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()

-----

References


-----

Python 量子運算（目錄）

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

-----

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標籤：

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：

-----

References


-----

Python 量子運算（目錄）

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

-----

張貼者： 於 沒有留言:
標籤：

2023年2月24日 星期五

坦尼沙羅比丘

 坦尼沙羅比丘

2023/01/11

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# 2023/01/11

230106 Standing Outside Your Thoughts

https://www.youtube.com/watch?v=vozbFNfb4XQ


# 2023/01/17

230105 Talking Among Your Selves 

https://www.youtube.com/watch?v=U2_8tlwONM4

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# 2014/04/15

(35) Ajaan Geoff: Dharma Talk - YouTube

https://www.youtube.com/watch?v=HNy323IsRaU

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References


[1] Ṭhānissaro Bhikkhu - Wikipedia

https://en.wikipedia.org/wiki/%E1%B9%ACh%C4%81nissaro_Bhikkhu


[2] (19) Dhamma Talks by Thanissaro Bhikkhu - YouTube

https://www.youtube.com/@DhammatalksOrg

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林居禪園筆記

http://mandhistory.blogspot.com/2022/01/blog-post_22.html

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張貼者： 於 沒有留言:
標籤：

2023年2月23日 星期四

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


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# 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()

解說：

-----

References


[1] Projection Operators and Completeness

https://quantummechanics.ucsd.edu/ph130a/130_notes/node185.html


[2] linear algebra - What is the idea behind a projection operator? What does it do? - Mathematics Stack Exchange

https://math.stackexchange.com/questions/1303977/what-is-the-idea-behind-a-projection-operator-what-does-it-do


[3] 黃子嘉 - 線代離散研究室: [線性代數] 請教4個的觀念

http://zjhwang.blogspot.com/2009/07/4.html


[4] 特殊矩陣 (5)：冪等矩陣 | 線代啟示錄

https://ccjou.wordpress.com/2009/09/29/%E7%89%B9%E6%AE%8A%E7%9F%A9%E9%99%A3-%E4%BA%94%EF%BC%9A%E5%86%AA%E7%AD%89%E7%9F%A9%E9%99%A3/

-----

Python 量子運算（目錄）

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

-----

張貼者： 於 沒有留言:
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2023年2月22日 星期三

Python 量子運算（二七）：三維直角坐標系

Python 量子運算（二七）：三維直角坐標系

2023/03/22

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Fig. 27.1. Three dimensional Cartesian coordinate system.

-----

代碼 27.1


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# Program 27.1：Three dimensional Cartesian coordinate system
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, 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():
    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'$\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, 0.45, s1, color='r', fontsize=70)  # equation 1

    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/0027_001.png')
plt.show()


解說：

-----

References


-----

Python 量子運算（目錄）

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

-----

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標籤：

Python 量子運算（二六）：機率幅

Python 量子運算（二六）：機率幅

2022/12/08

-----


Fig. 26.1. Probability amplitude.

-----

代碼 26.1


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# Program 26.1：Probability amplitude
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'$\frac{\theta}{2}\ \mapsto\ \theta$'
    )

    ax.text(0.15, 0.65, s1)
    ax.text(0.15, 0.30, s2, color='r')
    ax.text(0.41, 0.05, s3, color='r')

    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_1 = (
        r'$\vert 0 \rangle(\sigma_z=+1):$'
    )

    s1_2 = (
        r'$p_0 = \ $'
        r'$\vert \langle 0 \vert \psi \rangle \vert ^2 = \ $'
        r'$\cos^2 \frac{\theta}{2}$'
    )

    s2_1 = (
        r'$\vert 1 \rangle(\sigma_z=-1):$'
    )

    s2_2 = (
        r'$p_1 = \ $'
        r'$\vert \langle 1 \vert \psi \rangle \vert ^2 = \ $'
        r'$\sin^2 \frac{\theta}{2}$'
    )

    # string output
    ax.text(0.10, 0.75, s1_1)  # equation 1
    ax.text(0.10, 0.60, s1_2)  # equation 1
    ax.text(0.10, 0.35, s2_1)  # equation 2
    ax.text(0.10, 0.20, s2_2)  # equation 2

    plt.title('Probabilities', color='r')
    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/0026_001.png')
plt.show()


解說：

-----

References


[1] 機率幅 - 維基百科，自由的百科全書

https://zh.wikipedia.org/zh-tw/%E6%A9%9F%E7%8E%87%E5%B9%85


[2] 零與一之間的威力 量子電腦的原理 - 科學月刊Science Monthly

https://www.scimonth.com.tw/archives/5158


[3] 機率的量子力學起源─從各種詮釋談起 – 4分33秒的科哲天地

https://proscience2.wordpress.com/2019/09/16/%E6%A9%9F%E7%8E%87%E7%9A%84%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%B8%E8%B5%B7%E6%BA%90%E2%94%80%E5%BE%9E%E5%90%84%E7%A8%AE%E8%A9%AE%E9%87%8B%E8%AB%87%E8%B5%B7/


[4] 機率幅（Probability amplitude） | 科學Online

https://highscope.ch.ntu.edu.tw/wordpress/?p=17771


[5] [閒聊]關於繞射是甚麼東西的個人意見 - 看板 Physics - 批踢踢實業坊

https://www.ptt.cc/bbs/Physics/M.1462700720.A.BE0.html

-----

Python 量子運算（目錄）

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

-----

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