In this seminar I will discuss efficient quantum algorithms to prepare many-qubit states using highly expressive parameterized quantum circuits. This task is central in several applications, ranging from the simulation of physical and chemical systems to general tasks in machine learning and optimization. I will start by discussing the concept of Quantum Natural Gradient [1] and its efficient implementation [2] using the Simultaneous Perturbation Stochastic Approximation. This concept is a building block of several quantum algorithms for high-dimensional optimization, quantum machine learning, and variational imaginary-time evolution. In the context of simulating the real-time evolution of interacting quantum systems, I will also discuss an efficient variational quantum algorithm named "projected - Variational Quantum Dynamics" (p-VQD) realizing an iterative, global projection of the exact time evolution onto the parameterized manifold [3]. I will conclude highlighting the deep connection of these approaches with similarly motivated classical algorithms in variational Monte Carlo literature and will also highlight possible future improvements. [1] James Stokes, Josh Izaac, Nathan Killoran, and Giuseppe Carleo, Quantum 4, 269 (2020) [2] Julien Gacon, Christa Zoufal, Giuseppe Carleo, and Stefan Woerner, arXiv:2103.09232 (2021) [3] Stefano Barison, Filippo Vicentini, and Giuseppe Carleo, arXiv:2101.04579 (2021)