Generalized Performance of Concatenated Quantum Codes -- A Dynamical Systems Approach

Details Generalized Performance of Concatenated Quantum Codes -- A Dynamical Systems Approach Generalized Performance of Concatenated Quantum Codes -- A Dynamical Systems Approach

2004-09-15

arxiv.org/abs/quant-ph/0409084v2

IEEE Trans. on Automatic Control 51:448-459 (March 2006)

Physics

Quantum Physics

12 pages two-column format, no figures, slightly revised version

Scientific paper

10.1109/TAC.2006.871942

We apply a dynamical systems approach to concatenation of quantum error correcting codes, extending and generalizing the results of Rahn et al. [1] to both diagonal and nondiagonal channels. Our point of view is global: instead of focusing on particular types of noise channels, we study the geometry of the coding map as a discrete-time dynamical system on the entire space of noise channels. In the case of diagonal channels, we show that any code with distance at least three corrects (in the infinite concatenation limit) an open set of errors. For Calderbank-Shor-Steane (CSS) codes, we give a more precise characterization of that set. We show how to incorporate noise in the gates, thus completing the framework. We derive some general bounds for noise channels, which allows us to analyze several codes in detail.

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