As in classical coding theory, turbo product codes (TPCs) through serially concatenated block codes can achieve approximatively Shannon capacity limit and have low decoding complexity. However, special requirements in the quantum setting severely limit the structures of turbo product codes (QTPCs). To design a good structure for QTPCs, we present a new construction of QTPCs with the interleaved serial concatenation of [Formula: see text]-type quantum convolutional codes (QCCs). First, [Formula: see text]-type QCCs are proposed by exploiting the theory of CSS-type quantum stabilizer codes and QCCs, and the description and the analysis of encoder circuit are greatly simplified in the form of Hadamard gates and C-NOT gates. Second, the interleaved coded matrix of QTPCs is derived by quantum permutation SWAP gate definition. Finally, we prove the corresponding relation on the minimum Hamming distance of QTPCs associated with classical TPCs, and describe the state diagram of encoder and decoder of QTPCs that have a highly regular structure and simple design idea.
As in classical coding theory, turbo product codes (TPCs) through serially concatenated block codes can achieve approximatively Shannon capacity limit and have low decoding complexity. However, special requirements in the quantum setting severely limit the structures of turbo product codes (QTPCs). To design a good structure for QTPCs, we present a new construction of QTPCs with the interleaved serial concatenation of [Formula: see text]-type quantum convolutional codes (QCCs). First, [Formula: see text]-type QCCs are proposed by exploiting the theory of CSS-type quantum stabilizer codes and QCCs, and the description and the analysis of encoder circuit are greatly simplified in the form of Hadamard gates and C-NOT gates. Second, the interleaved coded matrix of QTPCs is derived by quantum permutation SWAP gate definition. Finally, we prove the corresponding relation on the minimum Hamming distance of QTPCs associated with classical TPCs, and describe the state diagram of encoder and decoder of QTPCs that have a highly regular structure and simple design idea.
Quantum communication is a growing interdisciplinary field which combines classical communications and quantum mechanics. Quantum error correction coding is one of the key techniques in quantum communication. Nearly all of the classical error correction coding schemes have been transplanted to the domain of quantum communication, and the quantum counterparts of classical error correction coding techniques have been found. Based on the classical turbo product codes (TPCs) which is one of the most outstanding schemes in classical coding region, a new structure of the CSS-type quantum convolutional codes (QCC) as stabilizer sub-code of the quantum turbo product codes (QTPC) is presented. Firstly, CSS-type QCC stabilizer generator is constructed with the help of group theory and the basic principle of stabilizer coders, and the corresponding networks are described. Secondly, the interleaved coded matrix of the QTPC is obtained by quantum permutation SWAP gate definition. Finally, the corresponding relation between the quantum trace distance of QTPC decoding and the distance of classical TPCs decoding is obtained, and the scheme of QTPCs coding and decoding is completed. The coding and decoding of QTPCs have a highly regular structure and a simple design idea, and the networks are easy to realize.
We present a theory of quantum serial turbo-codes and study their performance numerically on a depolarization channel. These codes can be considered as a generalization of classical serial turbo-codes. As their classical cousins, they can be iteratively decoded and with well chosen constituent convolutional codes, we observe an important reduction of the word error rate as the number of encoded qubits increases. Our construction offers several advantages over quantum LDPC codes. First, the Tanner graph used for decoding can be chosen to be free of 4-cycles that deteriorate the performances of iterative decoding. Secondly, the iterative decoder makes explicit use of the code's degeneracy. Finally, there is complete freedom in the code design in terms of length, rate, memory size, and interleaver choice. We address two issues related to the encoding of convolutional codes that are directly relevant for turbo-codes, namely the character of being recursive and non-catastrophic. We define a quantum analogue of a state diagram that provides an efficient way to verify these properties on a given quantum convolutional encoder. Unfortunately, we also prove that all recursive quantum convolutional encoder have catastrophic error propagation. In our constructions, the convolutional codes have thus been chosen to be non-catastrophic and non-recursive. While the resulting families of turbo-codes have bounded minimum distance, from a pragmatic point of view the effective minimum distances of the codes that we have simulated are large enough for not degrading iterative decoding performance up to reasonable word error rates and block sizes.
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.
In the early nineties, the introduction of turbo codes revolutionized classical error correction. The idea was mainly applied on two types of codes: convolutional turbo codes and turbo product codes. The first type of codes was adapted to quantum error correction which initiated the theory of quantum serial turbo codes. In this paper, we present a theory for quantum block turbo codes, the quantum analog of the second type of turbo codes. We describe their iterative decoding algorithm and simulate their performances on a depolarizing channel for different constituent codes.
In this paper, we present a theory of quantum serial turbo codes, describe their iterative decoding algorithm, and study their performances numerically on a depolarization channel. Our construction offers several advantages over quantum low-density parity-check (LDPC) codes. First, the Tanner graph used for decoding is free of 4-cycles that deteriorate the performances of iterative decoding. Second, the iterative decoder makes explicit use of the code's degeneracy. Finally, there is complete freedom in the code design in terms of length, rate, memory size, and interleaver choice. We define a quantum analogue of a state diagram that provides an efficient way to verify the properties of a quantum convolutional code, and in particular, its recursiveness and the presence of catastrophic error propagation. We prove that all recursive quantum convolutional encoders have catastrophic error propagation. In our constructions, the convolutional codes have thus been chosen to be noncatastrophic and nonrecursive. While the resulting families of turbo codes have bounded minimum distance, from a pragmatic point of view, the effective minimum distances of the codes that we have simulated are large enough not to degrade the iterative decoding performance up to reasonable word error rates and block sizes. With well-chosen constituent convolutional codes, we observe an important reduction of the word error rate as the code length increases.
We present a theory of quantum serial turbo-codes, describe their iterative decoding algorithm, and study their performances numerically on a depolarization channel. Our construction offers several advantages over quantum LDPC codes. First, the Tanner graph used for decoding is free of 4-cycles that deteriorate the performances of iterative decoding. Secondly, the iterative decoder makes explicit use of the code's degeneracy. Finally, there is complete freedom in the code design in terms of length, rate, memory size, and interleaver choice. We define a quantum analogue of a state diagram that provides an efficient way to verify the properties of a quantum convolutional code, and in particular its recursiveness and the presence of catastrophic error propagation. We prove that all recursive quantum convolutional encoder have catastrophic error propagation. In our constructions, the convolutional codes have thus been chosen to be non-catastrophic and non-recursive. While the resulting families of turbo-codes have bounded minimum distance, from a pragmatic point of view the effective minimum distances of the codes that we have simulated are large enough not to degrade the iterative decoding performance up to reasonable word error rates and block sizes. With well chosen constituent convolutional codes, we observe an important reduction of the word error rate as the code length increases.