Optimization after a partial trace?

[will-atom]

Is it possible to perform GRAPE optimization on only some quantum numbers of a system, i.e. after tracing out some degrees of freedom?

[goerz]

Sure! You can define any functional you want.

[goerz]

I would also question whether you need any of that, though. Just because you are working with a logical subspace embedded in a larger Hilbert space doesn't mean that you need to trace out anything. Typically (for the standard functionals), it's just that your target states are in the logical subspace, and that takes care of everything automatically. Entangling quantum gates for coupled transmon qubits is an example for optimizing for a two-qubit gate in a larger physical Hilbert space.

[goerz]

Could you describe what you’re actually trying to do, i.e., what does the functional you are interested in look like as an equation?

[will-atom]

Thanks for all the details! Yes, very simple to describe: My system is a tensor product of a 2-level system (spin up/down) and a harmonic oscillator (quantum number n=0,1,2...). I would like to optimize state transfer in the 2-level subspace independent of the state of the harmonic oscillator. The Rabi frequency for driving between states of the 2-level system depends on the harmonic oscillator state like ~|g><e|exp(ik*x) + h.c. for some k, where x is the position operator of the harmonic oscillator.

[goerz]

So that would be this, with $\vert\Psi(t=0)\rangle = \vert \uparrow 0\rangle$?

$$J_T = 1 - \sum_n \vert \langle \Psi(T) \vert \downarrow n \rangle \vert^2 = 1 - \langle \Psi(T) \vert \underbrace{\left(\sum_n \vert\downarrow n\rangle\langle\downarrow n \vert \right)}_{\equiv \hat{P}} \vert \Psi(T) \rangle$$

In that case, you'd indeed have to implement a function J_T_op to handle minimizing the expectation value of an operator (the projector $\hat{P}$ in this case).

In a general form, that might be nice addition to the collection of functionals in QuantumControl.Functions.

[will-atom]

Thanks very much!

[will-atom]

Sorry, still confused about the nature of the trajectories in this case. In this instance does it then not really matter what I set the target states to be in the trajectory?

[goerz]

Yes, you would have a single trajectory here, and not define any target_state

[goerz]

Would #86 clarify things for you? See the online preview