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IEEE International Conference on Quantum Computing and Engineering (QCE)

Quantum Convolutional Codes Constructed from Classical Self-Orthogonal Convolutional Codes

Vahid Nourozi, David G. M. Mitchell

Published Aug 30, 2025• 3 citations
Publisher

Abstract

In this paper, we construct new quantum convolutional codes (QCCs) from classical self-orthogonal convolutional (CSOC) codes. We develop a methodology that achieves the required code orthogonality and allows for code performance optimization, generalizing to the full algebraic structure. This approach leverages the significant body of work in classical CSOCs to achieve a specified free distance.

DOI
10.1109/qce65121.2025.10482

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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.
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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.
A new class of quantum stabilizer codes is presented.The problem of finding stabilizer quantum-error-correcting codes can be transformed into the problem of finding self-orthogonal codes over the Galois field GF(4) under a trace inner product.Based on this connection,the necessary and sufficient condition is proved under which classical convolutional codes over GF(4) are self-orthogonal with respect to the trace inner product,and then corresponding codes satisfying the condition are found.Tail-biting codes with a simple block structure are obtained by tail-biting transformation,which are self-orthogonal with respect to the trace inner product.Finally,relative quantum stabilizer codes are constructed.The code construction is simple,and the minimum distance of the code approaches the upper bound.