A QAOA-inspired circuit for Grover's unstructured search using a transverse field

Inspired by a class of algorithms proposed by Farhi et al., 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 introduced a novel technique, a spin-coherent-state approach, 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 the first example of a QAOA circuit that demonstrates a quantum advantage with 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.