  Quantum optimal control plays a crucial role in quantum computing by
providing the interface between compiler and hardware. Solving the optimal
control problem is particularly challenging for multi-qubit gates, due to the
exponential growth in computational complexity with the system's dimensionality
and the deterioration of optimization convergence. To ameliorate the
computational complexity of time-integration, this paper introduces a
multiple-shooting approach in which the time domain is divided into multiple
windows and the intermediate states at window boundaries are treated as
additional optimization variables. This enables parallel computation of state
evolution across time-windows, significantly accelerating objective function
and gradient evaluations. Since the initial state matrix in each window is only
guaranteed to be unitary upon convergence of the optimization algorithm, the
conventional gate trace infidelity is replaced by a generalized infidelity that
is convex for non-unitary state matrices. Continuity of the state across window
boundaries is enforced by equality constraints. A quadratic penalty
optimization method is used to solve the constrained optimal control problem,
and an efficient adjoint technique is employed to calculate the gradients in
each iteration. We demonstrate the effectiveness of the proposed method through
numerical experiments on quantum Fourier transform gates in systems with 2, 3,
and 4 qubits, noting a speedup of 80x for evaluating the gradient in the
4-qubit case, highlighting the method's potential for optimizing control pulses
in multi-qubit quantum systems.
