  Quantum Approximate Optimization Algorithm (QAOA) is a leading candidate
algorithm for solving combinatorial optimization problems on quantum computers.
However, in many cases QAOA requires computationally intensive parameter
optimization. The challenge of parameter optimization is particularly acute in
the case of weighted problems, for which the eigenvalues of the phase operator
are non-integer and the QAOA energy landscape is not periodic. In this work, we
develop parameter setting heuristics for QAOA applied to a general class of
weighted problems. First, we derive optimal parameters for QAOA with depth
$p=1$ applied to the weighted MaxCut problem under different assumptions on the
weights. In particular, we rigorously prove the conventional wisdom that in the
average case the first local optimum near zero gives globally-optimal QAOA
parameters. Second, for $p\geq 1$ we prove that the QAOA energy landscape for
weighted MaxCut approaches that for the unweighted case under a simple
rescaling of parameters. Therefore, we can use parameters previously obtained
for unweighted MaxCut for weighted problems. Finally, we prove that for $p=1$
the QAOA objective sharply concentrates around its expectation, which means
that our parameter setting rules hold with high probability for a random
weighted instance. We numerically validate this approach on general weighted
graphs and show that on average the QAOA energy with the proposed fixed
parameters is only $1.1$ percentage points away from that with optimized
parameters. Third, we propose a general heuristic rescaling scheme inspired by
the analytical results for weighted MaxCut and demonstrate its effectiveness
using QAOA with the XY Hamming-weight-preserving mixer applied to the portfolio
optimization problem. Our heuristic improves the convergence of local
optimizers, reducing the number of iterations by 7.4x on average.
