Metaplectic anyons, Majorana zero modes, and their computational power

Abstract

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 SO(m)2 Chern-Simons theory. We show how they can arise in a simple scenario for electron fractionalization and give a complete account of their quasiparticles 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, Fm) ⋉ H(2n − 2, Fm), where Sp(2n − 2, Fm) is the symplectic group over the finite field Fm and H(2n − 2, Fm) is the extra special group (also called the (2n − 1)-dimensional Heisenberg group) over Fm. Moreover, the braiding of fundamental quasiparticles can be efficiently simulated classically. However, computing the result of braiding a certain type of composite quasiparticle is #P -hard, although it is not universal for quantum computation because it has a finite braid group image. This 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.PACS numbers:

Majorana zero modes and topological quantum computation

“…SU(2) and all levels 5 and higher also have dense braiding but seem physically impractical. SU (2) 4 is an anomaly; it is potentially related to metaplectic anyonic systems 148 with a proposed realisation, 5,149-151 but braiding alone does not furnish a dense gate set. However, recent unpublished work 152 has demonstrated that SU(2) 4 becomes universal when braiding is combined with interferometric measurement.…”

Qubit representations of the braid groups from generalized Yang–Baxter matrices

“…, σ n−1 . We consider a representation ρ R : B n → End((C 2 ) ⊗(n+1) ) considered in [3]. We define ρ R and express it using the standard operators.…”

“…The matrices U i−1,i,i+1 in [3] correspond to our ρ R (σ i−1 ); we follow their convention for the sake of symmetry. For the remainder of the paper, we take i = 2, .…”

“…Let ν = −1 if m = 3, and ν = +1 if m ≥ 5. Then R (which was denoted by R Y 1 in [3]) is the 8 × 8 gYB matrix…”

“…An interesting class of weakly-integral anyons are those from the metaplectic modular categories related to parfermion zero modes [4]. In [3], the authors considered the braid group representations from the anyon types Y i in the metaplectic modular categories SO(m) 2 , m ≥ 3 odd. But the authors did not exactly identify the images of the resulting qubit representations of the braid groups.…”