mat_maker

Outline

This function generates the matrix that sets the coefficients of the linear combination into the ancillary qubit part.

Since the matrix used in the quantum circuit must be unitary, the construction process is nontrivial; however, by simply providing the first column elements and the corresponding coefficients, the function automatically generates a valid unitary matrix.

Index List

argument name	type	role
elelist	list(element:float)	Coefficient List for the Linear Combination
coeff_list	list(element:float)	Parameters Used in Matrix Computation

Return

mat_ele_list（np.array）:

np.array storing the generated unitary matrix (elements of type float).

Python code

def mat_maker(elelist, coeff_lst):
    mat_ele_list = []
    # append the first colum of the desired matrix
    mat_ele_list.append(elelist)
    # search the border between non-zero elements and zero elements
    for i in range(len(elelist)//2):
        zero_num = i*2
        if np.abs(elelist[i*2]) == 0:
            break
    if elelist[zero_num] != 0:
        zero_num = len(elelist)
    # copy list of coeffcients to use ".reverse()"
    cf_copy = copy.deepcopy(coeff_lst)
    cf_copy.reverse()
    for i in range(len(coeff_lst)):
        # copy elements of matrix, this copy will be append to returned matrix after some process
        mat_copy = copy.deepcopy(mat_ele_list)
        for j in range(len(mat_copy)):
            # calculate non-zero elements of unitary matrix as list
            app_list = vec_make(mat_copy[j], cf_copy[i])
            # append elemnets zero to list of non-zero elements
            if i < (len(coeff_lst) - 1.1):
                if zero_num < (len(elelist) - 0.1):
                    if zero_num%2**(i+2) != 0:
                        input_ele = 0
                        for s in range(2**(i+1) + 2*j):
                            input_ele += mat_copy[i][(zero_num//(2**(i+2)))*(2**(i+2)) + s]**2
                            app_list[(zero_num//(2**(i+2)))*(2**(i+2)) + s] = 0
                        app_list[zero_num] = np.sqrt(input_ele)
                        zero_num += 2
            mat_ele_list.append(np.array(app_list))
    base_mat = [[0, -1], [1, 0]]
    unit = np.eye(2)
    convert = np.kron(unit, base_mat)
    # exchange ith line and i+1th line (i = 0, 1, 2, ..., len(coeff_lst) - 1)
    for i in range(len(coeff_lst) - 1):
        convert = np.kron(unit, convert)
    mel_len = len(mat_ele_list)
    if (np.log2(len(elelist)) > 1):
        for i in range(mel_len):
            mat_ele_list.append(np.dot(convert, mat_ele_list[i]))
    if (np.log2(len(elelist)) < 1.1):
        mat_ele_list = [[elelist[0], elelist[1]], [elelist[1], -elelist[0]]]
    # transpose the matrix to make it the optimal shape for sandwiching between quantum states
    return np.array(mat_ele_list).T

Sample Run

import numpy as np
import pitbe

ele_lst = [0.5, 0.433, 0.25, 0.433, 0.433, 0.3536, 0., 0.]
cf_lst = [[1.483], [1.322, 0.5590]]
unit_mat = pitbe.mat_maker(ele_lst, cf_lst)
print(np.round(unit_mat, 4))

[[ 0.5    -0.3782 -0.3372  0.255  -0.433   0.3275  0.292  -0.2209]
 [ 0.433  -0.3275 -0.292   0.2209  0.5    -0.3782 -0.3372  0.255 ]
 [ 0.25    0.3305 -0.1686 -0.2229 -0.433  -0.5724  0.292   0.386 ]
 [ 0.433   0.5724 -0.292  -0.386   0.25    0.3305 -0.1686 -0.2229]
 [ 0.433   0.      0.6421  0.     -0.3536  0.     -0.5244  0.    ]
 [ 0.3536  0.      0.5244  0.      0.433   0.      0.6421  0.    ]
 [ 0.      0.559   0.      0.8291  0.      0.      0.      0.    ]
 [ 0.      0.      0.      0.      0.      0.559   0.      0.8291]]

The unitarity of the constructed matrix can be verified as follows.

# Norm of all eigenvalues are one.
eigenvalue = np.linalg.eig(unit_mat)[0]
for i in range(len(eigenvalue)):
    print(np.round(np.linalg.norm(eigenvalue[i]), 3))

1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0

# The product with the complex conjugate of the matrix is the identity matrix.
print(np.round(np.dot(unit_mat.T, unit_mat), 3))

[[ 1. -0. -0.  0.  0.  0. -0. -0.]
 [-0.  1.  0.  0.  0.  0.  0.  0.]
 [-0.  0.  1. -0.  0.  0.  0.  0.]
 [ 0.  0. -0.  1.  0.  0. -0.  0.]
 [ 0.  0.  0.  0.  1. -0. -0.  0.]
 [ 0.  0.  0.  0. -0.  1.  0.  0.]
 [-0.  0.  0. -0. -0.  0.  1. -0.]
 [-0.  0.  0.  0.  0.  0. -0.  1.]]