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Is there a way to create a rotation gate which has half the angle of some implementable gate?

I am looking to implement a gate on Quirk which allows for standard time-dependent rotations

$$R_x(\theta=t)=e^{iXt}$$

but I'd like to construct the same gate but with $\theta=\frac{t}{2}$ so I can have two different oscillation frequencies.

Is there some standard way to do this?

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(This answer is specific to the context of the question, which is about doing this construction out of Quirk's time dependent gates.)

It's not possible to have proper half-speed time-dependent rotations in Quirk. Behind the scenes there is actually a value $t$ that is varying from 0 to 1 and then cycling. If you managed to find a construction to cut the angle in half, your rotation would jump from 180 degrees to 0 degrees when $t$ went from 1 to 0.

If that's acceptable, then the simplest way to achieve it is by using the parameterized rotation gates and a counting gate:

circuit

An alternative trade-off do is define double-speed rotations by using pairs of time-dependent rotation gates, and then work in units of $2t$ instead of $t$. You can use custom circuit gates to fuse them into a single block. If custom gates annoy you (they are by far the weakest part of the UX), the time-dependent gradient gate gives you a series of faster and faster rotations:

gradient gate

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