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.
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.
Random permutation circuits were recently introduced as minimal models for local many-body dynamics that can be interpreted both as classical and quantum. Standard dynamical complexity indicators such as damage spreading and out-of-time-order correlators (OTOCs), show that these systems exhibit sensitivity to initial conditions in the classical setting and operator scrambling in the quantum setting. Here, we address their quantum chaoticity—a stricter property—by studying the time evolution of local operator entanglement (LOE). We show that the behavior of LOE in random permutation circuits depends on the dimension of the local configuration space q . When q = 2 , i.e., the circuits act on qubits, random permutations are Clifford and the LOE of any local operator is bounded by a constant, indicating that they are not truly chaotic. On the other hand, when the dimension of the local configuration space exceeds two, the LOE grows linearly in time. We prove this in the limit of large q and present numerical evidence that a three-dimensional local configuration space is sufficient for a linear growth of LOE. Our findings highlight that quantum chaos can be produced by essentially classical dynamics. Moreover, we show that LOE can be defined also in the classical realm and put it forward as a universal indicator chaos, both quantum and classical.
Out-of-time order correlators (OTOCs) are crucial tools for studying quantum chaos as they show distinct scrambling behavior for chaotic Hamiltonians. We calculate the OTOC and analyze the quantum information scrambling in atom-field and spin-spin interaction models, which are open-system models and exhibit non-Markovian behavior. We also examine the Loschmidt echo for these models and comment on their chaotic nature. The commutator growth of two local operators, which is upper bounded by the Lieb-Robinson bound, is studied for these models, and the patterns of scrambling are investigated.
Information scrambling, the process by which quantum information spreads and becomes effectively inaccessible, is central to modern quantum statistical physics and quantum chaos. These lecture notes provide an introduction to information scrambling from both static and dynamical perspectives. The spectral properties of reduced density matrices arising from Haar-random states are developed through the geometry of the unitary group and the universal results of random matrix theory. This geometric framework yields universal, model-independent predictions for entanglement and spectral statistics, capturing generic features of quantum chaos without reference to microscopic details. Dynamical diagnostics such as the spectral form factor and out-of-time-ordered correlators further reveal the onset of chaos in time-dependent evolution.
The notes are aimed at advanced undergraduate and graduate students in physics, mathematics, and computer science who are interested in the connections between quantum chaos, information dynamics, and quantum computing. Concepts such as entanglement growth, Haar randomness, random-matrix statistics, and unitary t-designs are introduced through their realization in random quantum circuits and circuit complexity. The same mathematical framework also underpins modern quantum-device benchmarking, where approximate unitary designs and random circuits translate geometric ideas into quantitative tools for assessing fidelity, noise, and scrambling efficiency in real quantum processors.
Thermalization of chaotic quantum many-body systems under unitary time evolution is related to the growth in complexity of initially simple Heisenberg operators.Operator growth is a manifestation of information scrambling and can be diagnosed by out-of-time-order correlators (OTOCs).However, the behavior of OTOCs of local operators in generic chaotic local Hamiltonians remains poorly understood, with some semiclassical and large-N models exhibiting exponential growth of OTOCs and a sharp chaos wave front and other random circuit models showing a diffusively broadened wave front.In this paper, we propose a unified physical picture for scrambling in chaotic local Hamiltonians.We construct a random time-dependent Hamiltonian model featuring a large-N limit where the OTOC obeys a Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) type equation and exhibits exponential growth and a sharp wave front.We show that quantum fluctuations manifest as noise (distinct from the randomness of the couplings in the underlying Hamiltonian) in the FKPP equation and that the noise-averaged OTOC exhibits a crossover to a diffusively broadened wave front.At small N, we demonstrate that operator growth dynamics, averaged over the random couplings, can be efficiently simulated for all time using matrix product state techniques.To show that time-dependent randomness is not essential to our conclusions, we push our previous matrix product operator methods to very large size and show that data for a timeindependent Hamiltonian model are also consistent with a diffusively broadened wave front.
We study quantum information scrambling in spin models with both long-range all-to-all and short-range interactions. We argue that a simple global, spatially homogeneous interaction together with local chaotic dynamics is sufficient to give rise to fast scrambling, which describes the spread of quantum information over the entire system in a time that is logarithmic in the system size. This is illustrated in two tractable models: (1) a random circuit with Haar random local unitaries and a global interaction and (2) a classical model of globally coupled nonlinear oscillators. We use exact numerics to provide further evidence by studying the time evolution of an out-of-time-order correlator and entanglement entropy in spin chains of intermediate sizes. Our results pave the way towards experimental investigations of fast scrambling and aspects of quantum gravity with quantum simulators.
