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arXiv · 1702.02577

Near-optimal quantum circuit for Grover's unstructured search using a transverse field

Zhang Jiang·Eleanor G. Rieffel·Zhihui Wang
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Abstract

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 $Θ(1/\sqrt N)$, and the overlaps are of order $Θ(1)$, yielding a nearly optimal query complexity of $T\simeq \sqrt N (π/2\sqrt 2\,)$. 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.

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Zhang Jiang, Eleanor G. Rieffel, Zhihui Wang. 2017-06-12. Near-optimal quantum circuit for Grover's unstructured search using a transverse field. https://doi.org/10.1103/physreva.95.062317

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