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QuantumConvolutional Codes Derived from GeneralizedReed-Solomon Codes

Salah A. Aly, Andreas Klappenecker, Pradeep KiranSarvepalli

Published Jan 1, 2007• 0 citations
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Abstract

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

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Asia-Pacific Conference on Communications (APCC)•Dec 13, 2005•2 citations
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
Quantum error correction codes were introduced as a means to protect quantum information from decoherance and operational errors. Based on their approach to error control, error correcting codes can be divided into two different classes: block codes and convolutional codes. There has been significant development towards finding quantum block codes, since they were first discovered in 1995. In contrast, quantum convolutional codes remained mainly uninvestigated. In this thesis, we develop the stabilizer formalism for quantum convolutional codes. We define distance properties of these codes and give a general method for constructing encoding circuits, given a set of generators of the stabilizer of a quantum convolutional stabilizer code, is shown. The resulting encoding circuit enables online encoding of the qubits, i.e., the encoder does not have to wait for the input transmission to end before starting the encoding process. We develop the quantum analogue of the Viterbi algorithm. The quantum Viterbi algorithm (QVA) is a maximum likehood error estimation algorithm, the complexity of which grows linearly with the number of encoded qubits. A variation of the quantum Viterbi algorithm, the Windowed QVA, is also discussed. Using Windowed QVA, we can estimate the most likely error without waiting for the entire received sequence.
AbstractA good quantum code corrects for a linear number of errors. It has been shownthat random codes attains the QGVB with a relative distance H −1q 2 ((1−R)/2) wherethe code is q-ary and R is the rate of the code (number of encoded systems / blocklength). However,randomcodeshavelittlestructure. Inthispaper,westudyafamilyofconcatenatedq-arystabilizercodes. Eachoftheinnercodesisarandomrate1q-arystabilizer code of block length n. The outer code is a quantum MDS code with blocklengthq n andalphabetsizeq n ,anarbitraryrateR≤1,anddistanceN(1−R)/2+1thatmeets the Quantum Singleton bound. Fixing the outer code rate and letting n grow,the concatenated stabilizer code has a distance that almost surely attains the QGVB.This partially generalizes Thommesen’s result, where heshowed thatthedistanceof aconcatenated code with a Reed-Solomon outer code and random inner linear codes ofarbitraryrateattainstheGilbert-Varshamovboundalmostsurely. 1 Introduction A family of q-ary quantum codes of increasing block length is defined to be good if the ratioof its distance to its block length approaches a non-zero constant. Designing good quantumcodes is highly nontrivial, just as it is in the classical case. The quantum Gilbert-Varshamovbound (QGVB) is a lower bound on an achievable relative distance of a quantum code ofa fixed rate. Explicit constructions of good quantum codes for q ≤ 7 have been studied[1, 2, 3], but they all fail to satisfy the QGVB. The QGVB bound is attainable for the familyof all random quantum codes [4], the family of random stabilizer codes [5], and the familyof random nondegenerate stabilizer codes [6], and the family of random degenerate stabilizercodes [6]. We show that concatenated quantum codes, with a quantum outer code havinga known structure and being efficiently decodable, and randomly chosen independent innerquantum codes also attains the QGVB. Thus our family of random quantum codes has morestructure than previously studied examples.In this paper, we generalize a special case of Thommesen’s result [7] to the quantumcase. He showed that a code from the family of binary concatenated codes made with a1
Rate-(n-2)/n unrestricted and CSS-type quantum convolutional codes with up to 4096 states and minimum distances up to 10 are constructed as stabilizer codes from classical self-orthogonal rate-1/n F4-linear and binary linear convolutional codes, respectively. These codes generally have higher rate and less decoding complexity than comparable quantum block codes or previous quantum convolutional codes. Rate-(n-2)/n block stabilizer codes with the same rate and error-correction capability and essentially the same decoding complexity are derived from these convolutional codes via tail-biting
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