Bounding Spectral Properties of Many-Body Quantum Systems

Abstract

Simulating quantum many-body systems with large size is a considerable challenge in physics. Often, understanding complex many-body phenomena requires information about the spectrum of their Hamiltonian. Apart from performing the exact diagonalisation, which is not possible for systems with tens of particles, there are some scalable methods for obtaining information about specific energy regions (e.g., ground state properties, spectral gap). The main methods that address the problem of characterising properties of the spectrum are variational methods and quantum Monte Carlo methods. These methods were mostly used for low energy states, performing poorly for excited states. On the other hand, semidefinite programming is emerging as a complementary tool, able to yield certified results for quantities of interest. In this thesis, we explore the capabilities of semidefinite programming in the context of quantum many-body systems. Recent works regarding the ground state [WSF+24], [WJF+26] showcased the potential of these programs and we proceed to focus on their performance in the context of highly entangled states. The proposed semidefinite program can determine parts of the spectrum with no eigenstate and provide bounds for the expectation values of a target eigenstate