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Topological quantum computation within the anyonic system the Kauffman–Jones version of SU(2) Chern–Simons theory at level 4

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Abstract

By braiding and measuring anyons, we realize irrational qubit and qutrit phase gates within the anyonic system the Kauffman-Jones version of SU(2) Chern-Simons theory at level 4. We obtain universality on 1-qubit and 1-qutrit gates. In the qubit case, we also provide a protocol for realizing the controlled NOT gate, thus leading to universality on n-qubit gates.

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Acknowledgments

The author is delighted to heartedly thank Michael Freedman for kind, exciting and generous guidance and for many inspiring discussions. Working with him and Station Q on these topics is a great chance and a fabulous experience. She thanks Matthew Hastings for enlightening discussions. She is also quite pleased to thank Bela Bauer, Stephen Bigelow, Parsa Bonderson, Meng Cheng, Spiros Michalakis, Chetan Nayak and Zhenghan Wang for helpful discussions.

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Authors and Affiliations

  1. Paris, France

    Claire Levaillant

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  1. Claire Levaillant
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Correspondence to Claire Levaillant.

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Claire Levaillant worked with the Microsoft Research Station Q team.

Appendices

Appendix 1: Mathematica notebook for the useful symbols

figure ba

The Mathematica command Bu[i, j, k, l, m, n] computes the unitary 6j-symbol

$$\begin{aligned} \left\{ \begin{array}{ccc}i&{}j&{}m\\ k&{}l&{}n\end{array}\right\} ^u \end{aligned}$$

The Mathematica command r[a, b, c] computes the phase associated with the R-move

figure bb

Appendix 2: The Freedman fusion operations

The operation was originally introduced for the qutrit (figure to the LHS).

The same operation can also be applied to the qubit (figure to the RHS).

figure bc

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Levaillant, C. Topological quantum computation within the anyonic system the Kauffman–Jones version of SU(2) Chern–Simons theory at level 4. Quantum Inf Process 15, 1135–1188 (2016). https://doi.org/10.1007/s11128-016-1249-4

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