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Efficient decoders for qudit topological codes

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If you have a question about this talk, please contact Mustapha Amrani.

Mathematical Challenges in Quantum Information

For a quantum error correcting code, a decoder is a (classical) algorithm, which given the set of measurement outcomes of the stabiliser generators of the code (the syndrome) outputs a set of unitarites which may be applied to correct the error. Since the mapping between errors and syndromes is not one-to-one, decoders attempt to output an operator which is most likely to correct the error, given the underlying error model.

For the qubit toric code, the most widely used decoder is the Minimum Weight Perfect Matching algorithm. This decoder utilises some very special properties of the qubit toric code, it is not applicable, in general, to other topological codes. In [1] we introduce two decoders for the qudit (d>2) toric code. One of them is a generalisation of the decoder introduced by Bravyi and Haah [2] for the cubic code, and the other is a generalisation of an RG-based algorithm proposed by Duclos-Cianci and Poulin [3]. I will focus on the former in my talk, introducing the decoder, its limitations, and how those limitations can be overcome to produce an efficient and effective decoder for high d qudit toric codes. I will finish my talk by comparing the thresholds achieved with these different decoder strategies with a conjectured optimal threshold for these codes.

[1] H. Anwar, B. Browne, E. T. Campbell, D.E. Browne, Efficient Decoders for Qudit Topological Codes, (to appear on the arxiv shortly). [2] Sergey Bravyi, Jeongwan Haah, Analytic and numerical demonstration of quantum self-correction in the 3D Cubic Code, arXiv:1112.3252 [3] Guillaume Duclos-Cianci, David Poulin, Fast Decoders for Topological Quantum Codes, Phys. Rev. Lett. 104 050504 (2010)

This talk is part of the Isaac Newton Institute Seminar Series series.

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