Efficient Z gates for quantum computing

For superconducting qubits, microwave pulses drive rotations around the Bloch sphere. The phase of these drives can be used to generate zero-duration arbitrary virtual $Z$ gates, which, combined with two ${X}_{ensuremath{pi}/2}$ gates, can generate any SU(2) gate. Here we show how to best utilize these virtual $Z$ gates to both improve algorithms and correct pulse errors. We perform randomized benchmarking using a Clifford set of Hadamard and $Z$ gates and show that the error per Clifford is reduced versus a set consisting of standard finite-duration $X$ and $Y$ gates. $Z$ gates can correct unitary rotation errors for weakly anharmonic qubits as an alternative to pulse-shaping techniques such as derivative removal by adiabatic gate (DRAG). We investigate leakage and show that a combination of DRAG pulse shaping to minimize leakage and $Z$ gates to correct rotation errors realizes a 13.3 ns ${X}_{ensuremath{pi}/2}$ gate characterized by low error $[1.95(3)ifmmodetimeselsetexttimesfi{}{10}^{ensuremath{-}4}]$ and low leakage $[3.1(6)ifmmodetimeselsetexttimesfi{}{10}^{ensuremath{-}6}]$. Ultimately leakage is limited by the finite temperature of the qubit, but this limit is two orders of magnitude smaller than pulse errors due to decoherence.