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.Comment: 24 pages, 15 figures, Published versio