The Parisi value (approximately -0.763166...) represents the ground state energy density of the SK model in the thermodynamic limit. The main conjecture is whether QAOA, as depth p approaches infinity, can achieve this optimal value.
One approach is to prove that QAOA can simulate approximate message passing, which optimizes the SK Hamiltonian under the widely believed no-OGP assumption.
See some related work: * Joao Basso, Edward Farhi, Kunal Marwaha, Benjamin Villalonga, and Leo Zhou developed an efficient iterative formula to evaluate QAOA performance on the SK model at arbitrary depth p, with computational complexity O(p² 4^p). This improved upon previous methods and allowed numerical evaluation up to p=20. * Boulebnane and collaborators provided numerical evidence that QAOA obtains a (1-ε) approximation to the optimal SK energy with circuit depth O(n/ε^1.13) in the average case. They achieved ε ≈ 2.2% at p=160 in the infinite-size limit. This was enabled by mapping QAOA energy evaluation to simulating a spin-boson system using matrix product states.