Implement approximate $Ry(2\pi /3)$ with gate set {$Rx(\pi),Ry(\pi),Rz(\pi),Rx(\pm \pi/2),Ry(\pm \pi/2),Rz(\pm \pi/2),T,S,H,CZ$}

Question

With gate sets {$Rx(\pi),Ry(\pi),Rz(\pi),Rx(\pm \pi/2),Ry(\pm \pi/2),Rz(\pm \pi/2),T,S,H,CZ$}, I want to implement an approximate $2\pi/3$ rotation around Y axis. An ancilla qubit can be used as well. Is there any efficient way to do this?

Answer 1

This is a bit of a circular answer. But... if you need to do a lot of these operations, you can catalyze it. Given the phase gradient state $|0\rangle + e^{i 2 \pi / 3} |1\rangle + e^{i 4 \pi / 3} |2\rangle$ you can apply a controlled mod 3 increment to get phase kickback of exactly 120 degrees onto the control.

If you decompose the Toffolis inline, using 7 T gates each, two T gates will cancel and you'll use 12 T gates. If you can do measurement with feedback you can instead decompose the Toffolis using 4 T gates each.

Of course, to get the phase gradient state in the first place, you have to have done the operation you're trying to do. My point is that, instead of directly applying the operation each time, you can do it the hard way twice to make this particular state and then the cost suddenly drops because catalysis becomes possible.