Quantum complexity in high-energy many-body systems

Abstract

The study of quantum many-body systems is a fundamental problem in modern physics, with direct connections to quantum simulation, condensed matter, high-energy physics, and quantum information. A central challenge is to understand which many-body states can be efficiently represented and simulated classically, and which ones require genuinely quantum computational resources. This question is naturally tied to the notion of quantum complexity, which can be probed through diagnostics such as entanglement entropy and non-Gaussianity. In this thesis, we study complexity markers in interacting fermionic quantum many-body systems. We first analyze a model interpolating between the chaotic Sachdev–Ye–Kitaev (SYK) model and the integrable transverse-field Ising model. Using the half-chain von Neumann entanglement entropy and the fermionic antiflatness, a recently introduced measure of fermionic non-Gaussianity, we characterize the transition between chaotic and integrable regimes in both ground states and excited states. We also discuss symmetryprotected effects in the SYK limit, where antiunitary symmetries can lead to the exact vanishing of the fermionic covariance matrix for specific system sizes. We then apply these ideas to the lattice Schwinger model, where tensornetwork methods are used to study the ground-state phase transition and realtime dynamics. In particular, we investigate how fermionic antiflatness and entanglement behave near criticality, and how their finite-size scaling can be used to extract universal features of the transition. Overall, this thesis shows that fermionic magic and entanglement provide complementary information about the structure and complexity of quantum many-body states.