This part of our project is focused on developing Path Integral approach to continuous Tensor Networks [1] for time-dependent states and exploring constraints on such Tensor Networks from Einstein equations [2,3]. It is expected that geometry of entanglement, that is represented by such dynamical continuous Tensor Networks, should be related to time-dependent solutions of Einstein Gravity in one higher dimension. 

The dynamical states that we study are classified in the framework of the so-called quantum quenches. Up to date, these setups were studied extensively in 1+1 dimensions analytically as well as numerically. In holographic conformal field theories, their dual geometries correspond to e.g. shock waves propagating in Anti-de Sitter backgrounds and we analyze various holographic probes in these geometries.

In the second part, we are developing precise measures of computational complexity in quantum field theories and explore their roles played in holographic duality. At the moment, we are studying „Path Integral Complexity” [1] and Krylov basis methods:  „Krylov Complexity” of the quantum operator growth [4,5] as well as „Spread Complexity” of quantum states [6].

The motivation for studying complexity, similarly to other concepts in holography and AdS/CFT itself, comes from black holes. Namely, in the framework of AdS/CFT, there exist black hole solutions in AdS spacetime connected by Einstein-Rosen bridge (wormhole) that grows with time. In such spacetimes, on the contrary to what we initially thought, entanglement entropy (RT) is not sufficient to fully characterize the dual geometry. By developing precise complexity measures we not only enlarge the holographic dictionary but also explore the late time evolution of black holes and physics of their interiors. It is also expected that complexity of CFT states should be capturing the full dynamics of gravity and we try to make this dream precise [7].