Quantum optimal control via gradient ascent in function space and the time-bandwidth quantum speed limit

A gradient ascent method for optimal quantum control synthesis is presented that employs a gradient derived with respect to the coefficients of a functional basis expansion of the control. Restricting the space of allowable controls to weighted sums of the Slepian sequences efficiently parameterizes the control in terms of bandwidth, resolution and pulse duration. A bound showing minimum time evolutions scaling with the inverse of the control bandwidth [S. Lloyd and S. Montangero, PRL, 113, 010502, (2014)] is recovered and the method is shown numerically to achieve the bound on entangling two-qubit quantum gates.