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arXiv:1004.0174

Efficient ML Decoding for Quantum Convolutional Codes

Peiyu Tan, Jing Li

Published Apr 1, 2010• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1004.0174

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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.
We present a decoding algorithm for quantum convolutional codes that finds the class of degenerate errors with the largest probability conditioned on a given error syndrome. The algorithm runs in time linear with the number of qubits. Previous decoding algorithms for quantum convolutional codes optimized the probability over individual errors instead of classes of degenerate errors. Using Monte Carlo simulations, we show that this modification to the decoding algorithm results in a significantly lower block error rate.
We present a decoding algorithm for quantum convolutional codes that finds the class of degenerate errors with the largest probability conditioned on a given error syndrome. The algorithm runs in time linear with the number of qubits. Previous decoding algorithms for quantum convolutional codes optimized the probability over individual errors instead of classes of degenerate errors. Using Monte Carlo simulations, we show that this modification to the decoding algorithm results in a significantly lower block error rate.
This research proposes a new method of decoding convolutional codes using quantum computers. The proposed method obtains maximum likelihood (ML) estimate of the transmitted codeword using quantum annealing (QA). The performance of the proposed method is assessed by its error performance and compared with the conventional Viterbi decoder on classical computers. The results verify the feasibility of QA for decoding convolutional codes. Furthermore, the execution time of both classical and quantum computers for decoding are compared and discussed.
Quantum error correction codes were introduced as a means to protect quantum information from decoherance and operational errors. Based on their approach to error control, error correcting codes can be divided into two different classes: block codes and convolutional codes. There has been significant development towards finding quantum block codes, since they were first discovered in 1995. In contrast, quantum convolutional codes remained mainly uninvestigated. In this thesis, we develop the stabilizer formalism for quantum convolutional codes. We define distance properties of these codes and give a general method for constructing encoding circuits, given a set of generators of the stabilizer of a quantum convolutional stabilizer code, is shown. The resulting encoding circuit enables online encoding of the qubits, i.e., the encoder does not have to wait for the input transmission to end before starting the encoding process. We develop the quantum analogue of the Viterbi algorithm. The quantum Viterbi algorithm (QVA) is a maximum likehood error estimation algorithm, the complexity of which grows linearly with the number of encoded qubits. A variation of the quantum Viterbi algorithm, the Windowed QVA, is also discussed. Using Windowed QVA, we can estimate the most likely error without waiting for the entire received sequence.
A quantum error correction scheme is proposed to protect a flow of transmitted quantum information. It is largely inspired by the theory of classical convolutional codes and the construction of the linear quantum code for error correction using the stabilizer formalism. Also, we provide an explicit encoding circuit to describe the process of the online encoding and the quantum error propagation. To decode the construction of the code, a quantum Viterbi algorithm is put forward for decoding of quantum signals received at the decoder