We provide a further exploration of a type of nonadiabatic rapid passage known as twisted rapid passage (TRP). This class of rapid passage pulses allows a qubit to be driven through resonance multiple times during a single TRP sweep. The multiple resonances give rise to controllable quantum interference effects that provide direct control over qubit transitions so that transitions can be greatly enhanced or suppressed. These quantum interference effects have recently been observed experimentally. We examine here a number of new TRP pulse profiles and show that they can be used to implement a quantum NOT gate that operates both nonadiabatically and with sufficient reliability to surpass the accuracy threshold needed for the gate to be used as part of a fault-tolerant scheme of quantum computation. These new TRP pulse profiles are shown to provide performance advantages over TRP pulses previously considered in the literature.