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arXiv:quant-ph/9712029 · Physical Review A

Quantum convolutional error-correcting codes

H. F. Chau

Published Aug 1, 1998• 38 citations
arXivHTMLPDF

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 can correct one quantum error for every eight consecutive quantum registers.

DOI
10.1103/physreva.58.905

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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.
In this paper, design of quantum convolutional codes and their encoder architectures have been investigated. We claim that rate-1/(n+1) quantum systematic convolutional codes can be constructed from rate-1/n classical nonsystematic convolutional codes, where n is greater than or equal to 2. The free distances (d/sub free/) of proposed rate-1/(n+1) quantum systematic convolutional codes are larger than that of original rate-1/n classical nonsystematic convolutional codes. A quantum convolutional code encoder can be implemented by using quantum linear feed-forward shift registers and quantum exclusive-OR (controlled-NOT: CNOT) gates. A quantum memory may be used as a quantum state delay element of a quantum register. It is also shown that different encoder architectures one needed for quantum nonsuperposition and superposition state inputs. For quantum superposition state input, additional Hadamard gates should be used in conjunction with a quantum convolutional code encoder for quantum nonsuperposition state input.
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-convolutional code was introduced recently as an alternative way to protect vital quantum information. To complete the analysis of the quantum-convolutional code, I report a way to decode certain quantum-convolutional codes based on the classical Viterbi decoding algorithm. This decoding algorithm is optimal for a memoryless channel. I also report three simple criteria to test if decoding errors in a quantum-convolutional code will terminate after a finite number of decoding steps whenever the Hilbert space dimension of each quantum register is a prime power. Finally, I show that certain quantum-convolutional codes are in fact stabilizer codes. And hence, these quantum stabilizer convolutional codes have fault-tolerant implementations.