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# 数学代写|微分拓扑作业代写differential topology代考|State Spaces and Fiber Bundles

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## 数学代写|微分拓扑作业代写differential topology代考|Qbits

In quantum computing one often talks about qbits. As opposed to an ordinary bit, which takes either the value 0 or the value 1 representing “false” and “true” respectively, a qbit, or quantum bit, is represented by a complex linear combination “superposition” in the physics parlance of two states. The two possible states of a bit are then often called $|0\rangle$ and $|1\rangle$, and so a qbit is represented by the “pure qbit state” $\alpha|0\rangle+\beta|1\rangle$, where $\alpha$ and $\beta$ are complex numbers and $|\alpha|^{2}+|\beta|^{2}=1$ (since the total probability is 1 , the numbers $|\alpha|^{2}$ and $|\beta|^{2}$ are interpreted as the probabilities that a measurement of the qbit will yield $|0\rangle$ and $|1\rangle$ respectively.
Note that the set of pairs $(\alpha, \beta) \in \mathbf{C}^{2}$ satisfying $|\alpha|^{2}+|\beta|^{2}=1$ is just another description of the sphere $S^{3} \subseteq \mathbf{R}^{4}=\mathbf{C}^{2}$. In other words, a pure qbit state is a point $(\alpha, \beta)$ on the sphere $S^{3}$.

However, for various reasons phase changes are not important. A phase change is the result of multiplying $(\alpha, \beta) \in S^{3}$ by a unit-length complex number. That is, if $z=e^{i \theta} \in S^{1} \subseteq \mathbf{C}$, the pure qbit state $(z \alpha, z \beta)$ is a phase shift of $(\alpha, \beta)$, and these should be identified. The state space is what you get when you identify each pure qbit state with the other pure qbit states you get by a phase change.

So, what is the relation between the space $S^{3}$ of pure qbit states and the state space? It turns out that the state space may be identified with the two-dimensional sphere $S^{2}$ (Figure $1.12$ ), and the projection down to state space $\eta: S^{3} \rightarrow S^{2}$ may then be given by $$\eta(\alpha, \beta)=\left(|\alpha|^{2}-|\beta|^{2}, 2 \alpha \bar{\beta}\right) \in S^{2} \subseteq \mathbf{R}^{3}=\mathbf{R} \times \mathbf{C}$$
Note that $\eta(\alpha, \beta)=\eta(z \alpha, z \beta)$ if $z \in S^{1}$, and so $\eta$ does indeed send all the phase shifts of a given pure qbit to the same point in state space, and conversely, any two pure qbits in preimage of a given point in state space are phase shifts of each other.

## 数学代写|微分拓扑作业代写differential topology代考|Moral

The idea is the important thing: if you want to understand some complicated model through some simplification, it is often so that the complicated model locally (in the simple model) can be built out of the simple model through multiplying with some fixed space.

How these local pictures are glued together to give the global picture is another matter, and often requires other tools, for instance from algebraic topology. In the $S^{3} \rightarrow S^{2}$ case, we see that $S^{3}$ and $S^{2} \times S^{1}$ cannot be identified since $S^{3}$ is simply connected (meaning that any closed loop in $S^{3}$ can be deformed continuously to a point) and $S^{2} \times S^{1}$ is not.

An important class of examples (of which the above is one) of locally trivial fibrations arises from symmetries: if M is some (configuration) space and you have a “group of symmetries” G e.g., rotations acting on $M$, then you can consider the space $M / G$ of points in $M$ where you have identified two points in $M$ if they can be obtained from each other by letting G act e.g., one is a rotated copy of the other. Under favorable circumstances $M / G$ will be a manifold and the projection $M \rightarrow M / G$ will be a locally trivial fibration, so that $M$ is built by gluing together spaces of the form $U \times G$, where U varies over the open subsets of $M / G$.

## Matlab代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。