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Communications in Mathematical Physics

, Volume 330, Issue 1, pp 45–68 | Cite as

On Metaplectic Modular Categories and Their Applications

  • Matthew B. Hastings
  • Chetan Nayak
  • Zhenghan WangEmail author
Article

Abstract

For non-abelian simple objects in a unitary modular category, the density of their braid group representations, the #P-hard evaluation of their associated link invariants, and the BQP-completeness of their anyonic quantum computing models are closely related. We systematically study such properties of the non-abelian simple objects in the metaplectic modular categories SO(m)2 for an odd integer m ≥ 3. The simple objects with quantum dimensions \({\sqrt{m}}\) have finite image braid group representations, and their link invariants are classically efficient to evaluate. We also provide classically efficient simulations of their braid group representations. These simulations of the braid group representations can be regarded as qudit generalizations of the Knill–Gottesmann theorem for the qubit case. The simple objects of dimension 2 give us a surprising result: while their braid group representations have finite images and are efficiently simulable classically after a generalized localization, their link invariants are #P-hard to evaluate exactly. We sharpen the #P-hardness by showing that any sufficiently accurate approximation of their associated link invariants is already #P-hard.

Keywords

Braid Group Simple Object Quantum Circuit Boltzmann Weight Link Invariant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Matthew B. Hastings
    • 1
  • Chetan Nayak
    • 1
    • 2
  • Zhenghan Wang
    • 1
    Email author
  1. 1.Microsoft Station QUniversity of CaliforniaSanta BarbaraUSA
  2. 2.Department of PhysicsUniversity of CaliforniaSanta BarbaraUSA

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