Optimized Single Qubit Gates via Filter Function Design

Error mitigation protocols represent a class of techniques that are meant to reduce gate error rates via specially designed control sequences. Here, we discuss a protocol that optimizes control sequences to combat temporally correlated noise, a class of noise that is known to be particularly detrimental to quantum error correction. We leverage the filter function formalism to transform the control problem into a filter design problem and show that the frequency response of a quantum system can be carefully tailored even in the presence of multi-axis noise. The control ansatz is specifically chosen to be a functional expansion of Slepians, a discrete time basis known to be optimally concentrated in time and frequency, and quite attractive when faced with experimental control hardware constraints. Using gradient ascent, we obtain optimized filter functions and investigate the relationship between filter function design, control bandwidth, and noise characteristics. We show that under certain noise conditions, it is possible to achieve high fidelity, arbitrary single qubit gate operations that simultaneously yield highpass or bandpass filter functions.