  Variational quantum algorithms are the centerpiece of modern quantum
programming. These algorithms involve training parameterized quantum circuits
using a classical co-processor, an approach adapted partly from classical
machine learning. An important subclass of these algorithms, designed for
combinatorial optimization on currrent quantum hardware, is the quantum
approximate optimization algorithm (QAOA). It is known that problem density - a
problem constraint to variable ratio - induces under-parametrization in fixed
depth QAOA. Density dependent performance has been reported in the literature,
yet the circuit depth required to achieve fixed performance (henceforth called
critical depth) remained unknown. Here, we propose a predictive model, based on
a logistic saturation conjecture for critical depth scaling with respect to
density. Focusing on random instances of MAX-2-SAT, we test our predictive
model against simulated data with up to 15 qubits. We report the average
critical depth, required to attain a success probability of 0.7, saturates at a
value of 10 for densities beyond 4. We observe the predictive model to describe
the simulated data within a $3\sigma$ confidence interval. Furthermore, based
on the model, a linear trend for the critical depth with respect problem size
is recovered for the range of 5 to 15 qubits.
