Parameter concentrations in quantum approximate optimization

The quantum approximate optimization algorithm (QAOA) has become a cornerstone of contemporary applications development. In QAOA, circuit is trained---by repeatedly adjusting parameters---to solve problem instance. Several recent findings have reported parameter concentration effects in QAOA and their presence one folklore: while empirically observed, the concentrations not been defined analytical approaches remain scarce, focusing on limiting system sizes neglecting scaling as size increases. We found that optimal parameters concentrate an inverse polynomial size, providing optimistic result for improving training. Our results are analytically demonstrated at $p=1,2$ (corresponding to 2 4 tunable parameters, respectively). technique also applicable higher depths wherein effect cross verified numerically. Parameter allow training fraction $w<n$ qubits assert these nearly $n$ qubits, thereby reducing time.