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I'm trying to implement the parametric $\text{iSWAP}$ gate, also known as $\text{XY}(\theta)$, in Qiskit.

$$ \text{XY}(\theta)= \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos{\theta/2} & i\sin{\theta/2} & 0 \\ 0 & i\sin{\theta/2} & \cos{\theta/2} & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}. $$

Furthermore, once I've implemented this gate I would like to use it in order to decompose quantum circuits in terms of single-qubit gates and $\text{XY}(\theta)$ as the only two-qubit gate.

What are the steps that I need to do in order to achieve this?

Cheers!

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You can implement an arbitrary unitary matrix as a Unitary Gate using the UnitaryGate class that qiskit provides. This class can be instantiated with an np.ndarray containing the definition of your valid unitary gate and a name for that unitary. It would look something like -

from qiskit.extensions import UnitaryGate
def U_theta(theta):
    arr = np.array([[1,0,0,0],
                    [0,np.cos(theta/2), 1j*np.sin(theta/2), 0],
                    [0,1j*np.sin(theta/2),np.cos(theta/2),0],
                    [0,0,0,1]])
    u_gate = UnitaryGate(data = arr,label = 'iSwap'+str(theta))
    return u_gate

gate1 = U_theta(0.5) # would return a gate with theta = 0.5

This gate can be used in your quantum circuits and attached to the relevant qubits using the compose function.

Regarding transpilation, I am not sure if qiskit allows us to give a custom basis set for transpilation but you could always keep a pre-transpiled version of your $iSWAP$ gate and just attach it to your current circuit.

Hope it helps!

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    $\begingroup$ This answer might helpl with the custom transpilation. $\endgroup$ – epelaaez Jun 29 at 15:13
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For the first part of your question, we have $$XY(\theta) = exp(-i {\frac{\theta}{2}} (X{\otimes}X + Y{\otimes}Y))$$ And since $X{\otimes}X$ and $Y{\otimes}Y$ commute, we can write it as $$XY(\theta) = exp(-i {\frac{\theta}{2}} X{\otimes}X) exp(-i {\frac{\theta}{2}} Y{\otimes}Y)$$ Qiskit already has these two gates: $$R_{XX}(\theta) = exp(-i {\frac{\theta}{2}} X{\otimes}X)$$ And, $$R_{YY}(\theta) = exp(-i {\frac{\theta}{2}} Y{\otimes}Y)$$

Hence, the implementation of $XY(\theta)$ as a parameterized gate in Qiskit will be as simple as

from qiskit import QuantumCircuit
from qiskit.circuit import Parameter

theta = Parameter('θ')

circuit = QuantumCircuit(2)
circuit.rxx(theta, 0, 1)
circuit.ryy(theta, 0, 1)
param_iswap = circuit.to_gate()

Another Solution

If you want to use more basic gates than rxx and ryy, you can use Qiskit's Operator Flow:

H = 0.5 * ((X^X) + (Y^Y))
theta = Parameter('θ')
evolution_op = (theta * H).exp_i() # exp(-iθH)

trotterized_op = PauliTrotterEvolution(trotter_mode = Suzuki(order = 1, reps = 1)).convert(evolution_op)
circuit = trotterized_op.to_circuit()
circuit.draw('mpl')

The composition: enter image description here And as before

param_iswap = circuit.to_gate()

For the second part of your question, I think the best answer you can have is the one mentioned in the comments by @epelaaez, as it is recent and from a member of Qiskit's development team.

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