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 $Sp(2n-2,{\mathbb{F}}_m)⋉H(2n-2,{\mathbb{F}}_m)$, 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 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.

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• Quantum phases of a chain of strongly interacting anyons
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• Disentangling (2+1)d Topological States of Matter with the Entanglement Negativity
• Quantum Fourier Transforms and the Complexity of Link Invariants for Quantum Doubles of Finite Groups
• Low-temperature environments for quantum computation and quantum simulation
• Parafermion stabilizer codes
• Chern-Simons-Higgs transitions out of Topological Superconducting Phases
• A new set of generators and a physical interpretation for the SU(3) finite subgroup D(9,1,1;2,1,1)
• Braid Representations from Unitary Braided Vector Spaces
• Ternary Logic Design in Topological Quantum Computing
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• Sub-symmetry Protected Topology in Topological Insulators and Superconductors
• Dihedral twist liquid models from emergent Majorana fermions
• Parameter-dependent Gaussian $(z,N)$-generalized Yang-Baxter operators
• ${\ell }_1$-norm and entanglement in screening out braiding from Yang-Baxter equation associated with ${\mathbb{Z}}_3$ parafermion
• Topological quantum compilation of metaplectic anyons based on the genetic optimized algorithms
• Partial fillings of the bosonic $E_8$ quantum Hall state
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• Complete set of unitary irreps of Discrete Heisenberg Group $H{W}_{2^s}$
• Signatures of Parafermion Zero Modes in Fractional Quantum Hall-Superconductor Heterostructures
• Dirac topological insulator in the d${}_{z^2}$ manifold of a honeycomb oxide
• SO(n) Affleck-Kennedy-Lieb-Tasaki states as conformal boundary states of integrable SU(n) spin chains
• Phase diagram of an extended parafermion chain