At the heart of physics lies the human fascination with natural phenomena and the wish to understand the underlying principles. A well-known example is the butterfly effect. It describes how small changes in the initial state can cause significantly different measurement outcomes at later times. Such sensitivity to initial conditions is a key signature of chaotic systems and reflects their instability under small perturbations. The principle underlying this phenomenon is information spreading: with time, information about the perturbation propagates through the system, changing measurement results even at distant locations. In quantum systems, information not only spreads but can also scramble, meaning that the precise details about the perturbation are hidden in highly nonlocal correlations. Information scrambling has emerged as a universal feature of quantum chaos and, together with other universal features, serves as an organizing principle for quantum systems. This motivates imposing structure like locality on quantum systems in order to probe the capabilities and limits of such organizing principles. There have been significant advances in both theoretical and experimental directions in recent years. The newly gained flexibility in designing experiments and simulations also inspires the analytical study of quantum systems from the bottom up, where tractable models offer insights into the generic features of quantum mechanics. In this thesis we build upon these advances to explore the implications of adding structure to chaotic quantum systems for the behavior of such features. In particular, we consider the measures entanglement entropy and Krylov complexity as probes of structure. These quantities are closely connected to information spreading. This motivates us to study the influence of structure on information spreading in the second part. In the first part, we begin with a detailed discussion of the behavior of entanglement in chaotic systems. We impose a minimal amount of local structure by separating the system into two parts, each modeled as a single spin. We validate the predicted entanglement for typical states in such systems and show that it saturates the Page limit as the Hilbert space dimension increases. We then study how different initially unentangled product states approach the Page limit under time evolution. We consider three types of product states, each representing a distinct case. Random product states represent generic states only restricted by the absence of entanglement. Random dipole and quadrupolar states represent two distinct types of semi-classical structure. The results on the entanglement generation in the three different types of states refine our earlier conclusion, demonstrating that the semi-classical limit can suppress the generation of extensive entanglement. This suppression does not occur for random product or quadrupolar states; where the imposed structure is insufficient. We conclude the second chapter by presenting two protocols that can stabilize the dynamics at finite spin sizes, similar to the semi-classical limit. In the following chapter, we move from measuring the complexity of time-evolved states using entanglement entropy to investigating time-evolved operators through the lens of Krylov complexity. After a short introduction to Krylov complexity for Hamiltonian dynamics, we generalize the framework for discrete time evolution. We identify the linear growth of Krylov complexity as a universal feature of generic systems and analyze its behavior as we transition from chaotic to increasingly structured systems. Dual-unitary circuits converge toward this universal behavior at rates that can be as fast as exponential, consistent with their classification as maximally chaotic. We then approximately restore energy conservation by considering trotterized systems, ranging from unstructured and chaotic to highly structured and integrable ones. We find these systems to thermalize on a scale set by the trotterization step, slowly approaching the universal linear Krylov complexity growth regime. Approaching the linear growth regime is a fundamental property of trotterized systems with an unbounded spectrum, while systems with a bounded energy spectrum can evade thermalization as probed by Krylov complexity. Instead, they have a nonanalytic transition as the Trotter step size increases. Our study of Krylov complexity establishes its universal properties for discrete time evolution. The presented results deepen our understanding of the thermalizing behavior of different classes of systems in relation to their spectral properties by introducing an intuitive notion of a bath with chaotic characteristics. We link this thermalization behavior to information spreading, which we explore in the second part of this thesis. Information scrambling, along with the associated spreading, is a key phenomenon in chaotic systems with local structure. In the fourth chapter, we investigate information spreading in proximity to an integrable system exhibiting spontaneous symmetry breaking, the transverse field Ising model (TFIM). As expected, integrability prevents the scrambling of information. At the same time, information spreading probed by quenched out-of-time ordered correlators (OTOCs) exhibits features reminiscent of the quantum phase transition present in the TFIM. We unravel the intricate behavior of these OTOCs and identify the underlying mechanism that stabilizes the long-time values of OTOCs: dynamical relocalization. We provide two variations of the TFIM in which the relocalization phenomenon is not directly related to the presence of a quantum phase transition. This shows that no direct, universal relation exists between the two. Finally, we discuss the effects of breaking the integrable structure of the TFIM. Our results show that for short times, chaotic systems in proximity of integrability can retain signatures of integrability. In contrast, we observe significant changes at large times. This suggests that information scrambling, as a universal feature of chaotic systems, is restored at large times—even in systems near integrability that exhibit order. Following the analysis of information spreading close to integrable regimes, we study the stability of information spreading in more generic systems. First, we consider charge-conserving random circuits. In these systems, charge-charge correlators exhibit large fluctuations across different circuit instances with a slow polynomial decay in time. At the same time, they remain stable against fine-tuned perturbations, which can be interpreted as dephasing noise in the eigenbasis of the charge density operators. Equivalently, charge-charge correlators are mostly insensitive to decoherence, with interference effects contributing only subleading corrections. In this sense, the charge information spreading probed by charge-charge correlators is effectively classical. We perform a detailed analysis and quantify the dephasing insensitivity of these correlators. This suggests that circuit-to-circuit fluctuations can be captured by considering small regions where “quantum” effects are included, embedded within otherwise “classical” dynamics. This observation motivates the development of an algorithm to compute charge-charge correlators for individual circuits on classical computers with high accuracy. We quantitatively relate the algorithm’s performance to the dephasing analysis. We demonstrate the algorithm’s accuracy numerically for charge-charge correlators. Then, we argue that in generic circuits without conservation laws, the information spreading, as probed by OTOCs, follows similar principles. We conclude by comparing OTOC values measured on quantum devices with up to 95 superconducting qubits and 1000 gates to predictions from the approximate algorithm. This cross-validates experimental measurements and theoretical predictions. In this thesis, we explore the implications of adding structure to chaotic quantum systems, contributing to a deeper understanding of the organizing principles governing quantum chaos. We establish that entanglement can be suppressed in the semiclassical limit and that Krylov complexity behaves differently in thermalizing or integrable time-dependent systems. We show that the phenomenon information scrambling is also present in chaotic quantum systems even in the vicinity of a quantum phase transition. Conversely, when charge conservation is added, information spreading is stabilized and becomes reminiscent of that in classical systems. This work opens up further research directions towards a better understanding of the quantum nature of chaotic quantum systems and the development of probes of chaos in time dependent systems with different types of structure.