Krotov’s method is a gradient-type optimization scheme with two distinguishing features. Its update is sequential: the correction to the control at time \(t\) uses the state already propagated under the updated control at earlier times. And it is monotonically convergent: every iteration is guaranteed not to make the figure of merit worse. It comes from V. F. Krotov’s global improvement methods in control theory. Tannor, Kazakov and coworkers adapted it to quantum dynamics in the early 1990s for laser control of molecules [1]. Together with GRAPE, it is one of the two classic workhorses of quantum optimal control.

The idea

As in all gradient methods, a figure of merit \(J_T\) evaluated at the final time is minimized with respect to control fields \(\epsilon_l(t)\) that enter the Hamiltonian. Krotov’s method minimizes the augmented functional

$$J = J_T\big[\{|\psi_k(T)\rangle\}\big] + \sum_l \int_0^T \frac{\lambda_{a,l}}{S_l(t)} \big[\epsilon_l^{(i)}(t) - \epsilon_l^{(i-1)}(t)\big]^2 dt ,$$

where \(i\) labels the iteration and \(\{|\psi_k\rangle\}\) are the states being steered: one state for state-to-state transfer, or a basis of states for a gate [2]. The running cost penalizes the change of the control from one iteration to the next, not the control itself. It therefore vanishes at convergence and does not bias the final solution. \(\lambda_{a,l}\) plays the role of an inverse step size, and the shape function \(S_l(t) \in [0,1]\) switches the update smoothly on and off — a simple way to keep the pulse’s beginning and end fixed.

The update

For a Hamiltonian linear in the controls and a functional that depends only on the final states, the first-order update reads (with \(\hbar = 1\))

$$\Delta\epsilon_l^{(i)}(t) = \frac{S_l(t)}{\lambda_{a,l}}\; \mathrm{Im} \sum_k \Big\langle \chi_k^{(i-1)}(t) \Big|\, \frac{\partial H}{\partial \epsilon_l}\, \Big| \psi_k^{(i)}(t) \Big\rangle .$$

Two sets of states appear:

  • the co-states \(|\chi_k^{(i-1)}(t)\rangle\), obtained by propagating the boundary condition \(|\chi_k(T)\rangle = -\partial J_T / \partial \langle \psi_k(T)|\) backward in time with the old controls;
  • the forward states \(|\psi_k^{(i)}(t)\rangle\), propagated forward with the new controls, as they are being constructed.

So each iteration is one backward propagation followed by one forward propagation, during which the new control is computed time step by time step. This sequential structure is what guarantees monotonic convergence. When the functional is not convex in the states, or the Hamiltonian depends non-linearly on the control, monotonicity requires an additional second-order term in the update [3].

Krotov and GRAPE

Both methods use the same two ingredients — forward-propagated states and backward-propagated co-states, as in Pontryagin's maximum principle — but they combine them differently:

Krotov GRAPE
Update sequential in time, uses the updated forward state concurrent: all time slices updated at once
Convergence monotonic by construction depends on step size / line search
Step size set by \(\lambda_a\) (fixed or adapted) line search, or a quasi-Newton method
Time grid controls on the propagation grid (continuous-time derivation) piecewise-constant slices, can be coarse

A commonly reported pattern is that sequential, Krotov-type updates make rapid progress in early iterations, while concurrent updates with a quasi-Newton method converge faster close to the optimum. The trade-offs, which depend on the problem, have been analysed within one common framework [4]. The two ideas can also be combined — for example by adding quasi-Newton updates to Krotov’s method [5].

Variants and extensions

  • Other monotonic schemes. Zhu and Rabitz derived a closely related rapidly converging, monotonic iteration [6]; Maday and Turinici unified several such schemes in one family [7].
  • Gates. Optimizing a unitary amounts to steering a set of basis states simultaneously [2]; see Quantum gate synthesis.
  • Non-linear dynamics. The method extends to the Gross–Pitaevskii (non-linear Schrödinger) equation, for example to load a Bose–Einstein condensate into an optical lattice [8]; see Ultracold atoms & BECs.
  • Open systems. Applied to density matrices with a Liouvillian instead of a Hamiltonian, the same update optimizes dissipative dynamics [3, 9]; see Control of open quantum systems.

Strengths and limitations

Strengths

  • Monotonic convergence without line searches.
  • Naturally produces time-continuous controls on the simulation grid; the shape function \(S(t)\) gives simple control over switch-on and switch-off.
  • Very general: states, gates, ensembles, open and non-linear dynamics.

Limitations

  • A local method, like all gradient approaches.
  • The step-size parameter \(\lambda_a\) needs tuning: too small gives erratic updates, too large slow progress.
  • Sequential updates require fine time steps, and constraints such as strict amplitude bounds or bandwidth limits are less natural to impose than in parametrized methods.