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

Abstract: We investigate signatures of quantum chaos and complexity in the quantum annealing Ising model on random Erd\H{o}s-R\'enyi graphs. By tuning the connectivity of the graph, the dynamics can be driven from a localized phase through a chaotic regime to an integrable limit. While this dynamical transition reflects in the spectral characteristics, we pursue a broader suite of quantum chaos indicators,some of which can be measured on near-term quantum devices. We study deep thermalization of a quantum state ensemble obtained from a natural unraveling of the subsystem density matrix as an indicator of chaotic dynamics. This extends the analysis of quantum chaos to the ensemble of quantum states. Furthermore, we analyze the eigenstate and eigenvalue correlations through the partial spectral form factor of subsystems and observe distinct signatures of the onset of chaos and its system size dependence, providing experimentally measurable indicators of the localization-to-chaos transition. As a locality-independent probe, we show that the Krylov complexity of operators is also maximized in the chaotic regime, providing a link between graph topology and information scrambling. Finally, we investigate a quantum analogue of the Mpemba effect, where initially "hotter" states can thermalize anomalously fast, a phenomenon most cleanly observed within the chaotic phase. However, away from the chaotic regime, the system is distinguished by multiple crossings across connectivity in its distance from the thermal ensemble with time. Collectively, this work presents a broad characterization of chaos, providing insight beyond spectral analysis and practical indicators for benchmarking near-term quantum devices.