Man and History: Python 量子運算（二八）：投影算子

Python 量子運算（二八）：投影算子

2023/02/23

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Fig. 28.1. Projection operator.

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代碼 28.1

# Program 28.1：Projection operator
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():
    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)
    ax.text(0.15, 0.20, s4)
    ax.text(0.15, 0.10, s5)

    ax.text(0.5, -0.055, '(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)

    # string setting
    s1 = r'$P = \vert \psi \rangle \langle \psi \vert $'

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

    s3 = (
        r'$=\ \frac{1}{2}$'
        r'$\begin{bmatrix}$'
        r'$1+\psi_z & \psi_x-i\psi_y \\$'
        r'$\psi_x+i\psi_y & 1-\psi_z$'
        r'$\end{bmatrix},$'
    )

    s4 = (
        r'$where\ P_{ij}(i,j=0,1) = \langle i \vert P \vert j \rangle .$'
    )

    # string output
    ax.text(0.10, 0.90, s1, color='r')
    ax.text(0.10, 0.65, s2, color='r', fontsize=32)
    ax.text(0.10, 0.35, s3, color='r')
    ax.text(0.10, 0.10, s4, 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/0028_001.png')
plt.show()