Matrix decomposition

This tutorial demonstrates how to use paulie to decompose a matrix in \(\mathbb{C}^{2^n \times 2^n}\) as a sum of Pauli strings of length \(n\).

The set of Pauli strings of length \(n\) given by \(\mathcal{P}=\{\bigotimes_{i=1}^{n}U_{i} \mid U_{i} \in (I, X, Y, Z)\}\) forms a basis of the vector space of complex matrices of dimensions \(2^n \times 2^n\) denoted by \(\mathbb{C}^{2^n \times 2^n}\).

paulie provides the method matrix_decomposition for decomposing general matrices which returns a weight vector:

import numpy as np
from paulie.application.matrix_decomposition import matrix_decomposition

A = np.array([[0. +0.j, -2.9-0.9j, 0.5+0.j, 0. +0.j]
              [3.1+1.5j, 0. +0.j, 0. +0.j, -0.5+0.j]
              [0.5+0.j, 0. +0.j, 0. +0.j, 3.1+1.5j]
              [0. +0.j, -0.5+0.j, -2.9-0.9j, 0. +0.j]])
decomp = matrix_decomposition(A)
print("Decomposition weight vector:")
print(decomp)

which outputs:

Decomposition weight vector:
[0. +0.j  0. +0.j  0.1+0.3j 0. +0.j  0. +0.j  0. +0.j  0. +0.j  1.2-3.j
 0. +0.j  0.5+0.j  0. +0.j  0. +0.j  0. +0.j  0. +0.j  0. +0.j  0. +0.j ]

From this weight vector, we can obtain the coefficient of a given Pauli string in the decomposition:

from paulie.common.pauli_string_factory import get_pauli_string as p

p1 = p("IX")
p2 = p("ZY")
p3 = p("XZ")

print("Coefficient of IX: ", np.round(p1.get_weight_in_matrix(decomp), 5))
print("Coefficient of ZY: ", np.round(p2.get_weight_in_matrix(decomp), 5))
print("Coefficient of XZ: ", np.round(p3.get_weight_in_matrix(decomp), 5))

which outputs:

Coefficient of IX:  (0.1+0.3j)
Coefficient of ZY:  (1.2-3j)
Coefficient of XZ:  (0.5+0j)

There is also a specialized method matrix_decomposition_diagonal for decomposing diagonal matrices:

from paulie.application.matrix_decomposition import matrix_decomposition_diagonal

A = np.array([0.41+0.16j, 0.1 +0.72j, 0.31+0.12j, 0.28+0.67j])
decomp = matrix_decomposition_diagonal(A)
pauli_strs = ["II", "IZ", "ZI", "ZZ"]
for pauli_str in pauli_strs:
    pstr = p(pauli_str)
    coeff = pstr.get_weight_in_matrix(decomp)
    print(f"Coefficient of {pauli_str}: {np.round(coeff, 5)}")

which outputs:

Coefficient of II: (0.275+0.4175j)
Coefficient of IZ: (0.085-0.2775j)
Coefficient of ZI: (-0.02+0.0225j)
Coefficient of ZZ: (0.07-0.0025j)