Hansen, I; Seedhouse, AE; Saraiva, A; Dzurak, AS; Yang, CH
The filter function formalism from quantum control theory is typically used to determine the noise susceptibility of pulse sequences by looking at the overlap between the filter function of the sequence and the noise power spectral density. Importantly, the square modulus of the filter function is used for this method. In this work we show that by using the square modulus one neglects valuable information about the system dynamics. We take advantage of the full filter function by including information about the phase of the perturbation and the resulting rotation axis. By decomposing the filter function with phase preservation before taking the modulus, we are able to consider the contributions to x, y, and z rotations separately. Continuously driven systems provide noise protection in the form of dynamical decoupling by canceling low-frequency noise; however, generating control pulses synchronously with an arbitrary driving field is not trivial. Using the decomposed filter function we look at the controllability of a system under arbitrary driving fields, as well as the noise susceptibility, and also relate the filter function to the geometric formalism.