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_{π /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_{π /2} gate characterized by low error [1.95 (3 ) ×10^-4] and low leakage [3.1 (6 ) ×10^-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.