We examine the transformation of noise under a quantum error correcting code (QECC) concatenated repeatedly with itself, by analyzing the effects of a quantum channel after each level of concatenation using recovery operators that are optimally adapted to use error syndrome information from the previous levels of the code. We use the Shannon entropy of these channels to estimate the thresholds of correctable noise for QECCs and find considerable improvements under this adaptive concatenation. Similar methods could be used to increase quantum fault tolerant thresholds.
We study encodings that give the best-known thresholds for the nonzero capacity of quantum channels, i.e., the upper bound for correctable noise, using an entropic approach to calculation of the threshold values. Our results show that Pauli noise is correctable up to the hashing bound. For a depolarizing channel, this approach allows one to achieve a nonzero capacity for a fidelity ͑probability of no error͒ of f = 0.808 70.
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 non-diagonal 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.
Index Terms-Quantum error correction, quantum channels, quantum fault toleranceJ. Fern is with the
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