In this thesis we study aspects at the interplay of holography, quantum field theory and quantum simulation with tensor network and related techniques. The discussion is divided in three parts. In the first introductory part, we describe the necessary background of the gauge/gravity duality as well as tensor network methods and algorithms for our work.
The second part focuses on simulations of quantum spin chain models to extract properties of (1+1)-dimensional quantum field theories at zero and finite temperature. We review relevant aspects of quantum chromodynamics and heavy-ion collisions, which primarily motivate our work. Subsequently we present a new method that combines matrix product operator simulations with a signal analysis method, allowing us to make ab initio predictions about the thermal response in nonintegrable interacting quantum field theories. We then extend this line of research by employing scaling operators originating from an analytic wavelet solution of the multiscale entanglement renormalization ansatz. Based on this alternative discretization scheme, we calculate dynamical correlation functions in a coarsegrained system and compare them to the ones for the bare Ising model. Partially motivated by these considerations, we subsequently discuss the effect of meson melting, which describes the thermally induced breaking of nonperturbative bound states in a medium. Phenomenological approaches for its understanding from the quantum chromodynamics side as well as holographic models are reviewed. We introduce a new paradigm for the description of meson melting by analyzing entanglement entropies in a static and dynamical setting for the nonintegrable ferromagnetic phase of the Ising quantum field theory. We explain observed features at high enough temperatures through the fact that meson states in the quantum many-body system are melted and argue that the considered entanglement measures can serve as a witness of that process. In the last project of this part of the thesis, we explore the capabilities of analog quantum simulations with trapped ions to detect relativistic meson spectra, and present a method for its experimental realization on current devices via absorption spectroscopy.
The third part of this thesis deals with complexity as a quantum information quantity, which quantifies the difficulty of realizing a quantum circuit. We review its computational definition and recently proposed holographic interpretations of it. We then give an overview of two approaches, circuit complexity and path integral optimization, to understand complexity for quantum field theories. We unify these two concepts by showing that path integral complexity arises as an approximation to a particular choice in the circuit approach to complexity. We discuss this result in the context of quantum gravity through discrete tensor network interpretations of the gauge/gravity duality based on the multiscale entanglement renormalization ansatz.
