Connection between single-layer Quantum Approximate Optimization Algorithm interferometry and thermal distributions sampling

The Quantum Approximate Optimization Algorithm (QAOA) is an algorithm originally proposed to find approximate solutions to Combinatorial Optimization problems on quantum computers. However, the algorithm has also attracted interest for sampling purposes since it was theoretically demonstrated under reasonable complexity assumptions that one layer of the algorithm already engineers a probability distribution beyond what can be simulated by classical computers. In this regard, a recent study has shown as well that, in universal Ising models, this global probability distribution resembles pure but thermal-like distributions at a temperature that depends on internal correlations of the spin model. In this work, through an interferometric interpretation of the algorithm, we extend the theoretical derivation of the amplitudes of the eigenstates, and the Boltzmann distributions generated by single-layer QAOA. We also review the implications that this behavior has from both a practical and fundamental perspective.