  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.
