Parameterization and optimizability of pulse-level VQEs

In conventional variational quantum eigensolvers (VQEs), trial states are prepared by applying series of parameterized gates to a reference state, with the gate parameters being varied to minimize the energy of the target system. Recognizing that the gates are intermediates which are ultimately compiled into a set of control pulses to be applied to each qubit in the lab, the recently proposed ctrl-VQE algorithm takes the amplitudes, frequencies, and phases of the pulse as the variational parameters used to minimize the molecular energy. In this work, we explore how all three degrees of freedom interrelate with one another. To this end, we consider several distinct strategies to parameterize the control pulses, assessing each one through numerical simulations of a transmon-like device. For each parameterization, we contrast the pulse duration required to prepare a good ansatz, and the difficulty to optimize that ansatz from a well-defined initial state. We deduce several guiding heuristics to implement practical ctrl-VQE in hardware, which we anticipate will generalize for generic device architectures.