Entanglement-Assisted Quantum Codes from Cyclic Codes

Abstract

Entanglement-assisted quantum-error-correcting (EAQEC) codes are quantum codes which use entanglement as a resource. These codes can provide better error correction than the (entanglement unassisted) codes derived from the traditional stabilizer formalism. In this paper, we provide a general method to construct EAQEC codes from cyclic codes. Afterwards, the method is applied to Reed–Solomon codes, BCH codes, and general cyclic codes. We use the Euclidean and Hermitian construction of EAQEC codes. Three families have been created: two families of EAQEC codes are maximal distance separable (MDS), and one is almost MDS or almost near MDS. The comparison of the codes in this paper is mostly based on the quantum Singleton bound.

Stabilizer codes for Open Quantum Systems

“…From the clear relation between Eqs. (13)(14)(15), we only need to show that one of these functions is a symplectic form. Then,…”

“…For the operation in Eqs. (13)(14)(15) to be a symplectic form, the first point we need to show is that the image of ζ equipped with a proper additive operation forms an additive group.…”

“…Now, we can use the previous algebraic structure to show that the expression given in Eqs. (13)(14)(15)…”

“…As an important result, it was shown that stabilizer codes exist if and only if there exist classical linear block codes obeying an orthogonality rule; e.g., Euclidean and Hermitian self-orthogonality 6 . Such a duality between stabilizer codes and classical block codes has stimulated several works [6][7][8][9][10][11][12][13][14] . Nevertheless, there is still room for novelties.…”

On Optimal and Quantum Code Construction from Cyclic Codes over FqPQ with Applications

“…The literature extensively delves into the exploration of cyclic codes and their constructions across various finite rings, such as applications in constructing minimal codes in [ 4 ] and projective two-weight codes in [ 5 ]. Recently, Pereira and Mancini have given a general method to construct EAQEC codes from cyclic codes in [ 6 ]. A particular area of interest in recent years has been the study of codes over mixed alphabets.…”