How can I decompose gate $R_{y}(\theta)$ into gate $R_{x}(\alpha)$ and gate $R_{z}(\beta)$. And how to perform it on Qiskit, is there any function to do it.


Qiskit supports translating to different continuous basis sets by specifying the basis_gates in the transpile method. So in your case you could just do

>>> from qiskit import QuantumCircuit, transpile
>>> from qiskit.circuit import Parameter
>>> circuit = QuantumCircuit(1)
>>> circuit.ry(Parameter('theta'), 0)
>>> rxrz_basis = transpile(circuit, basis_gates=['rx', 'rz'])
>>> rxrz_basis.draw()
q_0: ┤ RX(π/2) ├┤ RZ(theta + π) ├┤ RX(π/2) ├┤ RZ(3π) ├

So you see that $$ R_Y(\theta) = R_X\left(\frac{\pi}{2}\right) R_Z(\theta + \pi) R_X\left(\frac{\pi}{2}\right) R_Z(\pi) $$ where I replaced the $3\pi$ by $\pi$ in the last RZ rotation since the operation is mod $2\pi$.


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