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Is it possible to create controlled gates with an exponent in Cirq? For example, a controlled $\sqrt Z$ gate.

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Yes, it is possible to create controlled gates with an exponent in Cirq.

For the specific case of the Z gate, Cirq includes a dedicated CZ gate that can be raised to a power:

cs = cirq.CZ**0.5

More generally, cirq.ControlledGate works on any gate. It's a bit clunkier than the dedicated gates, but it does support being raised to a power (as long as the gate it is being applied to can be raised to a power). The following two lines are equivalent:

controlled_sqrt_y = cirq.ControlledGate(cirq.Y**0.5)
controlled_sqrt_y = cirq.ControlledGate(cirq.Y)**0.5

You can also raise to a power after applying to qubits, which is syntactically convenient:

cs_on_ab = cirq.CZ(a, b)**0.5
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Looking at the documentation and the GitHub, there is a something called ControlledGate. This class is said to augment existing gates with a control qubit. You can look at the test file. I can see line 72 :

cxa = cirq.ControlledGate(cirq.X**cirq.Symbol('a'))

Could you try:

gate = cirq.ControlledGate(cirq.X**0.5) ?

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