Other Optimization Methods

Gradient-based optimization

At its core, Krotov’s method is a gradient-based optimization method, and most directly compares to GRadient Ascent Pulse Engineering (GRAPE), another gradient-based method widely used in quantum control [KhanejaJMR05]. Historically, Krotov’s method has been widely used in the control of atomic and molecular dynamics, whereas GRAPE has its origin in NMR. Generally, both methods can be used interchangeably, see Choosing an optimization method below for some of the subtleties.

GRAPE is included in QuTiP, see the section on Quantum Optimal Control in the QuTiP docs. It is used via the qutip.control.pulseoptim.optimize_pulse() function.

This function has some limitations compared to the optimize_pulses() provided by the Krotov package. Most importantly:

  • QuTiP’s GRAPE implementation only allows for a single control field (hence “pulse”, instead of “pulses” in optimize_pulses())
  • The optimization for multiple simultaneous objectives in GRAPE is limited to optimizing a quantum gate. Other uses of parallel objectives, such as optimizing for robustness, is not available.
  • The GRAPE implementation does not allow for an arbitrary guess-pulse. Guess pulses can only be chosen from a specific set of options (including “random”)
  • There is only a fixed number of choices for choosing a method for time-propagation. Supplying a problem-specific propagator is not possible.
  • The optimized pulses are defined on the intervals of the time grid, which does not fit in with other parts of QuTiP, specifically qutip.mesolve.mesolve().

All of these are restrictions of the implementation, not of the underlying method.

Note

There are plans to provide a wrapper-function that provides the interface of qutip.control.pulseoptim.optimize_pulse() for use with Krotov’s method.

Gradient-free optimization

In situations where the controls can be reduced to a relatively small number of controllable parameters, gradient-free optimization becomes feasible. Gradient-free optimization requires at most half of the numerical effort than either Krotov’s method or GRAPE, as they only require a single forward-propagation of the objectives in each iteration. Such a gradient-free optimization can be easily realized by following two simple steps:

  • Write a function that takes an array of optimization parameters as input and returns a figure of merit on output. This function would, e.g., construct a numerical control pulse from the control parameters, simulate the dynamics using qutip.mesolve.mesolve(), and compare to a target state by evaluating the state overlap as a figure of merit.
  • Pass the function to scipy.optimize.minimize() for gradient-free optimization.

The implementation in scipy.optimize.minimize() allows to choose between different optimization methods, with “Nelder-Mead simplex” being the the default.

A special case of gradient-free optimization is the Chopped RAndom Basis (CRAB) method [DoriaPRL11][CanevaPRA2011]. The essence of CRAB is in the specific choice of the parametrization in terms of a low-dimensional random basis, as the name implies. The optimization itself is performed by Nelder-Mead simplex using this parametrization.

An implementation of CRAB is included in QuTiP, see QuTiP’s documentation of CRAB, and uses the same qutip.control.pulseoptim.optimize_pulse() interface as the GRAPE method discussed above with the same limitations.

Choosing an optimization method

Whether to use a gradient-free optimization method, GRAPE, or Krotov’s method depends on the size of the problem (both the Hilbert space dimension and the number of control parameters), the requirements on the control pulse, and the optimization functional. Gradient-free methods should be used if propagation is extremely cheap (small Hilbert space dimension), the number of independent control parameters is relatively small, or the functional is of a form that does not allow to calculate gradients easily.

GRAPE should be used if the control parameters are discrete, such as on a coarse-grained time grid, and the derivative of \(J\) with respect to each control parameter is easily computable. Moreover, evaluation of the gradient should be ideally numerically inexpensive.

Krotov’s method should be used if the control is near-continuous, and if the derivative of \(J_T\) with respect to the states, Eq. (7), can be easily calculated. When these conditions are met, Krotov’s method gives excellent convergence, although it is often observed to slow down when getting close to the minimum of \(J\). It can be beneficial to switch from Krotov’s method to GRAPE with LBFGS-B in the final stage of the optimization which does not show such a slow-down due to inclusion of second-derivative information by the Hessian on top of the gradient.