  The performance of variational quantum algorithms relies on the success of
using quantum and classical computing resources in tandem. Here, we study how
these quantum and classical components interrelate. In particular, we focus on
algorithms for solving the combinatorial optimization problem MaxCut, and study
how the structure of the classical optimization landscape relates to the
quantum circuit used to evaluate the MaxCut objective function. In order to
analytically characterize the impact of quantum features on the critical points
of the landscape, we consider a family of quantum circuit ans\"atze composed of
mutually commuting elements. We identify multiqubit operations as a key
resource and show that overparameterization allows for obtaining favorable
landscapes. Namely, we prove that an ansatz from this family containing
exponentially many variational parameters yields a landscape free of local
optima for generic graphs. However, we further prove that these ans\"atze do
not offer superpolynomial advantages over purely classical MaxCut algorithms.
We then present a series of numerical experiments illustrating that
noncommutativity and entanglement are important features for improving
algorithm performance.
