Generalized Performance of Concatenated Quantum Codes—A Dynamical Systems Approach
Fern, J.
Kempe, J.
Simic, S.N.
Sastry, S.
This paper appears in: Automatic Control, IEEE Transactions on
Publication Date: March 2006
Volume: 51,
Issue: 3
On page(s): 448- 459
ISSN: 0018-9286
Digital Object Identifier: 10.1109/TAC.2006.871942
Current Version Published: 2006-03-13
Abstract
We apply a dynamical systems approach to concatenation of quantum error correcting codes, extending and generalizing the results of Rahn 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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