  We investigate the use of amplitude amplification on the gate-based model of
quantum computing as a means for solving combinatorial optimization problems.
This study focuses primarily on QUBO (quadratic unconstrained binary
optimization) problems, which are well-suited for qubit superposition states.
Specifically, we demonstrate circuit designs which encode QUBOs as `cost
oracle' operations $U_{\textrm{C}}$, which when combined with the standard
Grover diffusion operator $U_{\textrm{s}}$ lead to high probabilities of
measurement for states corresponding to the optimal and near optimal solutions.
In order to achieve these probabilities, a single scalar parameter
$p_{\textrm{s}}$ is required, which we show can be found through a variational
quantum-classical hybrid approach.
