低能下的环面代数与逻辑算符
Torus algebra and logical operators at low energy
AI总结：
本文基于模张量范畴构造并证明半单的环面代数，阐明其任意子跳跃与低能态模解释，并给出用任意子移动实现任意逻辑算符的公式，支撑拓扑 qudit 操控。
摘要：
给定一个模张量范畴 $\mathscr{C}$，我们构造了一个结合代数 $\mathrm{Tor({\mathscr{C}})}$，并称之为环面代数。通过显式构造所有简单模，我们证明了环面代数是半单的。假设由 $\mathscr{C}$ 描述的拓扑有序相被放置在环面上。从物理上看，$\mathrm{Tor({\mathscr{C}})}$ 上的每个简单模都由环面上带有一个任意子激发的低能态组成，或者等价地，由穿孔环面上的基态组成，其中任意子被穿孔所包围。$\mathrm{Tor({\mathscr{C}})}$ 中的元素在物理上可解释为环面上的任意子跳跃过程。我们给出了精确公式，说明环面低能态上的任意逻辑算符如何能够通过在环面上移动任意子来实现。因此，我们的工作提供了一个理论方案：环面上的低能态可以充当拓扑量子位（qudit），并且人们可以通过四处移动任意子来任意操控它们。
英文摘要：
Given a modular tensor category $\mathscr{C}$, we construct an associative algebra $\mathrm{Tor({\mathscr{C}}})$, which we call the torus algebra. We prove that the torus algebra is semisimple by explicitly constructing all the simple modules. Suppose that a topological ordered phase described by $\mathscr{C}$ is put on a torus. Physically, each simple module over $\mathrm{Tor({\mathscr{C}}})$ consists of the low energy states on the torus with one anyon excitation, or equivalently, the ground states on a punctured torus where the anyon is enclosed by the puncture. Elements in $\mathrm{Tor({\mathscr{C}}})$ can be physically interpreted as anyon hopping processes on the torus. We give the precise formula how an arbitrary logical operator on the low energy states on a torus can be realized by moving anyons on the torus. Our work thus provides a theoretical proposal that the low energy states on a torus can serve as topological qudits and one can arbitrarily manipulate them by moving anyons around.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
