Dynamically Corrected Quantum Control: A Geometrical Framework

Implementing high-fidelity quantum control and suppressing the unwanted environmental noise has been one of the essential challenges in developing quantum information technologies. In the past, driving pulse sequences based on Dirac delta functions or square wave functions, such as Hahn spin echo or CPMG, have been developed to dynamically correcting the noise effects. However, implementing these ideal pulses with high fidelity is a challenging task in real experiments.

In this thesis, we provide a new and simple method to explore the entire solution space of driving pulse shapes that suppress environmental noise in the evolution of the system. In this method, any single-qubit phase gate that is first-order robust against quasi-static transversal noise corresponds to a closed curve on a two-dimensional plane, and more general first-order robust single-qubit gates correspond to closed three-dimensional space curves. Second-order robust gates correspond to closed curves having the property that their projection onto any two-dimensional planes shall enclose a zero net area. The driving pulse shapes that implement the gates can be determined by the curvature, torsion, and the length of the curve. By utilizing the framework it is possible to obtain globally optimal solutions in pulse shaping in respect of experimental constraints by mapping them into geometrical optimization problems. One such problem we solved is to prove that the fastest possible single-qubit phase gates that are second-order noise-resistant shall be implemented using sign-flipping square functions. Since square waves are not experimentally feasible, we provide a method to smooth these pulses with minimal loss in gate speed while maintaining the robustness, based on the geometrical framework. This framework can also be useful in diagnosing the noise-cancellation properties of pulse shapes generated from numerical methods such as GRAPE. We show that this method for pulse shaping can significantly improve the fidelity of single-qubit gates through numerical simulation.