Quantum computers are expected to change computations as we know it. How are they supposed to do that? Essentially they allow us to perform a subpart of linear algebra (certain matrix-vector multiplications) on exponentially large vectors. A natural mathematical framework to understand what they do is the tensor network formalism. Conversely, tensor networks are becoming popular as tools that can take the place of quantum computers, yet run on perfectly classical hardware. To do so, they rely on a hidden underlying structure of some mathematical problems (a form of entanglement) that can be harvested to compress exponentially large vectors into small tensor networks. An increasing number of, apparently exponentially difficult, problems are getting solved this way. Tensor networks are also intimately linked to artificial intelligence. For instance, automatic differentiation – the core algorithm at the center of all neural network optimizations – amounts to the contraction of a tensor network.

This PhD lies at the intersection between theoretical quantum physics and applied mathematics. The goal will be to develop and apply new algorithms to “beat the curse of dimensionality”, i.e. to push the frontier of problems that we are able to access computationally. More specifically, we will develop an extension of the tensor cross interpolation (TCI) algorithm to tensor trees (aka loopless tensor networks). In its current form, TCI is an active learning algorithm that can map an input high dimensional function onto a tensor train (linear tensor network) [1]. Its extension to trees will significantly enhance the expressivity of the network. In a second step, we will apply this algorithm to compute a class of high dimensional integrals that arise in the context of Feynman diagram calculations [2]. The envisioned algorithms combine the normalization flow approach (from neural networks) with the tensor cross interpolation (from tensor networks). The goal is to be able to calculate the out-of-equilibrium phase diagram of various correlated models starting from double quantum dots (of high current interest due to their applications to qubits) in the Kondo regime to the propagation of voltage pulses in electronic interferometers.

[1] https://scipost.org/SciPostPhys.18.3.104
[2] https://journals.aps.org/prx/abstract/10.1103/PhysRevX.10.041038