Analytical Series Expansion for Efficient Gradient Evaluation in Multi-Qubit Optimal Control

The open-loop optimization of quantum dynamics using gradient-based quantum optimal control methods involves calculating the time-ordered propagator and its gradient. In this Letter, we present a unifying framework for gradient-based quantum optimal control with respect to any general pulse parameterization by deriving the formal solution from first principles. For the case of unitary propagators, we derive a series expansion involving time-independent commutators and time-dependent coefficients, significantly reducing the number of matrix exponentials needed to compute the gradient. The expansion highlights the connection between derivatives of the propagator and operator evolution in the Heisenberg picture. The method is particularly suited for simulating optimal control tasks in quantum systems with local interactions, which is a common situation in large multi-qubit platforms. We compare the computational cost required for the series with the Gradient Optimization of Analytic conTrols (GOAT) method, and, focusing on the problem of preparation of a GHZ state, demonstrate more than an order of magnitude speedup for a qubit ladder and a chain geometry.