  The resonator-induced phase gate is a multi-qubit controlled-phase gate for
fixed-frequency superconducting qubits. Through off-resonant driving of a bus
resonator, statically coupled qubits acquire a state-dependent phase. However,
photon loss leads to dephasing during the gate, and any residual entanglement
between the resonator and qubits after the gate leads to decoherence. Here we
consider how to shape the drive pulse to minimize these unwanted effects.
First, we review how the gate's entangling and dephasing rates depend on the
system parameters and validate closed-form solutions against direct numerical
solution of a master equation. Next, we propose spline pulse shapes that reduce
residual qubit-bus entanglement, are robust to imprecise knowledge of the
resonator shift, and can be shortened by using higher-degree polynomials.
Finally, we present a procedure that optimizes over the subspace of pulses that
leave the resonator unpopulated. This finds shaped drive pulses that further
reduce the gate duration. Assuming realistic parameters, we exhibit shaped
pulses that have the potential to realize ~212 ns spline pulse gates and ~120
ns optimized gates with ~6e-4 average gate infidelity. These examples do not
represent fundamental limits of the gate and in principle even shorter gates
may be achievable.
