  This paper presents the Maximum Amplification Optimisation Algorithm (MAOA),
a novel quantum algorithm designed for combinatorial optimisation in the
restricted circuit depth context of near-term quantum computing. The MAOA first
produces a quantum state in which the optimal solutions to a problem are
amplified to the maximum extent possible subject to a given restricted circuit
depth. Subsequent repeated preparation and measurement of this maximally
amplified state produces solutions of the highest quality as efficiently as
possible. The MAOA performs considerably better than other near-term quantum
algorithms, such as the Quantum Approximate Optimisation Algorithm (QAOA), as
it amplifies optimal solutions significantly more and does so without the
computationally demanding variational procedure required by these other
algorithms. Additionally, a restricted circuit depth modification of the
existing Grover adaptive search is introduced. This modified algorithm is
referred to as the restricted Grover adaptive search (RGAS), and provides a
useful comparison to the MAOA. The MAOA and RGAS are simulated on a practical
vehicle routing problem, a computationally demanding portfolio optimisation
problem, and an arbitrarily large problem with normally distributed solution
qualities. In all cases, the MAOA and RGAS are shown to provide substantial
speedup over classical random sampling in finding optimal solutions, while the
MAOA consistently outperforms the RGAS. The speedup provided by the MAOA is
quantified by demonstrating numerical convergence to a theoretically derived
upper bound.
