This paper presents a fundamental reinterpretation of quantum chaos, arguing that the connection between quantum complexity and operational failure—such as decoherence, information loss, or leakage from protected quantum states—has been poorly understood. The author proposes that quantum chaos should be viewed through the Trojan framework of thin and thick chaos, where projection onto operational variables plays the central role. The core insight is that a quantum system may display all the traditional signatures of chaos—Wigner-Dyson spectral statistics, rapid operator growth, entanglement spreading—yet remain operationally harmless. The key distinction is whether the projected quantum state, rather than the full quantum state, reaches a failure region. This is analogous to classical systems where local instability may remain confined (thin chaos) or may connect to failure (thick chaos). The paper establishes a Quantum Projection-Containment Theorem, which provides a rigorous sufficient condition for avoiding failure: as long as the projected quantum displacement remains smaller than the distance to the failure set, the system remains quantum-thin regardless of its spectral complexity. The Quantum Thinness Ratio is introduced as a quantitative diagnostic, comparing actual projected motion with the available safety margin. Several quantum protection mechanisms are identified that keep chaos thin: spectral gaps that suppress transitions from protected subspaces, symmetries and selection rules that create invariant sectors, localization that prevents delocalization failure, and quantum error-correcting codes that protect logical information even when physical degrees of freedom are chaotic. Quantum thickening occurs when these protections fail—through gap collapse, symmetry breaking, delocalization, or code failure—allowing quantum chaos to become operational failure. The paper provides a practical protocol for diagnosing quantum thinness and emphasizes that quantum chaos is not automatically decoherence. The difference between harmless quantum complexity and operational failure is reachability: chaos becomes thick only when information spreading or leakage reaches protected variables. This framework unifies classical and quantum understandings of stability while respecting the fundamental differences between trajectories and quantum states.
This paper presents a fundamental reinterpretation of quantum chaos, arguing that the connection between quantum complexity and operational failure—such as decoherence, information loss, or leakage from protected quantum states—has been poorly understood. The author proposes that quantum chaos should be viewed through the Trojan framework of thin and thick chaos, where projection onto operational variables plays the central role. The core insight is that a quantum system may display all the traditional signatures of chaos—Wigner-Dyson spectral statistics, rapid operator growth, entanglement spreading—yet remain operationally harmless. The key distinction is whether the projected quantum state, rather than the full quantum state, reaches a failure region. This is analogous to classical systems where local instability may remain confined (thin chaos) or may connect to failure (thick chaos). The paper establishes a Quantum Projection-Containment Theorem, which provides a rigorous sufficient condition for avoiding failure: as long as the projected quantum displacement remains smaller than the distance to the failure set, the system remains quantum-thin regardless of its spectral complexity. The Quantum Thinness Ratio is introduced as a quantitative diagnostic, comparing actual projected motion with the available safety margin. Several quantum protection mechanisms are identified that keep chaos thin: spectral gaps that suppress transitions from protected subspaces, symmetries and selection rules that create invariant sectors, localization that prevents delocalization failure, and quantum error-correcting codes that protect logical information even when physical degrees of freedom are chaotic. Quantum thickening occurs when these protections fail—through gap collapse, symmetry breaking, delocalization, or code failure—allowing quantum chaos to become operational failure. The paper provides a practical protocol for diagnosing quantum thinness and emphasizes that quantum chaos is not automatically decoherence. The difference between harmless quantum complexity and operational failure is reachability: chaos becomes thick only when information spreading or leakage reaches protected variables. This framework unifies classical and quantum understandings of stability while respecting the fundamental differences between trajectories and quantum states.
