Constructions of good entanglement-assisted quantum error correcting codes

Abstract

Entanglement-assisted quantum error correcting codes (EAQECCs) are a simple and fundamental class of codes. They allow for the construction of quantum codes from classical codes by relaxing the duality condition and using pre-shared entanglement between the sender and receiver. However, in general it is not easy to determine the number of shared pairs required to construct an EAQECC. In this paper, we show that this number is related to the hull of the classical code. Using this fact, we give methods to construct EAQECCs requiring desirable amount of entanglement. This leads to design families of EAQECCs with good error performance. Moreover, we construct maximal entanglement EAQECCs from LCD codes. Finally, we prove the existence of asymptotically good EAQECCs in the odd characteristic case. * K. Guenda is with the Faculty

Euclidean and Hermitian Hulls of MDS Codes and Their Applications to EAQECCs

“…When c = 0, then an [[n, k, d; c]] q EAQECC is a stabilizer quantum code and is usually denoted as [[n, k, d]] q , which is the most widely studied class of quantum codes. One of the constraints among the parameters n, k, d and c is the following wellknown quantum Singleton bound: Guenda et al [22] provided the relation between the value of rank(HH † ) and the dimension of the Hermitian hull of the linear code with parity check matrix H.…”

Vector-Circulant Matrices and Vector-Circulant Based Additive Codes over Finite Fields

“…Nested pairs of additive codes were used in constructing asymmetric quantum codes [2]. In [14], the hull of codes and complementary dual codes were applied for constructions of entanglement-assisted quantum codes. We note that the said properties for vector-circulant based additive codes can be determined directly from their corresponding vector-circulant matrices.…”

“…Based on these properties, quantum codes can be constructed using suitable vector-circulant based additive codes and methods in [2,5,14].…”

On the Construction of LCD Codes overF5

“…Since then, construction and applications on LCD codes were deeply and investigated [5][6][7][8][9][10][11][12][13][14][15] widely. For example, many maximal entanglement entanglement-assisted quantum codes were derived from good LCD codes [5][6][7][8]. Some optimal LCD codes are studied and constructed in the Refs.…”

“…Some optimal LCD codes are studied and constructed in the Refs. [6][7][8][9][10][11][12][13][14][15].…”