On Metaplectic Modular Categories and their applications

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 metaplectic modular categories, which are unitary modular categories with fusion rules of SO(m)_2 for an odd integer m \geq 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 simulation 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.

Many minor revisions including a definition of simulation in the sense of stabilizer formalism. To appear in Comm. Math. Phys

Cited by in corpus (15)

• Universal Quantum Computation with Metaplectic Anyons
• Quantum Computing with Parafermions
• Classification of Metaplectic Modular Categories
• From Three Dimensional Manifolds to Modular Tensor Categories
• Classification of Grothendieck rings of complex fusion categories of multiplicity one up to rank six
• Quantum Fourier Transforms and the Complexity of Link Invariants for Quantum Doubles of Finite Groups
• Multipartite $d-$level GHZ bases associated with generalized braid matrices
• Braid Representations from Unitary Braided Vector Spaces
• Ternary Logic Design in Topological Quantum Computing
• Classification of Metaplectic Fusion Categories
• Dihedral twist liquid models from emergent Majorana fermions
• Topological quantum compilation of metaplectic anyons based on the genetic optimized algorithms
• Partial fillings of the bosonic $E_8$ quantum Hall state
• SO(n) Affleck-Kennedy-Lieb-Tasaki states as conformal boundary states of integrable SU(n) spin chains
• Signatures of Parafermion Zero Modes in Fractional Quantum Hall-Superconductor Heterostructures