Compiling Arbitrary Single-Qubit Gates Via the Phase-Shifts of Microwave Pulses

We give an arbitrary single-qubit gate compilation scheme on superconducting processors that takes advantage of tuning the phase shift of microwave pulses to obtain a continuous gate set. This scheme is compatible with any two-qubit gate, and we only need to calibrate the $X_{\pi }$ and $X_{\pi \mathrm{/}2}$ pulses. We implement this on fluxonium and obtain state-of-the-art fidelities. We give two other schemes: the first requires one $X_{\pi }$ pulse and one pulse with a variable rotation angle, and the second requires four $X_{\pi \mathrm{/}2}$ pulses. We also find that if we can do virtual $Z$ gates, then we can also do virtual gates around any axis. Our results apply to any physical platform that natively supports virtual $Z$.