Rotation angles of unitary operator

Question

Given a complex unitary $2*2$ matrix $A$ that represents some quantum gate on a single qubit.

What is the formula to extract to $\theta_X, \theta_Y, \theta_Z $ rotations around each one of the axes in the Bloch Sphere?

I will mention that I know that :

$$U_3(\theta, \phi, \lambda) = \begin{pmatrix}\cos(\theta/2) & -e^{i\lambda}\sin(\theta/2) \\ e^{i\phi}\sin(\theta/2) & e^{i\lambda + i\phi}\cos(\theta/2)\end{pmatrix}$$

But while trying to use it, I struggled with $cos()=complex-value$ that gave me complex angles

Answer 1

The Euler angles that you obtain depend on the decomposition that you implement over your 1 qubit gate. i.e it is possible to decompose any 1 qubit gate $U$ as:

$$ U = e^{i\gamma}R_z(\theta)R_y(\phi)R_z(\lambda)$$

or

$$ U = e^{i\gamma}R_z(\theta)R_x(\phi)R_z(\lambda)$$

In those cases, you'd obtain 2 $\theta_Z$ angles and 1 $\theta_Y$ or 2 $\theta_Z$ angles and 1 $\theta_X$ respectively. You can fin the methodology for some of the decompositions in this page:

https://purva-thakre.github.io/purva-blog/gsoc/qutip/single-qubit-example/

Qiskit has a class to implement several types of decompositions:

https://qiskit.org/documentation/stubs/qiskit.quantum_info.OneQubitEulerDecomposer.html

You can see the source code to know how they work:

https://qiskit.org/documentation/_modules/qiskit/quantum_info/synthesis/one_qubit_decompose.html#OneQubitEulerDecomposer