2023年2月14日 星期二

Python 量子運算（二三）：外積

Python 量子運算（二三）：外積

2023/02/10

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Fig. 23.1. Outer product.

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「Connection with the Kronecker product

The outer product and Kronecker product are closely related; in fact the same symbol is commonly used to denote both operations.

與克羅內克積的關係

外積與克羅內克積很接近； 事實上，同一個符號通常用於表示這兩種操作。

」[1]。


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代碼 23.1


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# Program 23.1：Outer product
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt


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 Cube(ax, P1, P2):

    # parallel to x axis
    for z in range(P1.z, P2.z+1):
        S = Point(P1.x, P1.y, z)
        E = Point(P2.x, P1.y, z)
        Line(ax, S, E)
    for y in range(P1.y, P2.y+1):
        S = Point(P1.x, y, P2.z)
        E = Point(P2.x, y, P2.z)
        Line(ax, S, E)

    # parallel to y axis
    for x in range(P1.x, P2.x+1):
        S = Point(x, P1.y, P2.z)
        E = Point(x, P2.y, P2.z)
        Line(ax, S, E)
    for z in range(P1.z, P2.z+1):
        S = Point(P2.x, P1.y, z)
        E = Point(P2.x, P2.y, z)
        Line(ax, S, E)

    # parallel to z axis
    for x in range(P1.x, P2.x+1):
        S = Point(x, P1.y, P1.z)
        E = Point(x, P1.y, P2.z)
        Line(ax, S, E)
    for y in range(P1.y, P2.y+1):
        S = Point(P2.x, y, P1.z)
        E = Point(P2.x, y, P2.z)
        Line(ax, S, E)

    return


def Set(ax, lim):
    ax.set_xlim([0, lim])
    ax.set_ylim([0, lim])
    ax.set_zlim([0, lim])
    ax.set_axis_off()
    return


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

    # string setting
    s1 = (
        r'$\mathbf{u} \otimes \mathbf{v} = \ $'
        r'$\mathbf{u} \mathbf{v}^{\operatorname{T}} =$'
    )
    s2 = (
        r'$\begin{bmatrix}$'
        r'$u_{1} \mathbf{v}^{\operatorname{T}} \\$'
        r'$\vdots \\$'
        r'$u_{m} \mathbf{v}^{\operatorname{T}}$'
        r'$\end{bmatrix}$'
    )

    # string output
    ax.text(0.1, 0.65, s1)
    ax.text(0.1, 0.35, s2)

    plt.title('Outer Product')
    ax.text(0.5, 0.05, '(a)', fontsize=20)
    ax.set_axis_off()

    return


def Subplot_2():
    ax = fig.add_subplot(222, projection='3d')  # 3d subplot

    P1 = Point(0, 0, 0)
    P2 = Point(1, 1, 3)
    Cube(ax, P1, P2)
    Set(ax, 6)

    for i in range(P2.z):
        for j in range(P2.x):
            ax.text(j, 0, i, r'$\mathbf{v}^{\operatorname{T}}$')

    plt.title('Rank 2 Tensor', color='r')
    ax.text(P2.x, P2.y, P2.z, r'matrix $(\mathbf{u} \otimes \mathbf{v})$')
    ax.text(4, 0, -1, '(b)', fontsize=20)

    return


def Subplot_3():
    ax = fig.add_subplot(223, projection='3d')  # 3d subplot

    P1 = Point(0, 0, 0)
    P2 = Point(1, 1, 3)
    Cube(ax, P1, P2)
    Set(ax, 6)

    plt.title('Rank 1 Tensor', color='r')
    ax.text(P2.x, P2.y, P2.z, r'vector $\mathbf{u}$')
    ax.text(4, 0, -1, '(c)', fontsize=20)

    return


def Subplot_4():
    ax = fig.add_subplot(224, projection='3d')  # 3d subplot

    P1 = Point(0, 0, 0)
    P2 = Point(4, 1, 1)
    Cube(ax, P1, P2)

    P3 = Point(5, 0, 0)
    P4 = Point(6, 1, 4)
    Cube(ax, P3, P4)

    Set(ax, 6)

    plt.title("Rank 1 Tensor", color='r')
    ax.text(0, 0, 3, 'Row Vecor(shape 1x4)', fontsize=16)
    ax.text(3, 0, 5, 'Column Vector(shape 4x1)', fontsize=16)
    ax.text(-1, 1, 1, r'vector $\mathbf{v}^{\operatorname{T}}$')
    ax.text(P4.x, P4.y, P4.z, r'vector $\mathbf{v}$')
    ax.text(4, 0, -1, '(d)', fontsize=20)

    return


# figure setting
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/0023_001.png')
# plt.show()


解說：

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References


[1] Outer product - Wikipedia

https://en.wikipedia.org/wiki/Outer_product


[2] NumPy Illustrated: The Visual Guide to NumPy | by Lev Maximov | Better Programming

https://en.wikipedia.org/wiki/Outer_product


[3] matplotlib - Python - Plotting colored grid based on values - Stack Overflow

https://stackoverflow.com/questions/43971138/python-plotting-colored-grid-based-on-values

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Python 量子運算（目錄）

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

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