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