Accelerated Computational Paradigms for Quantum Applications

The rigorous study of electronic behavior at the fundamental scale relies on solving the Schrödinger equation. While this is commonly approached using differential methods, an equally powerful and often more advantageous strategy is the use of an integral formulation, transforming the partial differential equation into a Boundary Integral Equation (BIE) or Volume Integral Equation (VIE). These formulations naturally enforce boundary conditions and are especially powerful for systems confined to specific domains, requiring discretization of only said domains or their boundaries. However, the resulting integral equations lead to dense matrix systems due to the non-local nature of integral operators. For realistic systems with millions of degrees of freedom, the computational cost scales unfavorably, severely limiting the size and complexity of problems that can be addressed. Such prohibitive issue has long been studied in the discipline of Computational Electromagnetism (CEM), which has developed several fast solver techniques to properly approximate matrices, arising from linear integral operators, in such a way to achieve quasi-linear complexity. The present thesis first derives the integral equations of the one-electron time-independent Schrödinger equation, formulating the homogeneous potential case as a BIE suitable for the Boundary Element Method (BEM) and casting the more challenging problem of anisotropic mass within an inhomogeneous potential into a novel CEM-inspired Current Volume Integral Equation. Subsequently, some of the major CEM fast solvers, such as the Multilevel Adaptive Cross Approximation (MLACA), are presented and discussed. Finally, this thesis shows a proof-of-concept demonstration of acceleration in bound state and scattering computations, as a preliminary work for further studies in accelerated quantum solvers.