Man and History: Python 量子運算（一五）：內積

Python 量子運算（一五）：內積

2023/01/11

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Fig. 15.1. Inner product.

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代碼 15.1

# Program 15.1：Inner product
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt


class Vec:
    def __init__(self, t1, t2, v_r, a_r):
        self.t1 = t1    # theta 1
        self.t2 = t2    # theta 2

        self.v_r = v_r  # vector radius
        self.v_x = v_r * np.cos(t2)
        self.v_y = v_r * np.sin(t2)

        self.a = np.linspace(t1, t2, 100)  # arc
        self.a_r = a_r                     # arc radius
        self.a_x = a_r * np.cos(self.a)
        self.a_y = a_r * np.sin(self.a)


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

    # coordinates
    ax.plot([-0.1, 1], [0, 0], 'k')
    ax.plot([0, 0], [-0.1, 1], 'k')

    # vector setting
    v1 = Vec(0, np.pi/3, 1, 0.6)
    v2 = Vec(0, np.pi/12, 1, 0.5)
    v3 = Vec(np.pi/12, np.pi/3, 1, 0.4)

    # vector x
    plt.quiver(0, 0, v1.v_x, v1.v_y, scale=1.25, color='r')
    ax.plot([0, v1.v_x], [v1.v_y, v1.v_y], 'r', linestyle=':')
    ax.plot([v1.v_x, v1.v_x], [0, v1.v_y], 'r', linestyle=':')
    ax.text(v1.v_x+0.05, v1.v_y+0.05, r"$\mathbf{x}$", color='r', fontsize=32)
    ax.text(v1.v_x-0.05, -0.1, r"$x1$", color='r', fontsize=32)
    ax.text(-0.15, v1.v_y, r"$x2$", color='r', fontsize=32)

    # vector y
    plt.quiver(0, 0, v2.v_x, v2.v_y, scale=1.2, color='b')
    ax.plot([0, v2.v_x], [v2.v_y, v2.v_y], 'b', linestyle=':')
    ax.plot([v2.v_x, v2.v_x], [0, v2.v_y], 'b', linestyle=':')
    ax.text(v2.v_x+0.05, v2.v_y+0.05, r"$\mathbf{y}$", color='b', fontsize=32)
    ax.text(v2.v_x-0.05, -0.1, r"$y1$", color='b', fontsize=32)
    ax.text(-0.15, v2.v_y, r"$y2$", color='b', fontsize=32)

    # arc
    ax.plot(v1.a_x, v1.a_y, 'r')
    ax.text(0.36, 0.36, r"$\alpha$", color='r', fontsize=32)
    ax.plot(v2.a_x, v2.a_y, 'b')
    ax.text(0.41, 0.03, r"$\beta$", color='b', fontsize=32)
    ax.plot(v3.a_x, v3.a_y, 'k')
    ax.text(0.11, 0.11, r"$\alpha - \beta$", color='k', fontsize=32)

    ax.text(0.5, -0.34, '(a)', fontsize=20)
    ax.set_axis_off()

    return


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

    # string setting
    s1_1 = r'$\mathbf{x}=(x_1,\cdots,x_n),$'
    s1_2 = r'$\mathbf{y}=(y_1,\cdots,y_n).$'

    s2_1 = r'$\mathbf{x} \cdot \mathbf{y}$'
    s2_2 = r'$=x_1y_1+ \cdots +x_ny_n$'
    s2_3 = r'$=\sum_{i=1}^n x_iy_i.$'

    s3_1 = r'$\mathbf{x}^T\mathbf{y}$'
    s3_2 = r'$=\begin{bmatrix}x_1 \cdots x_n\end{bmatrix}$'\
           r'$\begin{bmatrix}y_1\\ \vdots \\y_n\end{bmatrix}$'
    s3_3 = r'$=\sum_{i=1}^n x_iy_i$.'

    # string output
    ax.text(0.1, 0.75, s1_1)
    ax.text(0.1, 0.63, s1_2)

    ax.text(0.1, 0.45, s2_1)
    ax.text(0.3, 0.45, s2_2)
    ax.text(0.3, 0.33, s2_3)

    ax.text(0.1, 0.15, s3_1)
    ax.text(0.3, 0.15, s3_2)
    ax.text(0.3, 0.03, s3_3)

    ax.text(0.5, -0.15, '(b)', fontsize=20)
    ax.set_axis_off()

    return


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

    # string setting
    s1 = r'$\|\mathbf{x}\|=\sqrt{x_1^2+x_2^2},$'
    s2_1 = r'$\langle \mathbf{x}, \mathbf{y} \rangle$'
    s2_2 = r'$=\|\mathbf{x}\| \cos(\alpha - \beta) \|\mathbf{y}\|$'
    s2_3 = r'$=x_1y_1+x_2y_2.$'

    # string output
    ax.text(0.10, 0.75, s1)

    ax.text(0.20, 0.55, s2_1)
    ax.text(0.10, 0.35, s2_2)
    ax.text(0.10, 0.15, s2_3)

    ax.text(0.5, -0.15, '(c)', fontsize=20)
    ax.set_axis_off()

    return


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

    # string setting
    s1_1 = r'$\cos (\alpha - \beta)$'
    s1_2 = r'$=\cos\alpha\cos\beta+\sin\alpha\sin\beta$'
    s1_3 = r'$=\frac{x_1}{\|\mathbf{x}\|}\frac{y_1}{\|\mathbf{y}\|}$'\
           r'$+\frac{x_2}{\|\mathbf{x}\|}\frac{y_2}{\|\mathbf{y}\|}$'
    s1_4 = r'$=\frac{x_1y_1+x_2y_2}{\|\mathbf{x}\|\|\mathbf{y}\|}.$'

    # string output
    ax.text(0.20, 0.75, s1_1)
    ax.text(0.10, 0.55, s1_2)
    ax.text(0.10, 0.35, s1_3)
    ax.text(0.10, 0.15, s1_4)

    ax.text(0.5, -0.15, '(d)', fontsize=20)
    ax.set_axis_off()

    return


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

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

#plt.savefig('/content/drive/My Drive/pqc/0015_001.png')
plt.show()