  Inspired by a class of algorithms proposed by Farhi et al. (arXiv:1411.4028),
namely the quantum approximate optimization algorithm (QAOA), we present a
circuit-based quantum algorithm to search for a needle in a haystack, obtaining
the same quadratic speedup achieved by Grover's original algorithm. In our
algorithm, the problem Hamiltonian (oracle) and a transverse field are applied
alternately to the system in a periodic manner. We introduce a technique, based
on spin-coherent states, to analyze the composite unitary in a single period.
This composite unitary drives a closed transition between two states that have
high degrees of overlap with the initial state and the target state,
respectively. The transition rate in our algorithm is of order $\Theta(1/\sqrt
N)$, and the overlaps are of order $\Theta(1)$, yielding a nearly optimal query
complexity of $T\simeq \sqrt N (\pi/2\sqrt 2\,)$. Our algorithm is a QAOA
circuit that demonstrates a quantum advantage with a large number of iterations
that is not derived from Trotterization of an adiabatic quantum optimization
(AQO) algorithm. It also suggests that the analysis required to understand QAOA
circuits involves a very different process from estimating the energy gap of a
Hamiltonian in AQO.
