Quantum Computing

Basic

量子叠加（superposition）、量子纠缠（entanglement）、量子干涉（interference）

单个 qubit

基态（Deterministic State）

• $\ket{0}$
• $\ket{1}$

$\ket{}$ is called a ket.

叠加态（Superposition State）

$$\ket{\varphi}=a\ket{0}+b\ket{1}=\begin{pmatrix}a \\ b\end{pmatrix}$$

表示：对这个 qubit 进行「测量」操作的时候，有 $|a|^2$ 的概率得到 0，有 $|b|^2$ 的概率得到 1，其中 $|a|^2+|b|^2=1$。

两个常见的叠加态：

• $\ket{+}=\frac{1}{\sqrt{2}}\left(\ket{0}+\ket{1}\right)$
• $\ket{-}=\frac{1}{\sqrt{2}}\left(\ket{0}-\ket{1}\right)$

其他：

• $\ket{±}=\frac{1}{\sqrt{2}}\left(\ket{0}±\ket{1}\right)$
• $\ket{0}=\frac{1}{\sqrt{2}}\left(\ket{+}+\ket{-}\right)$
• $\ket{1}=\frac{1}{\sqrt{2}}\left(\ket{+}-\ket{-}\right)$
• $\ket{\mu}=\frac{1}{\sqrt{2}}\left(\ket{0}+i\ket{1}\right)$
• $\ket{\nu}=\frac{1}{\sqrt{2}}\left(\ket{0}-i\ket{1}\right)$

布洛赫球面（Bloch Sphere）

Qubit state: $\cos{\left(\theta/2\right)}\ket{0}+e^{i\varphi}\sin{\left(\theta/2\right)}\ket{1}$

Polar angle: $\theta$

Azimuthal Angle: $\varphi$

• $\ket{0}:\theta=0,\varphi=0$
• $\ket{1}:\theta=\pi,\varphi=0$
• $\ket{+}:\theta=\pi/2,\varphi=0$
• $\ket{-}:\theta=\pi/2,\varphi=\pi$
• $\ket{\mu}:\theta=\pi/2,\varphi=3\pi/2$
• $\ket{\nu}:\theta=\pi/2,\varphi=\pi/2$

单量子门

• Hadamard Gate - H
• Identity Gate - I
• 非门 - X
• Z Gate - Z
• S Gate - S ($S=\sqrt{Z}$)

Every quantum gate must always be reversible.

$$H=\begin{bmatrix}1&1\\1&-1\end{bmatrix}, I=\begin{bmatrix}1&0\\0&1\end{bmatrix}, X=\begin{bmatrix}0&1\\1&0\end{bmatrix}, Z=\begin{bmatrix}1&0\\0&-1\end{bmatrix}, S=\begin{bmatrix}1&0\\0&i\end{bmatrix}$$

• $H\ket{0}=\ket{+}$, $H\ket{1}=\ket{-}$, $H\ket{+}=\ket{0}$, $H\ket{-}=\ket{1}$
• $H\cdot H=I$
• $I\ket{0}=\ket{0}$, $I\ket{1}=\ket{1}$, $I\ket{+}=\ket{+}$, $I\ket{-}=\ket{-}$
• $X\ket{0}=\ket{1}$, $X\ket{1}=\ket{0}$
• $Z\ket{0}=\ket{0}$, $Z\ket{1}=-\ket{1}$, $Z\ket{+}=\ket{-}$, $Z\ket{-}=\ket{+}$
• $S\ket{+}=\ket{\mu}$, $S\ket{-}=\ket{\nu}$

The phase gate S and Z are 90° and 180° rotations around the vertical axis, often referred to as the z-axis.

The quantum NOT gate X is a 180° rotation around the horizontal axis between the Hadamard states, often referred to as the x-axis.

The Hadamard gate H is a 180° rotation around a diagonal between the x and z axes.

$$R_x(\theta)=\begin{bmatrix} \cos(\theta/2) & -i \sin(\theta/2) \\ -i \sin(\theta/2) & \cos{\theta/2} \end{bmatrix}$$

$$R_z(\varphi)=\begin{bmatrix} 1 & 0 \\ 0 & e^{i\varphi/2} \end{bmatrix}$$

Entanglement 纠缠

The Bell state is the prototypical example of an entangled state.

$$\ket{\varphi}_{bell}=\frac{1}{\sqrt{2}}\left(\ket{00}+\ket{11}\right)$$