On Metaplectic Modular Categories and their applications

TL;DR

This paper explores the properties of non-abelian simple objects in metaplectic modular categories, revealing their braid group representations, link invariants, and quantum computational capabilities, with new results on their classical simulability and computational hardness.

Contribution

It systematically analyzes the properties of simple objects in metaplectic modular categories, providing new insights into their braid representations, link invariants, and classical simulation methods.

Findings

Simple objects with quantum dimension √m have finite braid group image representations.

Their link invariants are classically efficiently evaluable.

Objects of dimension 2 have finite braid images but #P-hard link invariants.

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.