Performance of quantum convolutional coding with memory constraints

Quantum convolutional error-correcting codes [1] are often said to outperform block codes in terms of code rate and decoding complexity. However, there is no general proof of this. For example many promising examples of convolutional codes turn out to have bad properties, such as being catastrophic and thus spreading errors infinitely. The question we address here is the memory requirements of the encoding and decoding procedures for quantum convolutional codes. We introduce a description of convolutional stabilizer codes (CSC) by Clifford channels with memory that enables us to relate properties of the encoders to simple conditions on the classical representation of the channel. For example, the encoder of a CSC is non-catastrophic, if and only if the channel describing it is strictly forgetful, i.e., the influence of the original memory input vanishes after finitely many uses of the channel. This is in turn true, if and only if the memory-to-memory reduction of the channel is a nilpotent matrix. Using this description, we present a Hamming bound for convolutional stabilizer codes and discuss it with respect to coding under memory constraints. We show that convolutional codes are especially suitable in asymmetric setups, because, to use the power of the convolutional scheme, the decoder requires more memory than the encoder. Finally we illustrate our findings with examples of convolutional stabilizer codes.

[1] Harold Ollivier and Jean-Pierre Tillich. Quantum convolutional codes: fundamentals. preprint, Jan 2004; arXiv:quant-ph/0401134