Controlling quantum systems

No big surprises on the second day of Paraty quantum information school: Good lectures and relaxed atmosphere. I guess this combination is a defining aspect of this school.

The first lecture was given by Ivan Deustch, discussing how to control quantum systems in a optimal way.

Quantum control, measurement and tomography — Lecture I, Ivan Deutsch

Ivan started fixing two paradigms of control. In the first one, called closed-loop control, one adjusts the systems on the go. Think for instance on a plane that gets various informations to adjust its orientation and balance. The other paradigm is that of open-loop optimal control. In this strategy one uses the best prior information to design an optimal path — like when you ask Google maps to find an optimal route from A to B.

The first paradigm is the one Ivan focused on for quantum systems. The general idea is trying to generate some arbitrary unitary operation given that you control in time some Hamiltonians that act on your system. The simplest case is that of a spin-1/2 particle under the influence of a tunable magnetic field. One wants to generate an arbitrary unitary matrix in SU(2), which can be expressed as U=\exp[i \theta \vec{n}.\vec{S}]=\exp[i\theta (n_x \sigma_x + n_y \sigma_y + n_z \sigma_z)], by changing in time the magnetic field. The Hamiltonian governing the system is H(t) =-\mu_B\vec{S}.\vec{B}(t). Therefore, a simple way to obtain U is to choose \vec{B}(t) = B(t) \vec{n}. In this way we have evolution

e^{- i \int_0^t dt^\prime  \mu_B B(t^\prime) \vec{S}.\vec{n}}

The target unitary is then obtained by setting -\int_0^t dt^\prime  \mu_B B(t^\prime) = \theta.

That is a simple case. Things get more interesting when we don’t have access to all the generators of the group. For example when, for whatever reason, we can only apply magnetic fields on the x and y directions. Is it still possible to generate any unitary in SU(2)? It not difficult to convince one self that it is still possible. Ivan showed that in general the following theorem holds:

A system with a time dependent Hamiltonian of the form

H = H_0 + \sum_{i=1}^K \lambda_i(t) H_i

can generate any U\in SU(N) if the H_i and the commutators among them (and commutators of commutators,…) generate the whole associated algebra.

Given that we now know when we are able to generate any unitary, how can we optimize the control? A way to go about that is to think on the following problem: given an initial state \psi_0 and a Hamiltonian H as above, we want to get as close as possible from the state \psi_t. To quantify this Ivan used the fidelity measure:

\mathcal{F}(\{\lambda_i(t)\},T) = |\langle{\psi_t}| U(\{\lambda_i(t)\},T)|{\psi_0}\rangle|^2

The maximization is over the control wave-forms \{\lambda_i(t)\} and possibly on the total amount of time T. Since each \lambda_i is a continuous function of time, one usually discretize time, and the Hamiltonian at each time step can be seen as a square-pulse. An intriguing result by Rabitz [Science 303, 1998 (2004)] seems to show that every local maxima of this optimization is also a global maxima! Therefore if you start in any point on the optimization space you can climb up any hill, and it will lead you to a global maximum. As the theorem above, this result should be taken with care, since there’s no constraint on the duration of the pulses nor on the shape, things that are usually not easy to have in a lab.

In the last part of the lecture Ivan showed how apply these ideas to control the dynamics of an atomic spin… but this post is already getting quite big, maybe due to my excitement with the lecture which was very clear, and I’ll stop here.

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