A Quantum Approximate Optimization Algorithm for continuous problems

We introduce a quantum approximate optimization algorithm (QAOA) for continuous optimization. The algorithm is based on the dynamics of a quantum system moving in an energy potential which encodes the objective function. By approximating the dynamics at finite time steps, the algorithm can be expressed as alternating evolution under two non-commuting Hamiltonians. We show that each step of the algorithm updates the wavefunction in the direction of its local gradient, with an additional momentum-dependent displacement. For initial states in a superposition over many points, this method can therefore be interpreted as a coherent version of gradient descent, i.e., 'gradient descent in superposition.' This approach can be used for both constrained and unconstrained optimization. In terms of computational complexity, we show how variants of the algorithm can recover continuous-variable Grover search, and how a single iteration can replicate continuous-variable instantaneous quantum polynomial circuits. We also discuss how the algorithm can be adapted to solve discrete optimization problems. Finally, we test the algorithm through numerical simulation in optimizing the Styblinski-Tang function.