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Quantum Information and Computation

New entanglement-assisted MDS quantum constacyclic codes

Hangyu Liu, Sujuan Huang, Liqi Wang, Shixin Zhu

Published Sep 1, 2024• 0 citations
Publisher

Abstract

Entanglement-assisted quantum error-correcting codes can be seen as a new-type of quantum error-correcting codes and can be constructed from arbitrary linear codes which should not satisfy the dual-containing condition by utilizing shared entangled states between the sender and the receiver in advance. In this paper, we construct several new classes of entanglement-assisted quantum maximum-distance-separable codes from constacyclic codes and cyclic codes by exploiting small pre-shared entangled states, respectively. These codes are new in the sense that they are not covered by the codes available in the literature.

DOI
10.26421/qic24.9-10-4

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The entanglement-assisted (EA) formalism allows arbitrary classical linear codes to transform into entanglement-assisted quantum error correcting codes (EAQECCs) by using pre-shared entanglement between the sender and the receiver. In this work, we propose a decomposition of the defining set of constacyclic codes. Using this method, we construct four classes of q-ary entanglement-assisted quantum MDS (EAQMDS) codes based on classical constacyclic MDS codes by exploiting less pre-shared maximally entangled states. We show that a class of q-ary EAQMDS have minimum distance upper bound greater than 3q−1. Some of them have much larger minimum distance than the known quantum MDS (QMDS) codes of the same length. Most of these q-ary EAQMDS codes are new in the sense that their parameters are not covered by the codes available in the literature.
Entanglement-assisted quantum error-correcting (EAQEC) codes make use of preexisting entanglement between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQEC code from any classical linear code, unlike standard quantum error-correcting codes, which can only be constructed from dual-containing codes. However, the number ccc of pre-shared maximally entangled states is usually calculated by computer search. In this paper, we first give a new formula for calculating the number ccc of pre-shared maximally entangled states. Then, using this formula, we construct three classes of new entanglement-assisted quantum error-correcting maximum-distance-separable ( EAQEC MDS) codes.
Entanglement-assisted quantum error correcting codes (EAQECCs) play a significant role in protecting quantum information from decoherence and quantum noise. Recently, constructing entanglement-assisted quantum maximum distance separable (EAQMDS) codes with flexible parameters has received much attention. In this work, four families of EAQMDS codes with a more general length are presented. And the method of selecting defining set is different from others. Compared with all the previously known results, the EAQMDS codes we constructed have larger minimum distance. All of these EAQMDS codes are new in the sense that their parameters are not covered by the quantum codes available in the literature.
Abstract Entanglement-assisted quantum error correction codes (EAQEECs) can improve the reliability of transmission, which make use of pre-existing entanglement between two parties. In this paper, we construct two classes of entanglement-assisted quantum maximum distance separable (EAQMDS) codes via generalized Reed-Solomon (GRS) codes. Consequently, the results show that some of these EAQMDS codes have much larger minimum distance than ones available in the literature. Meanwhile, many of these EAQMDS codes are new in the sense that the parameters of these codes are not covered by the previously known ones. (Abstract)
he entanglement-assisted stabilizer formalism provides a useful framework for constructing quantum error-correcting codes (QECC), which can transform arbitrary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using pre-shared entanglement between the sender and the receiver. In this paper, we construct five classes of entanglement-assisted quantum MDS (EAQMDS) codes based on classical MDS codes by exploiting one or more pre-shared maximally entangled states. We show that these EAQMDS codes have much larger minimum distance than the standard quantum MDS (QMDS) codes of the same length, and three classes of these EAQMDS codes consume only one pair of maximally entangled states.