Metaplectic Anyons, Majorana Zero Modes, and their Computational Power
Metaplectic Anyons, Majorana Zero Modes, and their Computational Power
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Metaplectic 任意子、马约拉纳零模式及其计算能力
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发表时间：
2012
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通讯作者：
Zhenghan Wang
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作者：
M. Hastings;C. Nayak;Zhenghan Wang
我们介绍和研究一类任意子模型，是一个自然的推广伊辛任意子和马约拉纳费米子零模式。这些模型结合了联合收割机的伊辛任意子扇区与$SO（m）_2$ Chern-Simons理论。我们展示了它们是如何在一个简单的情况下产生电子分馏，并给出了它们的准粒子类型，融合规则和编织的完整说明。我们证明了对于2n个基本准粒子的集合，辫子群的象是有限的，并且是Sp的亚代数表示的一个真子群（2n-2，mathbb{F}_m）l × H（2n-2，mathbb{F}_m）$，其中$Sp（2n-2，mathbb{F}_m）$是有限域$mathbb{F}_m$上的辛群，$H（2n-2，mathbb{F}_m）$是$mathbb{F}_m$上的额外特殊群（也称为$（2n-1）$维海森堡群）。此外，编织的基本准粒子可以有效地模拟经典。然而，计算编织某种类型的复合准粒子的结果是$# P$-困难的，尽管它对于量子计算不是通用的，因为它具有有限的编织群图像。这是一个罕见的拓扑相位的例子，它对于通过编织的量子计算不是通用的，但仍然具有$# P$-硬链接不变量。我们认为，我们的模型是密切相关的，最近的分析发现非阿贝尔的任何性质的量子霍尔系统中的缺陷，推广马约拉纳零模式准一维系统。
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 $Sp(2n-2,mathbb{F}_m)ltimes H(2n-2,mathbb{F}_m)$, 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.