Determining probability density functions with adiabatic quantum computing

The two main approaches to quantum computing are gate-based computation and analog computation, which are polynomially equivalent in terms of complexity, and they are often seen as alternatives to each other. In this work, we present a method for fitting one-dimensional probability distributions as a practical example of how analog and gate-based computation can be used together to perform different tasks within a single algorithm. In particular, we propose a strategy for encoding data within an adiabatic evolution model, which accomodates the fitting of strictly monotonic functions, as it is the cumulative distribution function of a dataset. Subsequently, we use a Trotter-bounded procedure to translate the adiabatic evolution into a quantum circuit in which the evolution time t is identified with the parameters of the circuit. This facilitates computing the probability density as derivative of the cumulative function using parameter shift rules.