Abstract

Gradient-based optimization methods play a central role in both quantum optimal control and variational quantum algorithms. Existing gradient estimation techniques, such as parameter-shift rules, often rely on restrictive assumptions regarding the spectral structure of the underlying generators and are typically formulated for finite-dimensional parameter spaces. In contrast, quantum control problems naturally involve time-dependent control functions, leading to optimization problems over infinite-dimensional function spaces.

This work develops a functional analytic and interpolation-based framework for the approximation of quantum control functionals and their variations from sampled data. The considered cost functionals are generated by time-dependent quantum dynamics and depend on continuous control pulses through the corresponding time-ordered evolution operators. By discretizing the dynamics and evaluating the functional for sampled perturbations of the control function, the problem is transformed into a data-driven interpolation problem.

The central construction of the thesis is based on polynomial surrogate models obtained through Vandermonde interpolation. First, local polynomial approximations are constructed from sampled functional evaluations corresponding to shifted control functions. Subsequently, a second interpolation procedure in time is introduced to derive time-dependent polynomial surrogate models. This yields a framework in which the functional can be approximated through a polynomial representation whose coefficients evolve continuously in time.

The developed approach is motivated by applications in quantum computing and quantum optimal control, where access to exact gradients may be limited by hardware constraints, noise, or inaccessible system information. The presented framework therefore aims to bridge concepts from functional analysis, numerical interpolation theory, and quantum control optimization toward experimentally applicable gradient approximation methods for quantum technologies.