Decomposition of unitary operator into rotations around Bloch sphere

Question

I apologize in advance for any mistakes as I am new to this field and come from a programming, rather than mathematical/physical background.

I am looking for a way to decompose a given operator $U$ into $ZYZ$ rotations. After searching on the internet, I wrote this Julia function to perform this operation (V[1, 1] represents $V_{11}$, etc.):

decompose(U::Matrix{ComplexF64}) = begin
    α = atan(imag(det(U)),real(det(U)))/2
    V = exp(-im*α)*U
    θ₁ = magnitude(V[1, 1])≥magnitude(V[1, 2]) ? 2*acos(magnitude(V[1, 1])) : 2*asin(magnitude(V[1, 2]))
    if cos(θ₁/2) == 0
        θ₀ = atan(imag(V[2, 1]/sin(θ₁/2)), real(mag(V[2, 1]/sin(θ₁/2))))
        θ₀ = -θ₂
    elseif sin(θ₁/2) == 0
        θ₀ = atan(imag(V[2, 2]/cos(θ₁/2)), real(V[2, 2]/cos(θ₁/2)))
        θ₂ = θ₀
    else
        θ₀ = atan(imag(V[2, 2]/cos(θ₁/2)), real(V[2, 2]/cos(θ₁/2)))+atan(imag(V[2, 1]/sin(θ₁/2)), real(V[2, 1]/sin(θ₁/2)))
        θ₂ = 2*atan(imag(V[2, 2]/cos(θ₁/2)), real(V[2, 2]/cos(θ₁/2)))-θ₀
    end
    return α, θ₀, θ₁, θ₂
end

I then plug these values back into $e^{i*\alpha}*R_z(\theta_0)*R_y(\theta_1)*R_z(\theta_2)$, to 'recompose' the gate and check if the gate decomposed correctly. Here is the compose function for reference:

R_x(θ) = [cos(θ/2) -im*sin(θ/2); -im*sin(θ/2) cos(θ/2)]
R_y(θ) = [cos(θ/2) -sin(θ/2); sin(θ/2) cos(θ/2)]
R_z(θ) = [exp(-im*θ/2) 0; 0 exp(im*θ/2)]
compose(α, θ₀, θ₁, θ₂) = exp(im*α)*R_z(θ₀)*R_y(θ₁)*R_z(θ₂)

This seems to work for the Hadamard, Pauli-X, and Pauli-Z gates when I test it, but when I test it for S, T, and Pauli-Y gates, it does not seem to work.

For S, I expect $\begin{bmatrix}1 & 0 \\ 0 & i\end{bmatrix}$, but I receive $\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$.

For T, I expect $\begin{bmatrix}1 & 0 \\ 0 & \frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i\end{bmatrix}$, but I receive $\begin{bmatrix}1 & 0 \\ 0 & \sqrt{2}\end{bmatrix}$.

For Pauli-Y, I expect $\begin{bmatrix}0 & -i \\ i & 0\end{bmatrix}$, but I receive $\begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix}$.

It seems that the 'recomposed' gate has the values of the original gate written as $\Re(z)+\Im(z)$ instead of $\Re(z)+\Im(z)i$. Does this make any difference functionally or are the gates still equivalent? If there is a difference, is it possible to decompose the original gate and then compose it once again while retaining the complex values?

Answer 1

The matrices should coincide when you check them. There is a mistake in the decompose function.

In the first if clause you define θ₀ twice, second time by using uninitialized θ₂.

BTW, instead of atan you could use the function angle of a complex number. And magnitude is just abs or norm from the LinearAlgebra package.