using different form of cost function instead of fidelity

[Leeseok-628]

Hi,

I have been wondering if I can use krotov to optimize different forms of cost function, other than fidelity.
For instance, would it be possible to optimize trace(rho_target - rho_ansatz) where rho_target is target density matrix and rho_ansatz is the time-evolved state under control parameters. If so, how?

Additionally, suppose we have some random operator A. Then is it possible to minimize the trace distance between the two density operators mutilplied by this operator A: i.e. trace(A(rho_target - rho_ansatz)). Does this also feasible via krotov?

[christianekoch]

Of course!
Your request corresponds to "Optimization of a Dissipative State-to-State Transfer in a Lambda System" listed as one of the examples in the package documentation.
In case of a mixed target state, please see https://doi.org/10.1002/qute.201800110 for potential pitfalls in defining your target functional.

[christianekoch]

The answer to your second question is also, yes. This would simply be the optimization of an expectation value difference which is one of the standard optimization tasks.

[Leeseok-628]

Thank you so much for your response! I read through the example, it helped me a lot.

How can I customize (or build) my own cost function that goes into krotov.optimize_pulses function? Regarding my second question earlier, as you mentioned above, the cost function is tr(D(rho_tar - rho_trial)) = <psi_trial|D|psi_trial> - <psi_target|D|psi_target>, for given D and psi_target. But I don't see how to customize/code the cost function in this form of summation of two expectation values. The objectives that goes into krotov.optimize_pulses requires one to set initial and final states within some Hamiltonian, but I'm not quite sure how to build objectives when I need to calculate the expectation value of given operator D as my cost function. I'm confused on this because, I looked into the details of krotov functional (https://qucontrol.github.io/krotov/v1.0.0/API/krotov.functionals.html#module-krotov.functionals), and it seems like all functionals don't deal with expectation values nor linear combinations of two distinct terms!

I'd really appreciate it if you can help me out with this :)

[goerz]

From a technical perspective, the only thing required for a new functional is to define a function that evaluates ∂J/∂⟨ϕₖ| and that you can pass as chi_constructor to the optimize_pulses routine, as outlined in the documentation of krotov.functionals.

I would also point out that J_T_hs / chis_hs is already defined and should cover your first case. For the second case of the expectation value, you'll have to write your own function, but it should be fairly straightforward to calculate the χ-states exactly analogous to the calculation shown in the chis_hs docstring.

If you find the matrix calculus notation unsettling, ∂J/∂⟨ϕₖ| is just shorthand for a Wirtinger derivative, but calculating it component-wise is extremely cumbersome. You should be fine just writing out your functional in terms of inner products in Liouville space (Tr[A⁺B] ≡ ⟨A|B⟩) and taking the derivative with respect to the co-state ⟨A| treated as an independent variable, as we've done for all the existing functionals.

[goerz]

First, you always want to use the square of the norm instead of just the norm, to get rid of the square root.. Then χ is definitely not going to be independent of C: you'll have (⟨ϕ(T)|O|ϕ(T)⟩ - C)(⟨ϕ(T)|O|ϕ(T)⟩ - C)^†, which you have to multiply out and take the derivative of w.r.t. ⟨ϕ(T)|

[Leeseok-628]

I see, thanks! So, since O is Hermitian, ∂J/∂⟨ϕ(T)| will be just 2(⟨ϕ(T)|O|ϕ(T)⟩-C) * O|ϕ(T)⟩ !

[goerz]

That looks right

[Leeseok-628]

Awesome, so I assume the reason of avoiding the square root is that then we can't take derivative w.r.t ⟨ϕ(T)|; so that setting J = |<ϕ(T)|O|ϕ(T)> - C| would be impossible to optimize through Krotov.

[goerz]

It's not impossible (you can take the derivative of the square root), but it's an unnecessary complication. Minimizing |x| and minimizing |x|² is mathematically equivalent, and gradients of |x|² are always going to be more straightforward