Quantum Serial Turbo Codes

Abstract : We present a theory of quantum serial turbo-codes, describe their decoding algorithm, and study their performances 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 there is no guarantee that the resulting families of turbo-codes have a minimum distance growing with the number of encoded qubits, 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.
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David Poulin, Jean-Pierre Tillich, Harold Ollivier. Quantum Serial Turbo Codes. IEEE Transactions on Information Theory, Institute of Electrical and Electronics Engineers, 2009, 55 (6), pp.2776-2798. ⟨10.1109/TIT.2009.2018339⟩. ⟨hal-02120562⟩

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