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arXiv:1405.7479 · Quantum Information Processing

A quantum algorithm for Viterbi decoding of classical convolutional codes

Jon R. Grice, DAVID A. MEYER

Published May 4, 2015• 2 citations
arXivHTMLPDF
DOI
10.1007/s11128-015-1003-3

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A novel decoding algorithm is developed for general quantum convolutional codes. Exploiting useful ideas from classical coding theory, the new decoder introduces two innovations that drastically reduce the decoding complexity compared to the existing quantum Viterbi decoder. First, the new decoder uses an efficient linear-circuits-based mechanism to map a syndrome to a candidate vector, whereas the existing algorithm relies on a non-trivial lookup table. Second, the new algorithm is cleverly engineered such that only one run of the Viterbi algorithm suffices to locate the most-likely error pattern, whereas the existing algorithm must run the Viterbi algorithm many times. The efficiency of the proposed algorithm allows us to simulate and present the first performance curve of a general quantum convolutional code.
Quantum convolutional codes are predicted by many to offer higher error correction performance than quantum block codes of equivalent encoding complexity. However, to decode a quantum convolutional code is challenging, because the decoder does not have a measurement of the received codeword (due to the quantum mechanic rule “measurement destroys quantum states”), but only a measurement of the syndrome. Although quantum Viterbi decoding (QVD) has been proposed, its high complexity makes it rather difficult to implement or simulate. Exploiting useful ideas from classical coding theory, this paper develops a practical quantum syndrome decoder (QSD) by introducing two innovations that drastically reduce the decoding complexity compared to the existing QVD. The new decoder uses an efficient linear-circuits-based mechanism to map a syndrome to a candidate vector, obviating the need of a cumbersome lookup table. It is also cleverly engineered such that only one run of the Viterbi algorithm suffices to locate the most-likely error pattern, rather than have to run many rounds as in the previous QVD algorithm. The efficiency of the new decoder allows us to simulate and present the first performance curve of a general quantum convolutional code.
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.