Ultrafast quantum gate

A quantum bit, or qubit, is described by a point on a sphere. We want to rotate the sphere such that the point on the south pole moves to the north pole and vice versa. This operation is called the quantum NOT gate. We may just rotate the sphere around the horitontal axis by 180 degree. However, similary to the Earth, the sphere already rotates around the z-axis at the qubit frequency. Thus, the direction of the angular velocity, denoted by the orange line on the left figure, always have some component along the z-direction. Still, there is a way to implement the NOT gate: periodically shifting the rotational axis around the z-axis as shown on the top left panel. On the top right panel, you can see that the trajectory on the sphere arrives at somewhere near the north pole. However, for this scheme to work, the rotating axis should not deviate much from the z-axis, otherwise the accuracy falls. For this, the driving strength, Ω, should be sufficiently smaller than the qubit frequency ω. On the top left panel, the Ω is smaller than ω. The weaker the Ω is, the better the accuracy of the operation. In this case, the operation should be long compared to the qubit period 2π/ω, which is not favorable since the qubit can degrade due to its interaction with its environment, known as the decoherence process. Can we make the operation faster while maintaining the accuracy? We can try a pulse which is shorter than the qubit period as shown in the bottom left panel. However, the result shown on the bottom right panel tells us that the accuracy should be improved so that qubit state can go the north pole. How should we drive the qubit to achieve high accuracy at such fast timescale?