  Variational Quantum Algorithms (VQAs) have received considerable attention
due to their potential for achieving near-term quantum advantage. However, more
work is needed to understand their scalability. One known scaling result for
VQAs is barren plateaus, where certain circumstances lead to exponentially
vanishing gradients. It is common folklore that problem-inspired ansatzes avoid
barren plateaus, but in fact, very little is known about their gradient
scaling. In this work we employ tools from quantum optimal control to develop a
framework that can diagnose the presence or absence of barren plateaus for
problem-inspired ansatzes. Such ansatzes include the Quantum Alternating
Operator Ansatz (QAOA), the Hamiltonian Variational Ansatz (HVA), and others.
With our framework, we prove that avoiding barren plateaus for these ansatzes
is not always guaranteed. Specifically, we show that the gradient scaling of
the VQA depends on the degree of controllability of the system, and hence can
be diagnosed through the dynamical Lie algebra $\mathfrak{g}$ obtained from the
generators of the ansatz. We analyze the existence of barren plateaus in QAOA
and HVA ansatzes, and we highlight the role of the input state, as different
initial states can lead to the presence or absence of barren plateaus. Taken
together, our results provide a framework for trainability-aware ansatz design
strategies that do not come at the cost of extra quantum resources. Moreover,
we prove no-go results for obtaining ground states with variational ansatzes
for controllable system such as spin glasses. Our work establishes a link
between the existence of barren plateaus and the scaling of the dimension of
$\mathfrak{g}$.
