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Quantum Convolutional Codes

H. F. Chau

Published Dec 16, 1997• 2 citations
OpenAlex

Abstract

I report two general methods to construct quantum convolutional codes for N-state quantum systems. Using these general methods, I construct a quantum convolutional code of rate 1/4, which outperforms the five register perfect code.

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I construct a quantum error correction code (QECC) in higher spin systems using the idea of multiplicative group character. Each $N$-state quantum particle is encoded as five $N$-state quantum registers. By doing so, this code can correct any quantum error arising from any one of the five quantum registers. This code generalizes the well-known five qubit perfect code in spin-1/2 systems and is shown to be optimal for higher spin systems. I also report a simple algorithm for encoding. The importance of multiplicative group character in constructing QECCs will be addressed.

arXiv:quant-ph/9702033•Physical Review A•Jul 1, 1997•42 citations
I report two general methods to construct quantum convolutional codes for NNN-state quantum systems. Using these general methods, I construct a quantum convolutional code of rate 1/4, which can correct one quantum error for every eight consecutive quantum registers.
I report two general methods to construct quantum convolutional codes for quantum registers with internal NNN states. Using one of these methods, I construct a quantum convolutional code of rate 1/4 which is able to correct one general quantum error for every eight consecutive quantum registers.
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.
I construct a quantum error correction code (QECC) in higher spin systems using the idea of multiplicative group character. Each NNN-state quantum particle is encoded as five NNN-state quantum registers. By doing so, this code can correct any quantum error arising from any one of the five quantum registers. This code generalizes the well-known five qubit perfect code in spin-1/2 systems and is shown to be optimal for higher spin systems. I also report a simple algorithm for encoding. The importance of multiplicative group character in constructing QECCs will be addressed.