nLab topological quantum computation with anyons -- references

Topological quantum computation with anyons

The idea of topological quantum computation via a Chern-Simons theory with anyon braiding defects is due to:

and via a Dijkgraaf-Witten theory (like Chern-Simons theory but with discrete gauge group):

Textbook accounts:

Review:

Focus on abelian anyons:

Realization in experiment:

  • Daniel Nigg, Markus Mueller, Esteban A. Martinez, Philipp Schindler, Markus Hennrich, Thomas Monz, Miguel A. Martin-Delgado, Rainer Blatt,

    Experimental Quantum Computations on a Topologically Encoded Qubit, Science 18 Jul 2014: Vol. 345, Issue 6194, pp. 302-305 (arXiv:1403.5426, doi:10.1126/science.1253742)

    (for quantum error correction)

Simulation of Ising anyons in a lattice of ordinary superconducting qbits:

  • T. Andersen et al. Observation of non-Abelian exchange statistics on a superconducting processor [[arXiv:2210.10255]]

Braid group representations (as topological quantum gates)

On linear representations of braid groups (see also at braid group statistics and interpretation as quantum gates in topological quantum computation):

Review:

in relation to modular tensor categories:

  • Colleen Delaney, Lecture notes on modular tensor categories and braid group representations, 2019 (pdf, pdf)

Braid representations from the monodromy of the Knizhnik-Zamolodchikov connection on bundles of conformal blocks over configuration spaces of points:

and understood in terms of anyon statistics:

Braid representations seen inside the topological K-theory of the braid group‘s classifying space:

See also:

  • R. B. Zhang, Braid group representations arising from quantum supergroups with arbitrary qq and link polynomials, Journal of Mathematical Physics 33, 3918 (1992) (doi:10.1063/1.529840)

As quantum gates for topological quantum computation with anyons:

Introduction and review:

Realization of Fibonacci anyons on quasicrystal-states:

Realization on supersymmetric spin chains:

  • Indrajit Jana, Filippo Montorsi, Pramod Padmanabhan, Diego Trancanelli, Topological Quantum Computation on Supersymmetric Spin Chains [[arXiv:2209.03822]]

See also:


Compilation to braid gate circuits

On approximating (cf. the Solovay-Kitaev theorem) given quantum gates by (i.e. compiling them to) cicuits of anyon braid gates (generally considered for su(2)-anyons and here mostly for universal Fibonacci anyons, to some extent also for non-universal Majorana anyons):

Approximating all topological quantum gates by just the weaves among all braids:

Last revised on November 2, 2023 at 05:46:07. See the history of this page for a list of all contributions to it.