Quantum chaos concerns the emergence of universal statistical and dynamical prop- erties in quantum systems whose classical counterparts exhibit chaotic behavior. Unlike classical chaos, which can be characterized by sensitivity to initial conditions and exponential divergence of trajectories, quantum chaos manifests itself through a variety of indirect diagnostics, including spectral correlations, operator growth, and information scrambling. Understanding how these signatures arise from microscopic dynamics, and how they can be accessed in realistic many-body systems, is a central challenge in modern physics. This thesis investigates quantum chaos from complementary theoretical and practical perspectives. On the theoretical side, it develops effective descriptions of chaotic dy- namics in many-body systems with approximate symmetries, focusing on how weak symmetry breaking controls the exploration of Hilbert space and the crossover to random-matrix universality. Using perturbative arguments, numerical simulations, and a nonlinear sigma-model formulation, we identify a regime of Hilbert-space dif- fusion in which chaotic dynamics unfolds through a slow, collective spreading across nearly decoupled symmetry sectors. On the practical side, the thesis addresses the question of how paradigmatic chaotic models can be realized and probed in controlled experimental platforms. In partic- ular, it explores quantum simulation schemes for Sachdev–Ye–Kitaev (SYK) models using cavity quantum electrodynamics. By employing time-dependent disorder and Trotterized evolution in single-mode optical cavities, we demonstrate how low-rank interactions can be systematically promoted to effective all-to-all randomness, re- producing the spectral and dynamical signatures of SYK physics. Extensions to open-system dynamics are discussed, including Lindbladian realizations and the role of dissipation. The thesis thus highlights the connections between effective descriptions of quantum chaos and their implementation in controlled quantum simulation settings.
This paper examines dynamical localization in periodically driven quantum systems through the lens of the thin-thick classification. The central finding is that dynamical localization—where quantum interference suppresses classical chaotic diffusion—is equivalent to a large Floquet thinness index under the natural momentum or principal quantum number projection. In systems like the quantum kicked rotor and microwave-driven hydrogen, the localization length precisely equals the diameter of the projected reachable set of Floquet eigenstates. Throughout this localized regime, the quantum coherence horizon exceeds the classical prediction horizon by a factor bounded below by a positive power of Planck's constant. This means quantum coherence persists far longer than classical chaos would predict. The system only becomes "thick"—with the thinness index collapsing to order unity—when the driving strength reaches the quantum resonance-overlap threshold, where multiphoton resonances in the quasi-energy spectrum begin to overlap. This threshold lies above the classical chaos border, creating an operational window where systems are classically chaotic yet remain dynamically localized. The results apply to Floquet systems with one-and-a-half or two degrees of freedom and hold for fixed positive Planck's constant. The practical implication is that keeping driving strengths below the quantum resonance-overlap threshold preserves coherence on timescales much longer than classical predictions.
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.
Quantum Information-Theoretic Chaos Theory explores the potential of quantum mechanics to fundamentally define and understand chaos, offering a novel approach to predicting and controlling its behavior. This paper proposes a formalization of chaos as a fundamental quantum phenomenon, leveraging quantum entanglement and decoherence to generate inherently chaotic patterns. We introduce a new set of rules governing these interactions, aiming to provide a mathematically rigorous framework for analyzing and manipulating chaotic systems. The core mechanism revolves around establishing a precise relationship between quantum states and chaotic dynamics, offering a potential pathway to enhanced predictive capabilities. This research addresses a long-standing challenge in quantum mechanics: the lack of a consistent mathematical description of chaos, presenting a bold new direction for theoretical investigation and practical application.
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.
In classical systems, chaos is clearly defined via the behavior of trajectories. In quantum systems with a classical analogue one finds that the transition from regular to chaotic dynamics is signified by a change in the spectral statistics. This has been found to remain true for quantum systems with no classical analogue, including many-body systems. Furthermore, quantum chaotic systems explore all the allowed configurations in the Hilbert space, i.e. they are ergodic, while integrable systems, and systems in the many-body localized phase, are restricted to a certain subspace of the available phase space, and hence strongly break ergodicity. In this dissertation, we study the intermediate behavior between ergodicity and localization, i.e. the weak breaking of ergodicity. The model examined is the PXP spin chain model, where spins are allowed to flip only under certain kinetic constraints. We start by reproducing some already established results. First, we explore the eigenstate thermalization hypothesis (ETH) for this model and demonstrate the existence of a small number of states, throughout the PXP spectrum, that violate the ETH. Then we study the level-spacing statistics of the model, a well-known quantum chaos diagnostic, which turns out to be close to semi-Poisson and approach Wigner--Dyson statistics for large system sizes. Moreover, we examine various aspects of the model that have not been studied before. For example, the eigenvector component statistics, another quantum chaos diagnostic, for the PXP model turn out to be non-Gaussian. Finally, we perform a quench, in order to study how the energy spreads throughout the system, and observe ballistic fronts.