# krotov.propagators module¶

Routines that can be passed as propagator to optimize_pulses()

The numerical effort involved in the optimization is almost entirely within the simulation of the system dynamics. In every iteration and for every objective, the system must be “propagated” once forwards in time and once backwards in time, see also krotov.parallelization.

The implementation of this time propagation must be inside the user-supplied routine propagator that is passed to optimize_pulses() and must calculate the propagation over a single time step. In particular, qutip.mesolve.mesolve() is not automatically used for simulating any dynamics within the optimization. The signature for any propagator must be the same as the “reference” expm() propagator:

>>> str(inspect.signature(krotov.propagators.expm))
'(H, state, dt, c_ops=None, backwards=False, initialize=False)'


The arguments are as follows (cf. Propagator):

• H is the system Hamiltonian or Liouvillian, in a nested-list format similar to that used by qutip.mesolve.mesolve(), e.g., for a Hamiltonian $$\Op{H} = \Op{H}_0 + c \Op{H}_1$$, where $$c$$ is the value of a control field at a particular point in time, propagator would receive a list [H0, [H1, c]] where H0 and H1 are qutip.Qobj operators. The nested-list for H used here, with scalar values for the controls, is obtained internally from the format used by mesolve(), with time-dependent controls over the entire time grid, via krotov.conversions.plug_in_pulse_values().

• state is the qutip.Qobj state that should be propagated, either a Hilbert space state, or a density matrix.

• dt is the time step (a float). It is always positive, even for backwards=True.

• c_ops is None, or a list of collapse (Lindblad) operators, where each list element is a qutip.Qobj instance (or possibly a nested list, for time-dependent Lindblad operators. Note that is generally preferred for H to be a Liouvillian, for dissipative dynamics.

• backwards (bool): If passed as True, the propagator should propagate backwards in time. In Hilbert space, this means using -dt instead of dt. In Liouville space, there is no difference between forward and backward propagation. In the context of Krotov’s method, the backward propagation uses the conjugate Hamiltonian, respectively Liouvillian. However, the propagator routine does not need to be aware of this fact: it will receive the appropriate H and c_ops.

• initialize (bool): A flag to indicate the beginning of a propagation over a time grid. If False in subsequent calls, the propagator may assume that the input state is the result of the previous call to propagator.

The routines in this module are provided with no guarantee to be either general or efficient. The expm() propagator is exact to machine precision, but generally extremely slow. For “production use”, it is recommended to supply a problem-specific propagator that is highly optimized for speed. You might consider the use of Cython. This is key to minimize the runtime of the optimization.

The initialize flag enables “stateful” propagators that cache data between calls. This can significantly improve numerical efficiency. DensityMatrixODEPropagator is an example for such a propagator. In general, any stateful propagator should be an instance of Propagator.

## Summary¶

Classes:

 DensityMatrixODEPropagator Propagator for density matrix evolution under a Lindbladian Propagator Abstract base class for stateful propagators

Functions:

 expm Propagate using matrix exponentiation

__all__: DensityMatrixODEPropagator, Propagator, expm

## Reference¶

krotov.propagators.expm(H, state, dt, c_ops=None, backwards=False, initialize=False)[source]

Propagate using matrix exponentiation

This supports H being a Hamiltonian (for a Hilbert space state) or a Liouvillian (for state being a density matrix) in nested-list format. Collapse operators c_ops are not supported. The propagator is not stateful, thus initialize is ignored.

class krotov.propagators.Propagator[source]

Bases: abc.ABC

Abstract base class for stateful propagators

abstract __call__(H, state, dt, c_ops=None, backwards=False, initialize=False)[source]

Evaluation of a single propagation step

Parameters
• H (list) – A Hamiltonian or Liouvillian in qutip’s nested-list format, with a scalar value in the place of a time-dependency. For example, [H0, [H1, u]] for a drift Hamiltonian H0, a control Hamiltonian H1, and a scalar value u that is a time-dependent control evaluated for a particular point in time.

• state (qutip.Qobj) – The state to propagate

• dt (float) – The time step over which to propagate

• c_ops (list or None) – A list of Lindblad operators. Using explicit Lindblad operators should be avoided: it is usually more efficient to convert them into a Lindbladian, passed as H

• backwards (bool) – Whether the propagation is forward in time or backward in time

• initialize (bool) – Whether the propagator should (re-)initialize for a new propagation, when the propagator is used to advance on a time grid, initialize should be passed as True for the initial time step (0 to dt in a forward propagation, or T to T-dt for a backward propagation), and False otherwise.

Note

A propagator may assume the propagation to be “sequential” when initialize is False. That is, the state to propagate is the result of the previous call to the propagator.

class krotov.propagators.DensityMatrixODEPropagator(method='adams', order=12, atol=1e-08, rtol=1e-06, nsteps=1000, first_step=0, min_step=0, max_step=0, reentrant=False)[source]

Propagator for density matrix evolution under a Lindbladian

See qutip.solver.Options for all arguments except reentrant. Passing True for the reentrant re-initializes the propagator in every time step.

Warning

By default, the propagator is not “re-entrant”. That is, you cannot use more than one instance of DensityMatrixODEPropagator in the same process at the same time. This limitation is due to scipy.integrate.ode with the “zvode” integrator not being re-entrant. Passing reentrant=True side-steps this problem by re-initializating scipy.integrate.ode in every time step. This makes it possible to use DensityMatrixODEPropagator in the optimization of multiple objectives, but creates a significant overhead.

__call__(H, state, dt, c_ops=None, backwards=False, initialize=False)[source]

Evaluation of a single propagation step

Parameters
• H (list) – A Liouvillian superoperator in qutip’s nested-list format, with a scalar value in the place of a time-dependency. For example, [L0, [L1, u]] for a drift Liouvillian L0, a control Liouvillian H1, and a scalar value u that is a time-dependent control evaluated for a particular point in time. If initialize is False, only the control values are taken into account; any operators are assumed to be identical to the internally cached values of H during initialization.

• state (qutip.Qobj) – The density matrix to propagate. The passed value is ignored unless initialize is given as True. Otherwise, it is assumed that state matches the (internally stored) state that was the result from the previous propagation step.

• dt (float) – The time step over which to propagate

• c_ops (list or None) – An empty list, or None. Since this propagator assumes a full Liouvillian, it cannot be combined with Lindblad operators.

• backwards (bool) – Whether the propagation is forward in time or backward in time. Since the equation of motion for a Liouvillian and conjugate Liouvillian is the same, this parameter has no effect. Instead, for the backward propagation, the conjugate Liouvillian must be passed for L.

• initialize (bool) – Whether to (re-)initialize for a new propagation. This caches H (except for the control values) and state internally.