Using Krotov with QuTiP

The krotov package is designed around QuTiP, a very powerful “Quantum Toolbox” in Python. This means that all operators and states are expressed as qutip.Qobj quantum objects. The optimize_pulses() interface for Krotov’s optimization method is closely linked to the interface of QuTiP’s central mesolve() routine for simulating the system dynamics of a closed or open quantum system. In particular, when setting up an optimization, the (time-dependent) system Hamiltonian should be represented by a nested list. This is, a Hamiltonian of the form \(\Op{H} = \Op{H}_0 + \epsilon(t) \Op{H}_1\) is represented as H = [H0, [H1, eps]] where H0 and H1 are Qobj operators, and eps is a function with signature eps(t, args), or an array of control values with the length of the time grid (tlist parameter in mesolve()). The operator can depend on multiple controls, resulting in expressions of the form H = [H0, [H1, eps1], [H2, eps2], ...].

The central routine provided by the krotov package is optimize_pulses(). It takes as input a list of objectives, each of which is an instance of Objective. Each objective has an initial_state, which is a qutip.Qobj representing a Hilbert space state or density matrix, a target (usually the target state that the initial_state should evolve into when the objective is fulfilled), and a Hamiltonian H in the nested-list format described above. For dissipative dynamics, H should be a Liouvillian, which can be obtained from the Hamiltonian and a set of Lindblad operators via krotov.objectives.liouvillian(). The Liouvillian again is in nested list format to express time-dependencies. Alternatively, each objective could also directly include a list c_ops of collapse (Lindblad) operators , where each collapse operator is a Qobj operator. However, this only makes sense if the time propagation routine takes the collapse operators into account explicitly, such as in the Monte-Carlo mcsolve(). Otherwise, the use of c_ops is strongly discouraged.

If the control function (eps in the above example) relies on the dict args for static parameters, those args can be specified via the pulse_options argument in optimize_pulses(). See How to use args in time-dependent control fields.

In order to simulate the dynamics of the guess control, you can use Objective.mesolve(), which delegates to qutip.mesolve.mesolve(). There is also a related method Objective.propagate() that uses a different sampling of the control values, see krotov.propagators.

The optimization routine will automatically extract all controls that it can find in the objectives, and iteratively calculate updates to all controls in order to meet all objectives simultaneously. The result of the optimization will be in the returned Result object, with a list of the optimized controls in optimized_controls. The optimized_objectives property contains a copy of the objectives with the optimized_controls plugged into the Hamiltonian or Liouvillian and/or collapse operators. The dynamics under the optimized controls can then again be simulated through Objective.mesolve().

While the guess controls that are in the objectives on input may be functions, or an array of control values on the time grid, the output optimized_controls will always be an array of control values.