Graph-theoretic approach to quantum error correction
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Abstract
We investigate a novel class of quantum error correcting codes to correct errors on both qubits and higher-state quantum systems represented as qudits. These codes arise from an original graph-theoretic representation of sets of quantum errors. In this new framework, we represent the algebraic conditions for error correction in terms of edge avoidance between graphs providing a visual representation of the interplay between errors and error correcting codes. Most importantly, this framework supports the development of quantum codes that correct against a predetermined set of errors, in contrast to current methods. A heuristic algorithm is presented, providing steps to develop codes that correct against an arbitrary noisy channel. We benchmark the correction capability of reflexive stabilizer codes for the case of single qubit errors by comparison to existing stabilizer codes that are widely used. In addition, we present two instances of optimal encodings: an optimal encoding for fully correlated noise which achieves a higher encoding rate than previously known, and a minimal encoding for single qudit errors on a four-state system.