Set guess control pulses

[Leeseok-628]

Hi,

I want to start a new optimization with a guess control field as the previously optimized control field. Is it possible?

Equivalently, is there any way to start my guess control with specific values?

[goerz]

Yes, both of these things are discussed in the documentation!

• How to continue from a previous optimization
• Controls can be arrays instead of functions

[goerz]

This is expected. It very much depends on the exact system. There is no theory behind it. It pays off to play around a little, both for the update shape and even more for the guess pulse directly. This is why optimal control can be a bit of an art.

[Leeseok-628]

Got it, thanks. Also, how do we deal with the Hamiltonian with nonlinear control fields (e.g. H = cos(e_1(t))*X + cos(e_2(t))*Z)? It seems like we have to calculate proper σ(t) for the second-order update, and I have to follow the calculation in equation 13, 14 in https://arxiv.org/pdf/1008.5126.pdf, which seems little unclear to me on how to choose appropriate parameters A, B, and C (or ε_i).

[christianekoch]

You don't need the second order for a non-linear dependence on the control, so A, B and C can still be zero if your target functional and equations of motion allow for that. The relevant equation to ensure monotonic convergence for non-linear dependencies on the control are (32) and (30) in the preprint you linked.

[goerz]

For nonlinear controls you may have to increase λₐ a bit. This is because technically, the value of the updated field will appear in the right-hand-side of the update equation. We deal that with by plugging in the value of the guess field instead, which implies small updates, or large λₐ.

[Leeseok-628]

I read through the paper step by step. Thanks for the comments :)

Suppose we have H=H0+e(t)Hx. When e(t) is non-linear such that e(t) = cos(t), I can simply set Hamiltonian as H = [H0, [Hx, (lambda t : np.cos(t)]]. This will optimize e(t) for each time grid. When e(t) is linear as e(t) = at, then H = [H0, [Hx, (lambda t: a*t)]] for given a. For any case, the initial pulse is equal to e(t) (i.e. cos(t) and at respectively) for t in the given time grid. After the optimization is done, the optimized pulses might look pretty different from the initial pulse, which sometimes look non-physical.

So I'm wondering if there's any way to optimize the pulse while perfectly maintaining its function form so that the optimized pulse look more physical. For instance, is it possible to optimize for A, w, and Φ in e(t) = A*cos(wt + Φ), so that the optimized pulse in every iterations maintain as cosine waves (but with different frequency, amplitudes, and phase)?

[goerz]

Suppose we have H=H0+e(t)Hx. When e(t) is non-linear such that e(t) = cos(t), I can simply set Hamiltonian as H = [H0, [Hx, (lambda t : np.cos(t)]]. This will optimize e(t) for each time grid. When e(t) is linear as e(t) = a_t, then H = [H0, [Hx, (lambda t: a*t)]]
for given a.

When we talk about linear, we mean linear in e, not linear in t. H=H0+e(t)Hx is linear for any e(t). A non-linear (quadratic) control would be H=H0+e²(t)Hx.

After the optimization is done, the optimized pulses might look pretty different from the initial pulse, which sometimes look non-physical.

What do you mean by non-physical? If it's too "spiky", increase λₐ.

So I'm wondering if there's any way to optimize the pulse while perfectly maintaining its function form so that the optimized pulse look more physical. For instance, is it possible to optimize for A, w, and Φ in e(t) = A*cos(wt + Φ), so that the optimized pulse in every iterations maintain as cosine waves (but with different frequency, amplitudes, and phase)?

Not with Krotov's method, no. If you have just a handful of free parameters, you can use gradient-free optimization. You might also be interested in GROUP/GOAT. Krotov's method targets mainly unconstrained controls e(t) (up to the restrictions implied by the update_shape). There are a few tricks to add some amplitude and spectral constraints, but these are already pushing the method. If you have more fundamental constraints, e.g., only allowing specific control functions, you have to use a different method.

[Leeseok-628]

I see, great, thank you. Can GRAPE also do such task as well?

[goerz]

No, GRAPE is pretty similar to Krotov's method in this respect. GROUP/GOAT are in some sense variations of GRAPE that deal with the kind of analytical pulses you have in mind. Did you see our discussion on how to choose optimization methods?