entanglement-assisted Reed–Solomon code, a burst error correcting code for qudits of dimension p that corrects a burst of k (t m - 1) qudits or less, where t m = ⌊ (d m - 1) / 2 ⌋ is provided. (c) Finally, we provide amenable circuits for encoding and error correction with quantum circuit complexity of O (n 2) , useful for practical implementations. [ABSTRACT FROM AUTHOR]