Man and History: Python 量子運算（二四）：張量積

Python 量子運算（二四）：張量積

2023/02/10

-----

Fig. 24.1. Tensor product.

-----

張量積的定義

「In mathematics, the tensor product V⊗ W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V × W → V ⊗ W that maps a pair (v,w), v ∈ V, w ∈ W to an element of V ⊗ W denoted v ⊗ w. An element of the form v ⊗ w is called the tensor product of v and w.」[1]。

張量積與外積

「If arranged into a rectangular array, the coordinate vector of x ⊗ y is the outer product of the coordinate vectors of x and y. Therefore, the tensor product is a generalization of the outer product.」[1]。

-----

張量積與克羅內克積

「Tensor Product

It is a more general case of the Kronecker product, it takes the right hand operand and duplicates it in every element of the left hand operand. Each of these copies is scalar multiplied by the corresponding element.

張量積

這是 Kronecker 乘積的更一般情況，它採用右邊運算元並將其複製到左邊運算元的每個元素中。這些副本中的每一個都是純量乘以相應的元素。

」[2]。

「

@classmethod

def _tensor(cls, a, b):

ret = copy.copy(a)

ret._op_shape = a._op_shape.tensor(b._op_shape)

ret._data = np.kron(a.data, b.data)

return ret

」[4]。

-----

代碼 24.1

# Program 24.1：Tensor 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{v} \otimes \mathbf{w} = \ $'
    )
    s2 = (
        r'$\begin{bmatrix}$'
        r'$v_{1} \mathbf{w} \\$'
        r'$\vdots \\$'
        r'$v_{m} \mathbf{w}$'
        r'$\end{bmatrix}$'
    )

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

    plt.title('Tensor 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'$\mathbf{w}$')

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

    plt.title("Rank 1 Tensor", color='r')
    ax.text(P2.x, P2.y, P2.z, r'vector $\mathbf{w}$')
    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/0024_001.png')
plt.show()