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Constructions ofQuantumConvolutional Codes

Martin Rötteler

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

We address theproblems ofconstructing quantum convolutional codes(QCCs)andofencoding them.Thefirst construction isaCSS-type construction whichallows ustofind QCCsofrate2/4.Thesecondconstruction yields a quantum convolutional codebyapplying a product codeconstruction toanarbitrary classical convolutional codeandanarbitrary quantumblockcode.We showthattheresulting codeshave highly structured andefficient encoders. Furthermore, weshow thattheresulting quantumcircuits havefinite depth, independent ofthelengths oftheinput stream, andshowthatthis depthis polynomial inthedegree andframesize ofthecode.

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We address theproblems ofconstructing quantum convolutional codes(QCCs)andofencoding them.Thefirst construction isaCSS-type construction whichallows ustofind QCCsofrate2/4.Thesecondconstruction yields a quantum convolutional codebyapplying a product codeconstruction toanarbitrary classical convolutional codeandanarbitrary quantumblockcode.We showthattheresulting codeshave highly structured andefficient encoders. Furthermore, weshow thattheresulting quantumcircuits havefinite depth, independent ofthelengths oftheinput stream, andshowthatthis depthis polynomial inthedegree andframesize ofthecode.
We address the problems of constructing quantum convolutional codes (QCCs) and of encoding them. The first construction is a CSS-type construction which allows us to find QCCs of rate 2/4. The second construction yields a quantum convolutional code by applying a product code construction to an arbitrary classical convolutional code and an arbitrary quantum block code. We show that the resulting codes have highly structured and efficient encoders. Furthermore, we show that the resulting quantum circuits have finite depth, independent of the lengths of the input stream, and show that this depth is polynomial in the degree and frame size of the code.
Quantum convolutional codes can be used to protect streams of quantum bits against errors when sending them along a quantum channel.We review the underlying error model, conditions for quantum convolutional codes, and present a product code construction that yields families of quantum convolutional codes.Concerning the structure of encoding circuits for general quantum convolutional codes, we review the notion of catastrophic errors and show that each quantum convolutional code has encoders which are non-catastrophic, i. e., which have good error propagation properties.We present an algorithm to compute a quantum circuit which can be used to perform non-catastrophic encoding and inverse encoding in constant depth and give several examples for illustration.
Quantum convolutional coding is a technique for encoding a stream of quantum information before transmitting it over a noisy quantum channel. Two important goals in the design of quantum convolutional encoders are to minimize the memory required by them and to avoid the catastrophic propagation of errors. In a previous paper, we determined minimal-memory, noncatastrophic, polynomial-depth encoders for a few exemplary quantum convolutional codes. In this paper, we elucidate a general technique for finding an encoder of an arbitrary quantum convolutional code such that the encoder possesses these desirable properties. We also provide an elementary proof that these encoders are nonrecursive. Finally, we apply our technique to many quantum convolutional codes from the literature.
We present a construction of self-orthogonal codes using product codes. From the resulting codes, one can construct both block quantum error-correcting codes and quantum convolutional codes. We show that from the examples of convolutional codes found, we can derive ordinary quantum error-correcting codes using tail-biting with parameters [42N, 24N, 3]2. While it is known that the product construction cannot improve the rate in the classical case, we show that this can happen for quantum codes: we show that a code [15, 7, 3]2is obtained by the product of a code [5,1,3]2with a suitable code
Quantum convolutional coding is a technique for encoding a stream of quantum information before transmitting it over a noisy quantum channel. Two important goals in the design of quantum convolutional encoders are to minimize the memory required by them and to avoid the catastrophic propagation of errors. In a previous paper, we determined minimal-memory, noncatastrophic, polynomial-depth encoders for a few exemplary quantum convolutional codes. In this paper, we elucidate a general technique for finding an encoder of an arbitrary quantum convolutional code such that the encoder possesses these desirable properties. We also provide an elementary proof that these encoders are nonrecursive. Finally, we apply our technique to many quantum convolutional codes from the literature. © 1963-2012 IEEE.