A Hierarchy of Anyon Models Realised by Twists in Stacked Surface Codes

T. R. Scruby and D. E. Browne

Dept. of Physics and Astronomy, University College London, London WC1E 6BT, UK

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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.

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[1] Paul Webster, Michael Vasmer, Thomas R. Scruby, and Stephen D. Bartlett, "Universal fault-tolerant quantum computing with stabilizer codes", Physical Review Research 4 1, 013092 (2022).

[2] Héctor Bombín, Chris Dawson, Ryan V. Mishmash, Naomi Nickerson, Fernando Pastawski, and Sam Roberts, "Logical Blocks for Fault-Tolerant Topological Quantum Computation", PRX Quantum 4 2, 020303 (2023).

[3] Paul Webster and Stephen D. Bartlett, "Fault-tolerant quantum gates with defects in topological stabilizer codes", Physical Review A 102 2, 022403 (2020).

[4] Nora M. Bauer, Elias Kokkas, Victor Ale, and George Siopsis, "Non-Abelian anyons with Rydberg atoms", Physical Review A 107 6, 062407 (2023).

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