Signatures of quantum chaos and complexity in the Ising model on random graphs

Abstract

We investigate signatures of quantum chaos in the mixed-field quantum Ising model on finite-size Erd\H{o}s-R\'enyi graphs using probes scalable on near-term quantum devices. Upon tuning the graph connectivity, the system exhibits a crossover from a localized regime at low connectivity, through a chaotic regime at intermediate connectivity, to a permutation-symmetric integrable limit near all-to-all connectivity. This crossover has possible implications for the performance and trainability of variational algorithms such as QAOA. We characterize this crossover in finite-size systems using complementary probes. First, deep thermalization of a projected ensemble starting from a product state reveals slow (fast) convergence to the Haar ensemble at extremal (intermediate) connectivities. Second, we analyze eigenstate and eigenvalue correlations using the partial spectral form factor, an experimentally scalable proxy for the spectral form factor with reduced resource overhead, and observe characteristic chaos signatures at intermediate connectivities and distinct deviations at extremal connectivities. Finally, we explore the Krylov complexity of operators, a locality-independent diagnostic that, although not directly experimentally accessible, serves as a tool for quantifying scrambling. We show that it is maximized deep in the chaotic regime, corroborating the signatures observed through the experimentally scalable probes. Our results provide finite-size benchmarks demonstrating robust signatures of chaos in scalable probes and suggest that these diagnostics can be implemented in current quantum platforms to access regimes beyond classical simulation.