  Variational quantum algorithms have become the de facto model for current
quantum computations. A prominent example of such algorithms -- the quantum
approximate optimization algorithm (QAOA) -- was originally designed for
combinatorial optimization tasks, but has been shown to be successful for a
variety of other problems. However, for most of these problems the optimal
circuit depth remains unknown. One such problem is unstructured search which
consists on finding a particular bit string, or equivalently, preparing a state
of high overlap with a target state. To bound the optimal QAOA depth for such
problem we build on its known solution in a continuous time quantum walk
(CTQW). We trotterize a CTQW to recover a QAOA sequence, and employ recent
advances on the theory of Trotter formulas to bound the query complexity
(circuit depth) needed to prepare a state of almost perfect overlap with the
target state. The obtained complexity exceeds the Grover's algorithm complexity
$O\left(N^\frac{1}{2}\right)$, but remains smaller than $O
\left(N^{\frac{1}{2}+c}\right)$ for any $c>0$, which shows quantum advantage of
QAOA over classical solutions. We verify our analytical predictions by
numerical simulations of up to 68 qubits, which demonstrate that our result
overestimates the number of QAOA layers resulting from a trotterized CTQW by at
most a polynomial factor.
