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IEEE International Symposium on Information Theory (ISIT)

Quantum Convolutional Codes Derived from Generalized Reed-Solomon Codes

Salah A. Aly, Andreas Klappenecker, Pradeep Kiran Sarvepalli

Published Jun 1, 2007• 10 citations
Publisher

Abstract

Convolutional stabilizer codes promise to make quantum communication more reliable with attractive online encoding and decoding algorithms. This paper introduces a new approach to convolutional stabilizer codes based on direct limit constructions. A quantum Singleton bound for pure convolutional stabilizer codes is given. A familiy of quantum convolutional codes is derived from generalized Reed-Solomon codes. These codes are shown to be optimal with respect to the (quantum) Singleton bound.

DOI
10.1109/isit.2007.4557326

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Convolutional stabilizer codes promise to make quantum communication more reliable with attractive online encoding and decoding algorithms. This paper introduces a new approach to convolutional stabilizer codes based on direct limit constructions. A quantum Singleton bound for pure convolutional stabilizer codes is given. A familiy of quantum convolutional codes is derived from generalized Reed-Solomon codes. These codes are shown to be optimal with respect to the (quantum) Singleton bound.
Convolutional stabilizer codespromise tomake quantumcommunication morereliable withattractive online encoding anddecoding algorithms. Thispaperintroduces anew approach toconvolutional stabilizer codesbasedondirect limit constructions. A quantumSingleton boundforpureconvolutional stabilizer codesisgiven. A familiy ofquantumconvolutional codesisderived fromgeneralized Reed-Solomon codes. These codesareshowntobeoptimal withrespect tothe(quantum) Singleton bound. I.INTRODUCTION A keyobstacle tothecommunication ofquantuminfor- mationisdecoherence, thespontaneous interaction ofthe environment withtheinformation-carrying quantumsystem. Theprotection ofquantuminformation withquantumerror- correcting codestoreduce orperhaps nearly eliminate the impact ofdecoherence hasledtoahighly developed theory of quantum error-correcting block codes. Somewhat surprisingly, quantum convolutional codeshavereceived less attention. Ollivier andTillich developed thestabilizer framework for quantumconvolutional codes, andaddressed encoding and decoding aspects ofsuchcodes(11), (12). Almediaand Palazzo constructed aconcatenated convolutional codeofrate 1/4withmemorym = 3(1). Forney andGuhaconstructed quantum convolutional codes withrate1/3(3). Also, inajoint workwithGrassl, theyderived rate(n- 2)/n convolutional stabilizer codes(2). Grassl andRotteler constructed quantum convolutional codesfromproduct codes(6), andtheygavea general algorithm toobtain non-catastrophic encoders (5). Inthis paper, wegiveanewapproach toquantum convo- lutional codes based onadirect limit construction, generalize someofthepreviously knownresults. We proveaSingleton boundforpureconvolutional stabilizer codes. We construct afamily ofquantumconvolutional codesbasedonclassical generalized Reed-Solomon codes. Thequantum convolutional codesderived fromthegeneralized Reed-Solomon codesare showntobeoptimal. ~~~~~~q 0< i<p.We canassociate tothegenerator matrix G(D)its semi-infinite coefficient matrix
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.
A quantum error correction scheme is proposed to protect a flow of transmitted quantum information. It is largely inspired by the theory of classical convolutional codes and the construction of the linear quantum code for error correction using the stabilizer formalism. Also, we provide an explicit encoding circuit to describe the process of the online encoding and the quantum error propagation. To decode the construction of the code, a quantum Viterbi algorithm is put forward for decoding of quantum signals received at the decoder