Abstract
Braiding defects in topological stabiliser codes can be used to fault-tolerantly implement logical operations. Twists are defects corresponding to the end-points of domain walls and are associated with symmetries of the anyon model of the code. We consider twists in multiple copies of the 2d surface code and identify necessary and sufficient conditions for considering these twists as anyons: namely that they must be self-inverse and that all charges which can be localised by the twist must be invariant under its associated symmetry. If both of these conditions are satisfied the twist and its set of localisable anyonic charges reproduce the behaviour of an anyonic model belonging to a hierarchy which generalises the Ising anyons. We show that the braiding of these twists results in either (tensor products of) the S gate or (tensor products of) the CZ gate. We also show that for any number of copies of the 2d surface code the application of H gates within a copy and CNOT gates between copies is sufficient to generate all possible twists.
Highlights
The question of which quantum gates can be performed fault-tolerantly in a particular quantum error correcting code is of vital importance if we wish to use the code in quantum computation
We show in appendix C that the braiding matrices of the models depend only on φ and not on f
In this appendix we show that column permutations that preserve the symmetry of a symmetric 2k × 2k Hadamard matrix must alter the trace by either 0 or ±2k
Summary
Dept. of Physics and Astronomy, University College London, London WC1E 6BT, UK arXiv:1908.07353v3 [quant-ph] 3 Mar 2020. We consider twists in multiple copies of the 2d surface code and identify necessary and sufficient conditions for considering these twists as anyons: namely that they must be self-inverse and that all charges which can be localised by the twist must be invariant under its associated symmetry. If both of these conditions are satisfied the twist and its set of localisable anyonic charges reproduce the behaviour of an anyonic model belonging to a hierarchy which generalises the Ising anyons. We show that for any number of copies of the 2d surface code the application of H gates within a copy and CNOT gates between copies is sufficient to generate all possible twists
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