DC 10 – Geometry of tensor network varieties for quantum condensed matter physics

Project Title: Geometry of tensor network varieties for quantum condensed matter physic
Advisors: I. Carusotto (INO-BEC, CNR), S. Telen (MPG) Mentor: S. Mansfield (Quandela)
Objectives: Tensor Networks (TN) are a convenient way to represent those strongly entangled quantum states that appear in interacting many-particle systems and compute their physical properties. The objective of this PhD fellowship is to make use of the geometrical structure of the tensor network variety to develop more efficient algorithms for outstanding problems in quantum condensed matter physics.
Taking inspiration from recent results based on TNs of bond dimension 1 –the so-called quantum Gutzwiller ansatz–, we will build a quantum ansatz for the quantum dynamics on higher tensor network varieties. This will require identifying a suitable parametrization of the variety in the neighborhood of the energy-minimum state and then promote the coefficients of the harmonic expansion to operator-valued quantities. Not only will this offer an accurate representation of the quantum many-body state at a much lower cost in terms of required bond dimension, but will also provide a physical picture of the collective excitations of complex phases of matter. A central tool in our research will be homotopy techniques, that allow to continuously follow the energy-minimum state and its geometrical neighborhood while the physical system is continuously tuned from a trivial uncorrelated state towards the desired complex state by acting on some external parameter of the Hamiltonian. During the secondment at Quandela, the DC will make get familiar on the use of tensor network techniques for simulating photonic processors. This skills will be instrumental for the final task of exploring the advantage of quantum interferometry schemes based on collective excitations for quantum sensing and quantum information processing.
Expected Results: Develop a new quantum theory of the dynamics of quantum condensed-matter systems that exploit the geometry of higher tensor network varieties to describe the quantum dynamics of collective excitations and apply homotopy continuation techniques to extract information on strongly correlated states starting from simple uncorrelated ones. Writing conference/journal papers and completion of a PhD degree by the DC.
Planned secondment(s): with S. Telen (MPG) to identify suitable homotopy flows (M8-13) and to implement them for most significant physical examples (M20-24); with S. Mansfield (Quandela) on the application of tensor network techniques to simulate quantum photonic processors (M27-29).
Joint degree: University of Trento, University of Leipzig

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