Is it true that $Ry(\pi/2)\sigma_zRy(-\pi/2)=\sigma_x$?

Question

I saw in a qiskit document that said $Ry(\pi/2)\sigma_zRy(-\pi/2)=\sigma_x$ To confirm this I decided to create the matrix representations of these operations and multiply them together to see if I get the correct matrix, but when I did this in numpy I didn't get the correct matrix

import numpy as np

sig_z = np.array([[1,0],[0,-1]])
rz = np.array([[np.exp(-1j*np.pi/4), 0],[0,np.exp(1j*np.pi/4)]])
print(np.matmul(rz, np.matmul(sig_z, -rz)))

This outputs

array([[-2.22044605e-16+1.j,  0.00000000e+00+0.j],
       [ 0.00000000e+00+0.j,  2.22044605e-16+1.j]])

While I was expecting

array([[0+0.j,  1+0.j],
       [1+0.j,  0+0.j]])

Am I doing something wrong? For my $Rz$ gate I used the following model $$ Rz(\theta)= \begin{pmatrix} \mathrm{e}^{-i\frac{\theta}{2}} & 0 \\ 0 & \mathrm{e}^{i\frac{\theta}{2}} \\ \end{pmatrix} $$ and the $\sigma_x$ and $\sigma_z$ are supposed to be just standard Pauli operators.

Answer 1

$Ry$ gate is defined as $$ Ry(\theta) = \begin{pmatrix} \cos(\theta/2) & -\sin(\theta/2) \\ \sin(\theta/2) & \cos(\theta/2) \end{pmatrix}. $$ Hence $$ Ry(\pi/2) = \begin{pmatrix} \cos(\pi/4) & -\sin(\pi/4) \\ \sin(\pi/4) & \cos(\pi/4) \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}. $$ Similarly for $Ry(-\pi/2)$ we have (sine is odd function, so we only change sign before the sine, cosine is even, hence we left the sign unchanged) $$ Ry(-\pi/2) = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}. $$ Now, by direct multiplication of the matrices we came to $$ Ry(\pi/2)\,Z\,Ry(-\pi/2)= \frac{1}{2} \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}= \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = X. $$

Here is calculation Octave (MatLab):

>> Ry = [cos(pi/4) -sin(pi/4); sin(pi/4) cos(pi/4)]
Ry =

   0.70711  -0.70711
   0.70711   0.70711

>> dgRy = [cos(-pi/4) -sin(-pi/4); sin(-pi/4) cos(-pi/4)]
dgRy =

   0.70711   0.70711
  -0.70711   0.70711

>> Z=[1 0; 0 -1];
>> Z
Z =

   1   0
   0  -1

>> Ry*Z*dgRy
ans =

  2.2204e-016  1.0000e+000
  1.0000e+000  -2.2204e-016

Answer 2

You are using rz in your code, while the identity you are asking about uses $R_y$

Here is a qiskit code to check the identity:

from qiskit import QuantumCircuit
from qiskit.quantum_info.operators import Operator
from qiskit.visualization import array_to_latex
import numpy as np

circ = QuantumCircuit(1)
circ.ry(-np.pi / 2, 0)
circ.z(0)
circ.ry(np.pi / 2, 0)

array_to_latex(Operator(circ))