Implementing controlled $\frac{\pi}{2}$ y and z rotation in Qiskit

Question

I'm trying to implement a controlled $\frac{\pi}{2}$ rotation along y axis and z axis (two individual gates) in Qiskit but I'm stuck. My thought was calculating the square root of the y and z pauli matrices (done), and implement using the CU3 gate. However, I wasn't able to find the correct parameters ($\theta, \phi, \lambda$) for the matrices I got. Is there a workaround for this? Thanks in advance.

Edit: I know if I remove the overall phase of $\sqrt{Y}$ then I can get the correct parameters, but I think since this is not an overall phase of all qubit so if I remove the phase I'll get a different gate?

Answer 1

The circuit for the controlled $R_y(t)$ that is true also for $R_z$ (just change the indexes). For $R_x$, instead of $cx$ we should use $cz = I \otimes H \cdot cx \cdot I \otimes H$:

The corresponding code:

t = np.pi/2 # for the question's case
circuit = QuantumCircuit(2)
circuit.ry(t/2, 1)
circuit.cx(0, 1)
circuit.ry(-t/2, 1)
circuit.cx(0, 1)

Here are the expressions for $cx$ and $R_y(t)$:

\begin{equation*} cx = |0\rangle \langle 0| I + |1\rangle \langle 1 | X \qquad R_y(\theta) = \cos(\theta/2)I - i \sin(\theta/2)Y \end{equation*}

By taking into account that $XYX = -Y$ and thus $XR_y(\theta)X = R_y(-\theta)$:

$$cx \cdot I \otimes R_y(-t/2) \cdot cx = |0\rangle \langle0|R_y(-t/2) + |1\rangle \langle 1| R_y(t/2)$$

and if we multiply this with $R_y(t/2)$ from the right side we will obtain:

$$|0\rangle \langle0| I + |1\rangle \langle 1| R_y(t) = cR_y(t)$$

because $R_y(\theta_1)R_y(\theta_2) = R_y(\theta_1 + \theta_2)$.

Answer 2

You can use gate $CU3$ directly on IBM Q and then leave the gate decomposition on transpiler.

There are relations between rotational gates and U3 gate:

$$ Ry(\theta) = U3(\theta, 0, 0) $$

$$ Rx(\theta) = U3(\theta, -\pi/2, \pi/2). $$

Answer 3

If you don't know the decomposition for this kind of thing you can just program it at a high level in Qiskit and then rely on the compiler to decompose it.

For this you can do two things. Either start with an RYGate(pi/2) then control it; or start with a YGate(), raise it to power 1/2, then control it.

(Note the answer from these two approaches will differ by a relative phase, as YGate and RYGate(pi) have a global phase difference in their matrix definitions in Qiskit).

For the first approach, here's the code:

import numpy as np
import qiskit as qk
from qiskit.circuit.library import RYGate

circ = qk.QuantumCircuit(2)

circ.append(RYGate(np.pi/2).control(), [0, 1])

print('before...')
print(circ.draw())

new_circ = qk.transpile(circ, basis_gates=['u3', 'cx'], optimization_level=3)

print('after...')
print(new_circ_1.draw(fold=100))

before...

q_0: ─────■──────
     ┌────┴─────┐
q_1: ┤ RY(pi/2) ├
     └──────────┘
after...

q_0: ──────────────────■─────────────────────■──────────────────────
     ┌──────────────┐┌─┴─┐┌───────────────┐┌─┴─┐┌──────────────────┐
q_1: ┤ U3(0,0,pi/2) ├┤ X ├┤ U3(-pi/4,0,0) ├┤ X ├┤ U3(pi/4,-pi/2,0) ├
     └──────────────┘└───┘└───────────────┘└───┘└──────────────────┘