By Llorenç Balada Gaggioli – April 2026
My second year of PhD is taking place in Toulouse, where I have been working in the Polynomial Optimization group at LAAS. After my first year in Prague working on quantum optimal control from the physics perspective, now I have been focusing on the optimization side of things. Together with Didier Henrion and Milan Korda we have been considering the global optimization of polynomials that have structures that could arise in physical models, neural networks, and others.
The structures we study are tensor networks, a very popular, active, and useful tool that is commonly used in physics, approximation theory, and many other areas. We now want to extend this range of pplicability also to polynomial optimization, where we have designed Moment-SOS hierarchies to optimize polynomials presenting a tensor network structure.
The start of the secondment has led us to introduce this new notion of structure in the field using a specific case of tensor decomposition: the CP decomposition. Now we have worked on polynomial models of composition structures, which in tensor format can be generally seen as tensor train polynomials. And, after this, we want to generalize these hierarchies for any polynomial written as a tensor network and show how the complexity of solving the optimization problem scales with the parameters of the network.
Toulouse has been an excellent place both for a good quality of life and for work. The atmosphere at LAAS is excellent, it is very dynamic and very easy to discuss ideas with everyone. We even have a TENORS office, where Younes, Jonas and I work, and this really facilitates things as we can help each other and collaborate on projects. I am really looking forward to the rest of my secondment and to continue working on the projects we have started here during my next year back in Prague, where I can bring back the results we have developed and use them to develop more advanced techniques in quantum control.
Having worked in two very different groups during my first years of PhD has been a great experience. I have worked with many different people in a quite diverse range of topics, giving8 me the chance to see how different people approach doing research. Both during my stays in Prague and Toulouse, and during the conferences and workshops I have attended I have developed a network of researchers that will help me continue my career and facilitate collaboration in many interesting projects. Having worked on both quantum control and polynomial optimization, I feel I am in a good intersection where we can explore many interesting problems to try to push the boundaries of what is currently possible with quantum technology.