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arXiv:quant-ph/0703113 · Canadian Workshop on Information Theory (CWIT)

Quantum Convolutional BCH Codes

Salah A. Aly, Markus Grassl, Andreas Klappenecker, Martin Rötteler, Pradeep Kiran Sarvepalli

Published Jun 1, 2007• 37 citations
arXivHTMLPDF

Abstract

Quantum convolutional codes can be used to protect a sequence of qubits of arbitrary length against decoherence. We introduce two new families of quantum convolutional codes. Our construction is based on an algebraic method which allows to construct classical convolutional codes from block codes, in particular BCH codes. These codes have the property that they contain their Euclidean, respectively Hermitian, dual codes. Hence, they can be used to define quantum convolutional codes by the stabilizer code construction. We compute BCH-like bounds on the free distances which can be controlled as in the case of block codes, and establish that the codes have non-catastrophic encoders.

DOI
10.1109/cwit.2007.375730

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Quantum convolutional codes can be used to protect a sequence of qubits of arbitrary length against decoherence. We introduce two new families of quantum convolutional codes. Our construction is based on an algebraic method which allows to construct classical convolutional codes from block codes, in particular BCH codes. These codes have the property that they contain their Euclidean, respectively Hermitian, dual codes. Hence, they can be used to define quantum convolutional codes by the stabilizer code construction. We compute BCH-like bounds on the free distances which can be controlled as in the case of block codes, and establish that the codes have non-catastrophic encoders.
Classical Bose–Chaudhuri–Hocquenghem (BCH) codes that contain their (Euclidean or Hermitian) dual codes can be used to construct quantum stabilizer codes; this correspondence studies the properties of such codes. It is shown that a BCH code of lengthnnncan contain its dual code only if its designed distanceδ=O(n)\delta =O(\sqrt {n})δ=O(n​), and the converse is proved in the case of narrow-sense codes. Furthermore, the dimension of narrow-sense BCH codes with small design distance is completely determined, and – consequently – the bounds on their minimum distance are improved. These results make it possible to determine the parameters of quantum BCH codes in terms of their design parameters.
It is conjectured that quantum computers are able to solve certain problems more \nquickly than any deterministic or probabilistic computer. For instance, Shor's algorithm \nis able to factor large integers in polynomial time on a quantum computer. \nA quantum computer exploits the rules of quantum mechanics to speed up computations. \nHowever, it is a formidable task to build a quantum computer, since the \nquantum mechanical systems storing the information unavoidably interact with their \nenvironment. Therefore, one has to mitigate the resulting noise and decoherence \neffects to avoid computational errors. \nIn this dissertation, I study various aspects of quantum error control codes - the key component of fault-tolerant quantum information processing. I present the \nfundamental theory and necessary background of quantum codes and construct many \nfamilies of quantum block and convolutional codes over finite fields, in addition to \nfamilies of subsystem codes. This dissertation is organized into three parts: \nQuantum Block Codes. After introducing the theory of quantum block codes, I \nestablish conditions when BCH codes are self-orthogonal (or dual-containing) \nwith respect to Euclidean and Hermitian inner products. In particular, I derive \ntwo families of nonbinary quantum BCH codes using the stabilizer formalism. I study duadic codes and establish the existence of families of degenerate quantum \ncodes, as well as families of quantum codes derived from projective geometries. \nSubsystem Codes. Subsystem codes form a new class of quantum codes in which \nthe underlying classical codes do not need to be self-orthogonal. I give an \nintroduction to subsystem codes and present several methods for subsystem \ncode constructions. I derive families of subsystem codes from classical BCH and \nRS codes and establish a family of optimal MDS subsystem codes. I establish \npropagation rules of subsystem codes and construct tables of upper and lower \nbounds on subsystem code parameters. \nQuantum Convolutional Codes. Quantum convolutional codes are particularly \nwell-suited for communication applications. I develop the theory of quantum \nconvolutional codes and give families of quantum convolutional codes based \non RS codes. Furthermore, I establish a bound on the code parameters of \nquantum convolutional codes - the generalized Singleton bound. I develop a \ngeneral framework for deriving convolutional codes from block codes and use it \nto derive families of non-catastrophic quantum convolutional codes from BCH \ncodes. \nThe dissertation concludes with a discussion of some open problems.
Classical Bose-Chaudhuri-Hocquenghem (BCH) codes C that contain their dual codes can be used to construct quantum stabilizer codes this chapter studies the properties of such codes. It had been shown that a BCH code of length n which contains its dual code satisfies the bound on weight of any non-zero codeword in C and converse is also true. One impressive difficulty in quantum communication and computation is to protect informationcarrying quantum states against undesired interactions with the environment. To address this difficulty, many good quantum errorcorrecting codes have been derived as binary stabilizer codes. We were able to shed more light on the structure of dual containing BCH codes. These results make it possible to determine the parameters of quantum BCH codes in terms of weight of non-zero dual codeword.
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.