2023年2月25日 星期六

Python 量子運算（二九）：兩極與赤道

Python 量子運算（二九）：兩極與赤道

2023/02/25

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Fig. 29.1. Two poles and equator.

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代碼 29.1


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# Program 29.1：Two poles and equator
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt

from qiskit.visualization import plot_bloch_vector


class Point:
    def __init__(self, x, y, z):
        self.x = x
        self.y = y
        self.z = z


def Line(ax, A, B):
    ax.plot([A.x, B.x], [A.y, B.y], [A.z, B.z], 'b')
    return


def Subplot_1():
    ax1 = fig.add_subplot(221, projection='3d')

    PO = Point(0, 0, 0)
    plot_bloch_vector([PO.x, PO.y, PO.z], ax=ax1)

    P1 = Point(0, 0, 1)
    P2 = Point(0, 0, -1)
    P3 = Point(0, -1, 0)
    P4 = Point(0, 1, 0)
    P5 = Point(1, 0, 0)
    P6 = Point(-1, 0, 0)

    ax1.text(P1.x, P1.y, P1.z, r'$\vert 0 \rangle$', color='r')
    ax1.text(P2.x, P2.y, P2.z, r'$\vert 1 \rangle$', color='r')
    ax1.text(P3.x, P3.y, P3.z, r'$\vert + \rangle$', color='r')
    ax1.text(P4.x, P4.y, P4.z, r'$\vert - \rangle$', color='r')
    ax1.text(P5.x, P5.y, P5.z, r'$\vert i \rangle$', color='r')
    ax1.text(P6.x, P6.y, P6.z, r'$\vert -i \rangle$', color='r')

    ax1.text(0, 0, -1.8, '(a)', fontsize=20)
    ax.set_axis_off()

    return


def Subplot_2():
    ax1 = fig.add_subplot(222, projection='3d')

    # P1 = Point(0, 0, 1)
    # plot_bloch_vector([P1.x, P1.y, P1.z], ax=ax1)
    # ax1.scatter(P1.x, P1.y, P1.z)

    # P2 = Point(0, 0, 0)
    # Line(ax1, P1, P2)

    theta = np.pi / 6
    rotation = (3/2) * np.pi  # transfer 3d to qiskit 3d
    phi_1 = np.pi / 3         # 3d
    phi_2 = rotation + phi_1  # qiskit 3d

    PO = Point(0, 0, 0)
    PB_1 = Point(np.sin(theta)*np.cos(phi_1), np.sin(theta)*np.sin(phi_1), 0)
    PB_2 = Point(np.sin(theta)*np.cos(phi_2), np.sin(theta)*np.sin(phi_2), 0)
    PA_1 = Point(PB_1.x, PB_1.y, np.cos(theta))
    PA_2 = Point(PB_2.x, PB_2.y, np.cos(theta))

    plot_bloch_vector([PA_1.x, PA_1.y, PA_1.z], ax=ax1)  # qiskit 3d

    Line(ax1, PO, PB_2)
    # Line(ax1, PO, PA_2)
    Line(ax1, PA_2, PB_2)

    # lables(psi)
    ax1.text(0.35, 0, 0.8, r"$\vert\psi\rangle$")
    ax1.text(0.05, 0, 0.4, r"$\theta$")
    ax1.text(-0.1, 0, -0.45, r"$\phi$")

    # curves: theta and phi
    theta_max = np.pi / 6  # angle between psi and z axis
    phi_max = np.pi / 3    # angle between psi and x axis
    phi_offset = -np.pi / 2  # xy coordinate rotation from matplotlib to qiskit
    curve_radius = 0.3
    n = 20

    c1 = np.linspace(0, theta_max, n)
    x1 = curve_radius * np.sin(c1) * np.cos(phi_max+phi_offset)
    y1 = curve_radius * np.sin(c1) * np.sin(phi_max+phi_offset)
    z1 = curve_radius * np.cos(c1)
    ax1.plot(x1, y1, z1, 'g', lw=2)  # curve theta

    c2 = np.linspace(phi_offset, phi_max+phi_offset, n)
    x2 = curve_radius * np.cos(c2)
    y2 = curve_radius * np.sin(c2)
    z2 = c2 * 0
    ax1.plot(x2, y2, z2, 'r', lw=2)  # curve phi

    ax1.text(0, 0, -1.8, '(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)

    # Notation
    nttn = ['Notation',
            '',
            r'$\vert 0 \rangle$',
            r'$\vert 1 \rangle$',
            r'$\vert + \rangle$',
            r'$\vert - \rangle$',
            r'$\vert i \rangle$',
            r'$\vert -i \rangle$',
            ]

    # Description
    dctn = ['Description',
            '',
            r'$(1,0) \mapsto (0,0,1)$',
            r'$(0,1) \mapsto (0,0,-1)$',
            r'$(\frac{1}{\sqrt 2},\frac{1}{\sqrt 2}) \mapsto (1,0,0)$',
            r'$(\frac{1}{\sqrt 2},-\frac{1}{\sqrt 2}) \mapsto (-1,0,0)$',
            r'$(\frac{1}{\sqrt 2},\frac{i}{\sqrt 2}) \mapsto (0,1,0)$',
            r'$(\frac{1}{\sqrt 2},-\frac{i}{\sqrt 2}) \mapsto (0,-1,0)$',
            ]

    for i in range(8):
        ax.text(0.05, 1-0.12*i, nttn[i], color='r', fontsize=32)
        ax.text(0.40, 1-0.12*i, dctn[i], color='r', fontsize=32)

    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/0029_001.png', facecolor='w')
plt.show()

解說：

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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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