Depth scaling of unstructured search via quantum approximate optimization

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.