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arXiv:quant-ph/9806032 · Physical Review A

Good quantum-convolutional error-correction codes and their decoding algorithm exist

H. F. Chau

Published Sep 1, 1999• 24 citations
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Abstract

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.

DOI
10.1103/physreva.60.1966

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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.
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
We describe the theory of quantum convolutional error correcting codes. These codes are aimed at protecting a flow of quantum information over long distance communication. They are largely inspired by their classical analogs which are used in similar circumstances in classical communication. In this article, we provide an efficient polynomial formalism for describing their stabilizer group, derive an on-line encoding circuit with linear gate complexity and study error propagation together with the existence of on-line decoding. Finally, we provide a maximum likelihood error estimation algorithm with linear classical complexity for any memoryless channel.
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.
We describe a quantum error correction scheme aimed at protecting a flow of quantum information over long distance communication. It is largely inspired by the theory of classical convolutional codes which are used in similar circumstances in classical communication. The particular example shown here uses the stabilizer formalism. We provide an explicit encoding circuit and its associated error estimation algorithm. The latter gives the most likely error over any memoryless quantum channel, with a complexity growing only linearly with the number of encoded qubits.
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.
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.