论文标题
ZX-Calculus作为拓扑量子计算的语言
The ZX-calculus as a Language for Topological Quantum Computation
论文作者
论文摘要
单一融合类别正式化拓扑量子计算的代数理论。这些类别自然地富含希尔伯特空间类别的子类别，并且通过查看此子类别，可以确定一个用于实现量子计算的生成器的集合。我们代表斐波那契和伊辛模型的这种生成器，即Qubits和相关的编织组表示的编码，以及ZX-Calculus，并表明，在这两种情况下，Yang-baxter方程都直接连接到完全ZX-Calculus中的重要规则，称为P-allculus，以p-callus为p-lule，以使一个阶段相互互换，以互相构造，以适应一个定义。在Ising情况下，这将减少到熟悉的规则，该规则将Hadamard Gate的两个不同的Euler分解为$π/2 $ Z-和X相门，而在FibonACCI情况下，我们给出了先前未经考虑的涉及黄金比率的P规则的精确解决方案。我们通过给出了斐波那契的单量辫子方程以及iSing anyons的单量辫子方程以及单量和双Qubit的编织方程来证明这些表示的实用性。此外，我们提供了一个完全图形的过程，用于使用斐波那契Anyons的ZX代理模拟和简化辫子。
Unitary fusion categories formalise the algebraic theory of topological quantum computation. These categories come naturally enriched in a subcategory of the category of Hilbert spaces, and by looking at this subcategory, one can identify a collection of generators for implementing quantum computation. We represent such generators for the Fibonacci and Ising models, namely the encoding of qubits and the associated braid group representations, with the ZX-calculus and show that in both cases, the Yang-Baxter equation is directly connected to an important rule in the complete ZX-calculus known as the P-rule, which enables one to interchange the phase gates defined with respect to complementary bases. In the Ising case, this reduces to a familiar rule relating two distinct Euler decompositions of the Hadamard gate as $π/2$ Z- and X-phase gates, whereas in the Fibonacci case, we give a previously unconsidered exact solution of the P-rule involving the Golden ratio. We demonstrate the utility of these representations by giving graphical derivations of the single-qubit braid equations for Fibonacci anyons and the single- and two-qubit braid equations for Ising anyons. We furthermore present a fully graphical procedure for simulating and simplifying braids with the ZX-representation of Fibonacci anyons.
