4 research outputs found
Quantum serial turbo-codes
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
Trellis Decoding And Applications For Quantum Error Correction
Compact, graphical representations of error-correcting codes called trellises are a crucial tool in classical coding theory, establishing both theoretical properties and performance metrics for practical use. The idea was extended to quantum error-correcting codes by Ollivier and Tillich in 2005. Here, we use their foundation to establish a practical decoder able to compute the maximum-likely error for any stabilizer code over a finite field of prime dimension. We define a canonical form for the stabilizer group and use it to classify the internal structure of the graph. Similarities and differences between the classical and quantum theories are discussed throughout. Numerical results are presented which match or outperform current state-of-the-art decoding techniques. New construction techniques for large trellises are developed and practical implementations discussed. We then define a dual trellis and use algebraic graph theory to solve the maximum-likely coset problem for any stabilizer code over a finite field of prime dimension at minimum added cost.
Classical trellis theory makes occasional theoretical use of a graph product called the trellis product. We establish the relationship between the trellis product and the standard graph products and use it to provide a closed form expression for the resulting graph, allowing it to be used in practice. We explore its properties and classify all idempotents. The special structure of the trellis allows us to present a factorization procedure for the product, which is much simpler than that of the standard products.
Finally, we turn to an algorithmic study of the trellis and explore what coding-theoretic information can be extracted assuming no other information about the code is available. In the process, we present a state-of-the-art algorithm for computing the minimum distance for any stabilizer code over a finite field of prime dimension. We also define a new weight enumerator for stabilizer codes over F_2 incorporating the phases of each stabilizer and provide a trellis-based algorithm to compute it.Ph.D