  Quantum optimal control is an important technology that enables fast state
preparation and gate design. In the absence of an analytic solution, most
quantum optimal control methods rely on an iterative scheme to update the
solution estimate. At present, the convergence rate of existing solvers is at
most superlinear. This paper develops a new general purpose solver for quantum
optimal control based on the PRojection Operator Newton method for Trajectory
Optimization, or PRONTO. Specifically, the proposed approach uses a projection
operator to incorporate the Schr\"odinger equation directly into the cost
function, which is then minimized using a quasi-Newton method. At each
iteration, the descent direction is obtained by computing the analytic solution
to a Linear-Quadratic trajectory optimization problem. The resulting method
guarantees monotonic convergence at every iteration and quadratic convergence
in proximity of the solution. To highlight the potential of PRONTO, we present
an numerical example that employs it to solve the optimal state-to-state
mapping problem for a qubit and compares its performance to a state-of-the-art
quadratic optimal control method.
