There has recently been considerable interest in studying quantum systems via dynamical Lie algebras (DLAs) – Lie algebras generated by the terms which appear in the Hamiltonian of the system. However, there are some important properties that are revealed only at a finer level of granularity than the DLA. In this work we explore, via the commutator graph, average notions of scrambling, chaos and complexity over ensembles of systems with DLAs that possess a basis consisting of Pauli strings. Unlike DLAs, commutator graphs are sensitive to short-time dynamics, and therefore constitute a finer probe to various characteristics of the corresponding ensemble. We link graph-theoretic properties of the commutator graph to the out-of-time-order correlator (OTOC), the frame potential, the frustration graph of the Hamiltonian of the system, and the Krylov complexity of operators evolving under the dynamics. For example, we reduce the calculation of average OTOCs to a counting problem on the graph; separately, we connect the Krylov complexity of an operator to the module structure of the adjoint action of the DLA on the space of operators in which it resides, and prove that its average over the ensemble is lower bounded by the average shortest path length between the initial operator and the other operators in the commutator graph.
There has recently been considerable interest in studying quantum systems via dynamical Lie algebras (DLAs) – Lie algebras generated by the terms which appear in the Hamiltonian of the system. However, there are some important properties that are revealed only at a finer level of granularity than the DLA. In this work we explore, via the commutator graph, average notions of scrambling, chaos and complexity over ensembles of systems with DLAs that possess a basis consisting of Pauli strings. Unlike DLAs, commutator graphs are sensitive to short-time dynamics, and therefore constitute a finer probe to various characteristics of the corresponding ensemble. We link graph-theoretic properties of the commutator graph to the out-of-time-order correlator (OTOC), the frame potential, the frustration graph of the Hamiltonian of the system, and the Krylov complexity of operators evolving under the dynamics. For example, we reduce the calculation of average OTOCs to a counting problem on the graph; separately, we connect the Krylov complexity of an operator to the module structure of the adjoint action of the DLA on the space of operators in which it resides, and prove that its average over the ensemble is lower bounded by the average shortest path length between the initial operator and the other operators in the commutator graph.
Chaos and complexity entail an entropic and computational obstruction to describing a system, and thus are intrinsically difficult to characterize. In this paper, we consider time evolution by Gaussian Unitary Ensemble (GUE) Hamiltonians and analytically compute out-of-time-ordered correlation functions (OTOCs) and frame potentials to quantify scrambling, Haar-randomness, and circuit complexity. While our random matrix analysis gives a qualitatively correct prediction of the late-time behavior of chaotic systems, we find unphysical behavior at early times including an O(1) scrambling time and the apparent breakdown of spatial and temporal locality. The salient feature of GUE Hamiltonians which gives us computational traction is the Haar-invariance of the ensemble, meaning that the ensemble-averaged dynamics look the same in any basis. Motivated by this property of the GUE, we introduce k-invariance as a precise definition of what it means for the dynamics of a quantum system to be described by random matrix theory. Here,we envision that the dynamical onset of approximate k-invariance will be a useful tool for capturing the transition from early-time chaos, as seen by OTOCs, to late-time chaos, as seen by random matrix theory.
We investigate the integrability-to-chaos transition and information scrambling in Ising spin networks via a graph-theoretic formulation. Modeling spins as vertices and interactions as edges encoded by adjacency matrices across path, Erdos-Renyi, and Watts-Strogatz topologies, we demonstrate that long-range couplings and heterogeneous degree distributions markedly accelerate quantum information propagation. The Hamiltonian comprises local and normalized non-local interactions; tuning the non-local coupling and field heterogeneity drives integrability breaking. To quantify scrambling, we employ bipartite mutual and tripartite information. Increasing non-local interactions drives tripartite information to large negative values, signaling deep information scrambling. The squared commutators constructed from out-of-time-order correlators (OTOCs) exhibit early-time exponential growth; equivalently, the OTOC four-point functions themselves decay exponentially, yielding quantum Lyapunov exponents that scale systematically with parameters governing the chaotic regime. Complementing this, Krylov complexity reveals rapid operator growth in the chaotic phase, synchronizing with OTOC and mutual information dynamics. Spectrally, the transition manifests as a shift from Poissonian to Wigner-Dyson level spacing statistics. The spectral form factor (SFF) exhibits the characteristic slope-dip-ramp-plateau structure, enabling the extraction of Thouless and Heisenberg times. Crucially, a reduced Thouless time strongly correlates with accelerated information and operator scrambling. Ultimately, this work establishes a unified framework bridging network topology with information-theoretic, operator, and spectral diagnostics, offering insights into thermalization and non-equilibrium dynamics in quantum many-body systems.
