Man and History: Python 量子運算（二二）：克羅內克積

Python 量子運算（二二）：克羅內克積

2023/02/09

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Fig. 22.1. Kronecker.

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目標是張量積，但進行之前，有必要先瞭解克羅內克積以及外積。

先看一下參考文獻一的結論：

外積是克羅內克積的特例 [2]。

對於向量而言，外積和是張量積等價的 [2]。

「In linear algebra, an outer product is the tensor product of two coordinate vectors, a special case of the Kronecker product of matrices.

在線性代數中，外積是兩個坐標向量的張量積，是矩陣的克羅內克積的特例。」[2]。

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代碼 22.1

# Program 22.1：Kronecker 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'$\mathbf{A} \otimes \mathbf{B} =$'
    s2 = (
        r'$\begin{bmatrix}$'
        r'$a_{11} \mathbf{B} & \cdots & a_{1n} \mathbf{B} \\$'
        r'$\vdots & \ddots & \vdots \\$'
        r'$a_{m1} \mathbf{B} & \cdots & a_{mn} \mathbf{B}$'
        r'$\end{bmatrix}$'
    )

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

    plt.title('Kronecker 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(3, 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, 'B')

    plt.title('Rank 2 Tensor', color='r')
    ax.text(P2.x, P2.y, P2.z, r'matrix $(\mathbf{A} \otimes \mathbf{B})$')
    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(3, 1, 3)
    Cube(ax, P1, P2)
    Set(ax, 6)

    plt.title('Rank 2 Tensor', color='r')
    ax.text(P2.x, P2.y, P2.z, 'matrix A')
    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, 3)
    Cube(ax, P1, P2)
    Set(ax, 6)

    plt.title('Rank 2 Tensor', color='r')
    ax.text(P2.x, P2.y, P2.z, 'matrix B')
    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/0022_001.png')
plt.show()