$\kappa$-Parametrized Boltzmann Distribution Sampling on Quantum Approximate Optimization Algorithm

Generative models use training data to produce new samples that resemble the original, with applications in classification and recommendation systems. Restricted Boltzmann Machines (RBMs) are at the core of generative models but suffer from the computational intractability of training, which relies on sampling an unknown Boltzmann Distribution (BD). The Quantum Approximate Optimization Algorithm (QAOA) can assist RBM training, potentially providing benefits over strictly classical strategies. One particular problem is determining the quantum circuit depth for an arbitrary problem size that yields the expected sampling results. In this study, we evaluate QAOA-assisted sampling of BDs using Qiskit's Aer Simulator by analysing the inverse effective temperature of the distribution, $\kappa$, for tractable problem sizes. The results provide numerical evidence, based on Qiskit Aer simulations, that parameterized annealing-scheduled QAOA (PAQ-QAOA) can approximate RBM Boltzmann sampling for the tractable instances considered, while hardware performance and scaling remain open questions.