A bstract Chaos and complexity entail an entropic and computational obstruction to describing a system, and thus are intrinsically difficult to characterize. In this paper, we consider time evolution by Gaussian Unitary Ensemble (GUE) Hamiltonians and analytically compute out-of-time-ordered correlation functions (OTOCs) and frame potentials to quantify scrambling, Haar-randomness, and circuit complexity. While our random matrix analysis gives a qualitatively correct prediction of the late-time behavior of chaotic systems, we find unphysical behavior at early times including an O(1) O 1 scrambling time and the apparent breakdown of spatial and temporal locality. The salient feature of GUE Hamiltonians which gives us computational traction is the Haar-invariance of the ensemble, meaning that the ensemble-averaged dynamics look the same in any basis. Motivated by this property of the GUE, we introduce k -invariance as a precise definition of what it means for the dynamics of a quantum system to be described by random matrix theory. We envision that the dynamical onset of approximate k -invariance will be a useful tool for capturing the transition from early-time chaos, as seen by OTOCs, to late-time chaos, as seen by random matrix theory.
Scrambling is a key concept in the analysis of nonequilibrium properties of quantum many-body systems. Most studies focus on its characterization via out-of-time-ordered correlation (OTOC) functions, particularly through the early-time decay of the OTOC. However, scrambling is a complex process which involves operator spreading and operator entanglement, and a full characterization requires one to access more refined information on the operator dynamics at several timescales. In this work we analyze operator scrambling by expanding the target operator in a complete basis and studying the structure of the expansion coefficients treated as a coarse-grained probability distribution in the space of operators. We study different features of this distribution, such as its mean, variance, and participation ratio, for the Ising model with longitudinal and transverse fields, kicked collective spin models, and random circuit models. We show that the long-time properties of the operator distribution display common features across these cases and discuss how these properties can be used as a proxy for the onset of quantum chaos. Finally, we discuss the connection with OTOCs and analyze the cost of probing the operator distribution experimentally using these correlation functions.
A bstract We investigate minimal two-body Hamiltonians with random interactions that generate spectra resembling those of Gaussian random matrices, a phenomenon we term quadratic quantum chaos . Unlike integrable two-body fermionic systems, the corresponding hard-core boson models exhibit genuinely chaotic dynamics, closely paralleling the Sachdev-Ye-Kitaev (SYK) model in its spin representation. This chaotic behavior is diagnosed through spectral statistics and measures of operator growth, including Krylov complexity and the late-time decay of higher-order out-of-time-ordered correlators (OTOCs); the latter reveals the emergence of freeness in the sense of free probability. Moreover, the fractal dimension and Stabilizer Rényi entropy of a representative mid-spectrum eigenstate show finite-size deviations yet converge toward Haar-randomness as the system size increases. This convergence, constrained by local interactions, highlights the “weakly chaotic” character of these eigenstates. Owing to their simplicity and bosonic nature, these minimal models may constitute promising and resource-efficient candidates for probing quantum chaos and information scrambling on near-term quantum devices.
Operator scrambling is a crucial ingredient of quantum chaos. Specifically, in the quantum chaotic system, a simple operator can become increasingly complicated under unitary time evolution. This can be diagnosed by various measures such as square of the commutator (out-of-time-ordered correlator), operator entanglement entropy etc. In this paper, we discuss operator scrambling in three representative models: a chaotic spin-1/2 chain with spatially local interactions, a 2-local spin model and the quantum linear map. In the first two examples, although the speeds of scrambling are quite different, a simple Pauli spin operator can eventually approach a "highly entangled" operator with operator entanglement entropy taking a volume law value (close to the Page value). Meanwhile, the spectrum of the operator reduced density matrix develops a universal spectral correlation which can be characterized by the Wishart random matrix ensemble. In the second example, we further connect the 2-local model into a one dimensional chain and briefly discuss the operator scrambling there. In contrast, in the quantum linear map, although the square of commutator can increase exponentially with time, a simple operator does not scramble but performs chaotic motion in the operator basis space determined by the classical linear map. We show that once we modify the quantum linear map such that operator can mix in the operator basis, the operator entanglement entropy can grow and eventually saturate to its Page value, thus making it a truly quantum chaotic model.