2023年2月26日 星期日

Python 量子運算（三０）：直角坐標系與和角

Python 量子運算（三０）：直角坐標系與和角

2023/02/26

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Fig. 30.1. Cartesian coordinate system and angle sum.

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# Program 30.1：Cartesian coordinate system and angle sum
import matplotlib as mpl
import matplotlib.pyplot as plt

from qiskit.visualization import plot_bloch_vector


def Subplot_1():
    ax = plt.subplot(221)

    s1 = (
        r'$\vert 0 \rangle \equiv $'
        r'$\begin{bmatrix} 1 \\ 0 \end{bmatrix} \mapsto (0,0,1)$'
    )

    s2 = (
        r'$\vert 1 \rangle \equiv $'
        r'$\begin{bmatrix} 0 \\ 1 \end{bmatrix} \mapsto (0,0,-1)$'
    )

    s3 = r'$\psi_x=\cos \phi \sin \theta$'
    s4 = r'$\psi_y=\sin \phi \sin \theta$'
    s5 = r'$\psi_z=\cos \theta$'

    ax.text(0.15, 0.90, s1)
    ax.text(0.15, 0.60, s2)
    ax.text(0.15, 0.30, s3, color='r')
    ax.text(0.15, 0.20, s4, color='r')
    ax.text(0.15, 0.10, s5, color='r')

    ax.text(0.5, -0.055, '(a)', fontsize=20)
    ax.set_axis_off()

    return


def Subplot_2():
    ax = plt.subplot(222)

    # string setting
    s1_1 = r'$\sin(\alpha+\beta)$'
    s1_2 = r'$=\sin\alpha\cos\beta+\cos\alpha\sin\beta$'
    s1_3 = r'$\sin \theta = 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}$'

    s2_1 = r'$\cos(\alpha+\beta)$'
    s2_2 = r'$=\cos\alpha\cos\beta-\sin\alpha\sin\beta$'
    s2_3 = r'$\cos \theta = \cos^2 \frac{\theta}{2} - \sin^2 \frac{\theta}{2}$'

    # string output
    ax.text(0.10, 0.95, s1_1)
    ax.text(0.20, 0.80, s1_2)
    ax.text(0.10, 0.65, s1_3, color='r')

    ax.text(0.20, 0.45, s2_1)
    ax.text(0.10, 0.30, s2_2)
    ax.text(0.10, 0.15, s2_3, color='r')

    ax.text(0.5, -0.055, '(b)', fontsize=20)
    ax.set_axis_off()

    return


def Subplot_3():
    ax = plt.subplot(223)

    # string setting
    s1_1 = r'$\vert\psi\rangle$'

    s1_2 = (
        r'$=\cos\frac{\theta}{2}\ \vert0\rangle'
        r'+e^{i\phi}\sin\frac{\theta}{2}\ \vert1\rangle$'
    )

    s2 = (
        r'$=\begin{bmatrix}\cos\frac{\theta}{2}\\$'
        r'$\ e^{i\phi}\sin\frac{\theta}{2}\ \end{bmatrix}$'
    )

    s3 = r'$(0\leq\theta\leq\pi,\ 0\leq\phi<2\pi)$'

    # string output
    ax.text(0.10, 0.75, s1_1)
    ax.text(0.25, 0.75, s1_2)
    ax.text(0.25, 0.45, s2)
    ax.text(0.10, 0.15, s3)

    ax.text(0.5, -0.055, '(c)', fontsize=20)
    ax.set_axis_off()

    return


def Subplot_4():
    ax = plt.subplot(224)

    # string setting
    s1 = r'$1 = \cos^2 \frac{\theta}{2} + \sin^2 \frac{\theta}{2}$'
    s2 = r'$e ^{i\phi} = \cos \phi + i \sin \phi$'

    s3 = (
        r'$\vert \psi \rangle =\ $'
        r'$\begin{bmatrix}$'
        r'$\sqrt \frac{1+\psi_z}{2} \\$'
        r'$\frac{\psi_x+i\psi_y}{\sqrt{2(1+\psi_z)}}$'
        r'$\end{bmatrix}$'
    )

    # string output
    ax.text(0.10, 0.95, s1, color='r')
    ax.text(0.10, 0.85, s2, color='r')
    ax.text(0.10, 0.35, s3, color='r', fontsize=70)

    ax.text(0.5, -0.055, '(d)', fontsize=20)
    ax.set_axis_off()

    return


# figure setting
mpl.rcParams['text.usetex'] = True
mpl.rcParams['text.latex.preamble'] = r'\usepackage{{amsmath}}'
mpl.rcParams['font.size'] = 40
fig, ax = plt.subplots(figsize=(16, 16))

Subplot_1()
Subplot_2()
Subplot_3()
Subplot_4()

# plt.savefig('/content/drive/My Drive/pqc/0030_001.png', facecolor='w')
plt.show()

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References


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Python 量子運算（目錄）

https://mandhistory.blogspot.com/2022/01/quantum-computing.html

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