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In this unit, the special gate RY is introduced to understand its property around y-axis. 在本單元中，分析特殊閘RY， 以瞭解其繞y-軸的特性。

In this unit, the special gate RY is introduced to understand its property around y-axis. 在本單元中，分析特殊閘RZ， 以瞭解其繞z-軸的特性。

In this unit, the properties of X, Y, Z gates are analyzed to reveal their computing abilities on Bloch Sphere. 在本單元中，分析 X、Y、Z 量子閘的性質，以了解它們在 Bloch Sphere 上的旋轉計算能力。

In this unit, the matrix tensor is introduced to integrate several quantum gates to a larger gate. 在本單元中，介紹矩陣張量積，可以將多個量子閘計算為一個更大量子閘。

In this unit, the canonical quantum state is formally defined, and the family of quantum states corresponding to the same canonical quantum state can be mapped into a small point on Bloch sphere. 在本單元中，正式定義正則量子態和對應於同一正則量子態的量子態族可以映射到 Bloch 球上的同一個點。

In this unit, each 1-qubit state can be shown in 3-D sphere with Hopf map. Directly, we can translate each canonical quantum state to its corresponding spherical coordinate system. 在本單元中，每個1次量子比特都可以使用 Hopf 映射到3-D 球體中。我們更可以直接將每個政則量子態轉換為其相應的球座標系。

In this unit, proofs of Euler’s formula based on Taylor’s expansion and differential equation are given. In addition, some formulas of trigonometric functions can be easily obtained due to the powerful property of Euler’s formula. 在本單元中，基於泰勒展開和微分方程的歐拉公式的證明被詳細導出。此外，由於歐拉公式的特性，可以很容易地獲得一些三角函數的公式。

In this unit, complex plane, Euler’s formula and phase difference are introduced. The polar coordinate is reviewed to emphasize the properties cos and sin in the geometrical space. 在本單元中，介紹複數平面、歐拉公式和相位差。複習了極座標進，以強調幾何空間中 cos 和 sin的效用。

In this unit, how to write an n-qubit input into the quantum circuit is defined. Readers can understand how to arrange the MLB and LSB in the computing order. 在本單元中，定義了如何將n次量子比特輸入到量子電路。讀者可以明確瞭解如何安排MLB 和 LSB的計算順序。

The operations of complex numbers are introduced and the linear combination with the standard basis { |00⟩, |01⟩, |10⟩, |11⟩ } is first shown in this series which is an important property to explain the concept of quantum superposition. 介紹複數的運算，並首次展示標準基底 { |00⟩, |01⟩, |10⟩, |11⟩ } 的線性組合式，這是後續解釋量子疊加概念的重要性質。

1-qubit and 2-qubit quantum superpositions are defined in this unit. Also, the operation of Hadamard gate is also shown to understand its property of operation. 本單元定義了 1次量子比特和 2次量子比特量子的疊加態。另外，哈達瑪閘的運作也被展示出來，以瞭解其運算的性質。