  Variational Quantum Algorithms (VQAs) and Quantum Machine Learning (QML)
models train a parametrized quantum circuit to solve a given learning task. The
success of these algorithms greatly hinges on appropriately choosing an ansatz
for the quantum circuit. Perhaps one of the most famous ansatzes is the
one-dimensional layered Hardware Efficient Ansatz (HEA), which seeks to
minimize the effect of hardware noise by using native gates and connectives.
The use of this HEA has generated a certain ambivalence arising from the fact
that while it suffers from barren plateaus at long depths, it can also avoid
them at shallow ones. In this work, we attempt to determine whether one should,
or should not, use a HEA. We rigorously identify scenarios where shallow HEAs
should likely be avoided (e.g., VQA or QML tasks with data satisfying a volume
law of entanglement). More importantly, we identify a Goldilocks scenario where
shallow HEAs could achieve a quantum speedup: QML tasks with data satisfying an
area law of entanglement. We provide examples for such scenario (such as
Gaussian diagonal ensemble random Hamiltonian discrimination), and we show that
in these cases a shallow HEA is always trainable and that there exists an
anti-concentration of loss function values. Our work highlights the crucial
role that input states play in the trainability of a parametrized quantum
circuit, a phenomenon that is verified in our numerics.
