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Entanglement-assisted quantum codes from algebraic geometry codes

Francisco Revson F. Pereira, Ruud Pellikaan, Giuliano G. La Guardia, Francisco de Assis

Published Jan 1, 2019• 24 citations
OpenAlex

Abstract

Quantum error correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error correcting codes to achieve higher rates than the one obtained via traditional stabilizer formalism. Such codes are called entanglement-assisted quantum (QUENTA) codes. In this paper, we use algebraic geometry codes to construct three families of QUENTA codes, where one of them has maximal entanglement and is maximal distance separable. At the end, we show that for any asymptotically good tower of algebraic function fields there is an asymptotically good family of maximal entanglement QUENTA codes with nonzero rate, relative minimal distance, and relative amount of entanglement

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Quantum error correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error correcting codes to achieve higher rates than the one obtained via traditional stabilizer formalism. Such codes are called entanglement-assisted quantum (QUENTA) codes. In this paper, we use algebraic geometry codes to construct three families of QUENTA codes, where one of them has maximal entanglement and is maximal distance separable. At the end, we show that for any asymptotically good tower of algebraic function fields there is an asymptotically good family of maximal entanglement QUENTA codes with nonzero rate, relative minimal distance, and relative amount of entanglement
Quantum error correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error correcting codes to achieve higher rates than the one obtained via traditional stabilizer formalism. Such codes are called entanglement-assisted quantum (QUENTA) codes. In this paper, we use algebraic geometry codes to construct two families of QUENTA codes, where one of them has maximal entanglement and is maximal distance separable. In the end, we show that for any asymptotically good tower of algebraic function fields there is an asymptotically good family of maximal entanglement QUENTA codes with nonzero rate, relative minimal distance, and relative amount of entanglement.
Quantum error-correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error-correcting codes to achieve higher rates than the one obtained by means of the traditional stabilizer formalism. Such codes are called entanglement-assisted quantum error-correcting (EAQEC) codes. In this paper, we utilize algebraic geometry codes to construct several families of EAQEC codes derived from the Euclidean and the Hermitian construction. Three families constructed here consist of codes whose quantum Singleton defect is equal to zero, one, or two. We also construct families of EAQEC codes with an encoding rate exceeding the quantum Gilbert-Varshamov bound. Additionally, asymptotically good towers of linear complementary dual codes are used to obtain asymptotically good families of EAQEC codes consuming maximal entanglement. Furthermore, a simple comparison with the quantum Gilbert-Varshamov bound demonstrates that, by utilizing the proposed construction, it is possible to generate an asymptotically family of EAQEC codes that exceeds this bound.
Entanglement-assisted concatenated quantum codes (EACQCs) are constructed by concatenating two entanglement-assisted quantum error-correcting codes (EAQECCs). By selecting the inner and outer component codes carefully, it is able to construct state-of-the-art EACQCs with parameters better than previous quantum codes. In this work, we use almost maximum-distance-separable (MDS) codes and ℏ\hbarℏ-MDS codes as the outer codes to construct EACQCs. Because the range of code length of almost MDS and ℏ\hbarℏ-MDS codes is much more free than that of the commonly used MDS codes. We derive several families of new EACQCs with parameters better than the previously best known EAQECCs and standard quantum error-correcting codes (QECCs) of the same length and net transmissions. Moreover, we demonstrate that EACQCs are with maximal entanglement if both the inner and outer component codes are with maximal entanglement. As a result, we construct three new maximal-entanglement EACQCs which have optimal parameters. In addition, we present several new maximal-entanglement EACQCs whose minimum distance is only one less than the minimum distance of the optimal codes. In particular, we propose two new families of asymptotically good maximal-entanglement EACQCs with explicit constructions by using entanglement-assisted quantum algebraic geometry codes as the outer codes. At last, we prove that EACQCs can attain the quantum Gilbert-Varshamov bound for EAQECCs asymptotically.
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.
Entanglement-assisted quantum error-correcting codes (EAQECCs) can be obtained from arbitrary classical linear codes based on the entanglement-assisted stabilizer formalism, which greatly promoted the development of quantum coding theory. In this paper, we construct several families of [Formula: see text]-ary entanglement-assisted quantum maximum-distance-separable (EAQMDS) codes of lengths [Formula: see text] with flexible parameters as to the minimum distance [Formula: see text] and the number [Formula: see text] of maximally entangled states. Most of the obtained EAQMDS codes have larger minimum distances than the codes available in the literature.