Metaplectic anyons, Majorana zero modes, and their computational power

We introduce and study a class of anyon models that are a natural generalization of Ising anyons and Majorana fermion zero modes. These models combine an Ising anyon sector with a sector associated with $\mathit{\text{SO}}{\left(m\right)}_2$ Chern-Simons theory. We show how they can arise in a simple scenario for electron fractionalization and give a complete account of their quasiparticle types, fusion rules, and braiding. We show that the image of the braid group is finite for a collection of $2n$ fundamental quasiparticles and is a proper subgroup of the metaplectic representation of $Sp(2n-2,{\mathbb{F}}_m)⋉H(2n-2,{\mathbb{F}}_m)$, where $Sp(2n-2,{\mathbb{F}}_m)$ is the symplectic group over the finite field ${\mathbb{F}}_m$ and $H(2n-2,{\mathbb{F}}_m)$ is the extra special group (also called the $(2n-1)$-dimensional Heisenberg group) over ${\mathbb{F}}_m$. Moreover, the braiding of fundamental quasiparticles combined with a restricted set of measurements can be efficiently simulated classically. However, computing the result of braiding a certain type of composite quasiparticle followed by fusion into the identity is $\#P$-hard. It is not universal for quantum computation because it has a finite braid group image. This is a rare example of a topological phase that is not universal for quantum computation through braiding but nevertheless has $\#P$-hard link invariants. We argue that our models are closely related to recent analyses finding non-Abelian anyonic properties for defects in quantum Hall systems, generalizing Majorana zero modes in quasi-1D systems.