Randomness is a resource that enables applications such as efficient probabilistic algorithms, numerical integration, simulation, and optimization. In the last few years it was realized that quantum devices can generate probability distributions that are inaccessible with classical means. Hybrid Quantum Computational models combine classical processing with these quantum sampling machines to obtain computational advantage in some tasks. Moreover, NISQ (Noisy, Intermediate-Scale Quantum) technology may suffice to obtain this advantage in the near term, long before we can build large-scale, universal quantum computers. PHOQUSING aims to implement PHOtonic Quantum SamplING machines based on large, reconfigurable interferometers with active feedback, and state-of-the-art photon sources based both on quantum dots and parametric down-conversion. We will overview the different architectures enabling the generation of these hard-to-sample distributions using integrated photonics, optimizing the designs and studying the tolerance to errors. We will build two quantum sampling machines with different technologies, as a way to do cross-checks while exploiting all advantages of each platform. These machines will establish a new state-of-the-art in photonic reconfigurability, system complexity, and integration. Finally, we plan to perform first, proof-of-principle demonstrations of Hybrid Quantum Computation applications in optimization, machine learning, and graph theory. The PHOQUSING team includes long-term scientific collaborators who were among the first to demonstrate quantum photonic samplers; two of the leading European start-ups in the relevant quantum technologies; and theoretical experts in photonics and quantum information science. This project will help establish photonics as a leading new quantum computational technology in Europe, addressing the science-to-technology transition towards a new industrial sector with a large foreseeable economic impact.