Quantum serial turbo-codes

Abstract

Abstract-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.

Quantum convolutional coding with shared entanglement: general structure

“…The techniques developed in this article represent a useful way for encoding quantum information. The next step should be to combine the theory in this article with Poulin et al's recent theory of quantum serial-turbo coding [20].…”

“…Quantum convolutional coding theory combined with other coding techniques may be one step along the way to finding quantum error-correcting codes that approach the capacity of a noisy quantum communication channel for sending quantum information [13,14,15,16,17,18,19]. Poulin et al have recently incorporated some of this well-developed theory of quantum convolutional coding into a theory of quantum serial-turbo coding [20] with the goal of designing quantum codes that come close to achieving capacity.…”

“…An important practical issue in the design of entanglement-assisted quantum convolutional codes is the characterization of performance in terms of some parameter like code distance or distance spectrum [20]. The focus of this paper is on the construction and design of encoding and decoding circuits for non-CSS entanglement-assisted quantum convolutional codes-not on the performance of these codes.…”

“…We take this approach because entanglement-assisted codes produced from classical codes inherit distance and error-correcting properties from the parent classical codes. If the entanglement-assisted code does not come from a classical code, then one can analyze distance of the code in the standard way as the minimum weight element of the normalizer modulo the stabilizer [22] or with the distance spectrum parameter of Poulin et al [20], but an analysis of this sort is beyond the scope of the present paper.…”

New construction of quantum error-avoiding codes via group representation of quantum stabilizer codes

“…Quantum error-correcting codes (QECCs) have become the most efficient way to combat decoherence [4]. However, QECCs are rather costly in quantum computing resources [5]. To protect a single quantum bit (qubit) of information from suffering general singlequbit errors, at least five qubits are needed [6].…”