基于核磁共振系统的拓扑物态量子模拟

QUANTUM SIMULATION OF TOPOLOGICAL MATTER STATES WITH NMR SYSTEM

张泽

11749028

硕士

物理学

鲁大为

2019-05-14

2019-07-11

哈尔滨工业大学

深圳

本文从量子计算的历史开始讲起，介绍了量子计算的诞生，量子计算以及量子模拟等相关概念，然后我们介绍了利用自旋进行量子计算的核磁共振量子计算的基本理论。之后，我们介绍了拓扑序的产生与发展，以及衍生出的各种拓扑序模型，以及拓扑序模型中的任意子概念及其性质，然后阐述了对任意子进行操作来实现拓扑量子计算的过程。拓扑序的发展过程中面对的一个基本的问题就是如何区分不同的拓扑序。最早的时候，物理学家们通过基态简并度来区分不同的拓扑序。随着研究的深入，人们发现不同的拓扑序也会有相同的基 态简并度。后来物理学家们发现通过模数据（S，T 矩阵）和任意子的融合规则可以区分不同的拓扑序。近来，有研究表明，不同的拓扑序也可以有相同的模数据，这样使得先前大家认为已经近乎完备的区分标准失效了。为了解决这个问题，我们提出了通过半编织操作来区分不同的拓扑序，通过对模型的简化，我们设计了三比特和四比特实验方案，经过半编织操作，量子态会产生一个非平凡的相位因子，之后我们进行了数值模拟，然后在核磁共振量子模拟器上完成了实验。实验结果证明了我们的预言是正确的，我们在实验室上首次证明了 半编织操作是可以观测的。 我们的实验为后面的近一步探索区分拓扑序的的最基本的标准打下坚实的基础。接下来，我们还将继续向前探索，我们可以用特殊的算符来替代之前实验中末态测量的量子态全息的方法。因为在之前的实验中，我们使用的阿贝尔的任意子，我们还可以去尝试设计一个非阿贝尔任意子模型的半编织方案，从而可以将阿贝尔和非阿贝尔的任意子模型统一用半编织实验来区分。现在理论 上计算出的有相同模数据的不同的拓扑序需要 49 个量子比特来验证，我们尚没有技术可以去实现，所以我们可以进一步去寻找更少比特的有相同模数据的不 同拓扑序。

Starting with quantum computing, this thesis introduces the birth of quantum computing, the basic concepts of quantum computing and quantum simulation, and then we introduce the basic theory of nuclear magnetic resonance quantum computing using spin. After that, we introduce the generation and development of topological order, various topological order models, and the concept and properties of arbitrary sub-models in the topological order model. Then we show that we can realize topological quantum computation by manipulating arbitrary sub-models. A basic problem in the development of topological order is how to distinguish different topological order. Earlier, we distinguished different topological orders by ground state degeneracy. With the further research, we find that different topological orders have the same ground state degeneracy. Then we find that different topological orders can be distinguished by the fusion rules of modular data (S, T matrix) and arbitrary elements. Recent studies have shown that different topological orders can also have the same modulus data, which makes the distinction criteria which we previously thought were almost complete invalid. In order to solve this problem, we propose to distinguish different topological orders from data obtained by semi-braiding operation. By simplifying the model, we design three-bit and four-bit experimental schemes. After semi-braiding operation, a non-trivial phase factor will be generated in the quantum state. Then we simulate it and complete the experiment on a nuclear magnetic resonance quantum simulator. The experimental results prove that our prediction is correct and that the semi-braiding operation is observable for the first time in the laboratory. Our experiments lay a solid foundation for the next step in exploring the most basic criteria for distinguishing topological orders. Next, we will continue to explore, we can use special operators to replace the previous experimental method of quantum state holography measurement of the final state. Because in previous experiments, we used Abel's arbitrary sub-model, we can also try to design a semi-braiding scheme of non-Abel's arbitrary sub-model, so that we can unify Abel's and non-Abel's arbitrary sub-model with semi-braiding experiment to distinguish them. Now the theoretically calculated topological order with the same modulus data needs 49 qubits, and we have no technology to implement it, so we can further search for different topological orders with the same modulus data with fewer bits.

量子计算 核磁共振 拓扑序 编织操作 模数据 任意子

quantum computation nuclear magnetism resonance topological order modular data anyon

中文

联合培养

南方科技大学

张泽. 基于核磁共振系统的拓扑物态量子模拟[D]. 深圳. 哈尔滨工业大学,2019.