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International Journal of Quantum Information

New optimal quantum convolutional codes

Shixin Zhu, Liqi Wang, Xiaoshan Kai

Published Apr 1, 2015• 6 citations
Publisher

Abstract

One of the most challenges to prove the feasibility of quantum computers is to protect the quantum nature of information. Quantum convolutional codes are aimed at protecting a stream of quantum information in a long distance communication, which are the correct generalization to the quantum domain of their classical analogs. In this paper, we construct some classes of quantum convolutional codes by employing classical constacyclic codes. These codes are optimal in the sense that they attain the Singleton bound for pure convolutional stabilizer codes.

DOI
10.1142/s0219749915500197

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One of the most challenges to prove the feasibility of quantum computers is to protect the quantum nature of information. Quantum convolutional codes are aimed at protecting a stream of quantum information in a long distance communication, which are the correct generalization to the quantum domain of their classical analogs. In this paper, we construct some classes of quantum convolutional codes by employing classical constacyclic codes. These codes are optimal in the sense that they attain the Singleton bound for pure convolutional stabilizer codes.
We describe the theory of quantum convolutional error correcting codes. These codes are aimed at protecting a flow of quantum information over long distance communication. They are largely inspired by their classical analogs which are used in similar circumstances in classical communication. In this article, we provide an efficient polynomial formalism for describing their stabilizer group, derive an on-line encoding circuit with linear gate complexity and study error propagation together with the existence of on-line decoding. Finally, we provide a maximum likelihood error estimation algorithm with linear classical complexity for any memoryless channel.
Quantum maximum-distance-separable (MDS) codes form an important family of quantum codes. Quantum MDS codes are optimal, which can attain the quantum Singleton bound. In this paper, using some classes of constacyclic codes and Hermitian constructions, we propose the constructions of optimal quantum convolutional codes and optimal asymmetric quantum codes in the sense of the quantum Singleton bound. Compared with the codes obtained in the literature, the parameters of the constructed quantum codes are new as well as the distance dz of optimal asymmetric quantum codes [[n,k,dz/dx]]q2 constructed in this paper is larger.
We describe a quantum error correction scheme aimed at protecting a flow of quantum information over long distance communication. It is largely inspired by the theory of classical convolutional codes which are used in similar circumstances in classical communication. The particular example shown here uses the stabilizer formalism. We provide an explicit encoding circuit and its associated error estimation algorithm. The latter gives the most likely error over any memoryless quantum channel, with a complexity growing only linearly with the number of encoded qubits.