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Published September 1, 2002 | public
Journal Article Open

Exact performance of concatenated quantum codes

Abstract

When a logical qubit is protected using a quantum error-correcting code, the net effect of coding, decoherence (a physical channel acting on qubits in the codeword) and recovery can be represented exactly by an effective channel acting directly on the logical qubit. In this paper we describe a procedure for deriving the map between physical and effective channels that results from a given coding and recovery procedure. We show that the map for a concatenation of codes is given by the composition of the maps for the constituent codes. This perspective leads us to an efficient means for calculating the exact performance of quantum codes with arbitrary levels of concatenation. We present explicit results for single-bit Pauli channels. For certain codes under the symmetric depolarizing channel, we use the coding maps to compute exact threshold error probabilities for achievability of perfect fidelity in the infinite concatenation limit.

Additional Information

©2002 The American Physical Society Received 31 October 2001; published 13 September 2002 This work was partially supported by the Caltech MURI Center for Quantum Networks and the NSF Institute for Quantum Information. B.R. acknowledges the support of the NSF and thanks J. Preskill and P. Parrilo for insightful discussions.

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Created:
August 21, 2023
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October 13, 2023