Understanding quantum chaos is of profound theoretical interest and carries significant implications for various applications, from condensed matter physics to quantum error correction. Recently, out-of-time ordered correlators (OTOCs) have emerged as a powerful tool to quantify quantum chaos. For a given quantum system, the OTOCs measure incompatibility between an operator evolved in the Heisenberg picture and an unevolved operator. In the first part of this thesis, we employ OTOCs to study the dynamical sensitivity of a perturbed non-Komogorov-Arnold-Moser (non-KAM) system in the quantum limit as the parameter that characterizes the $\textit{resonance}$ condition is slowly varied. For this purpose, we consider a quantized kicked harmonic oscillator (KHO) model that displays stochastic webs in the phase space. The OTOC analysis is followed by a study of quantum Fisher information (QFI) at the resonances and a comparison with the non-resonance cases. We shall show that scaling of the QFI in time is enhanced at the resonances, making the dynamics of the non-KAM systems good candidates for quantum sensing. In the following chapter, we study the OTOCs in a bipartite system of kicked coupled tops with a special focus on the mixed phase space OTOC dynamics. The last part of the thesis is devoted to the study of the emergence of quantum state designs as a signature of quantum chaos and the role of symmetries in this phenomenon. Recently proposed projected ensemble framework utilizes quantum chaos as a resource to construct approximate higher-order state designs. Despite being ubiquitous, the effects of symmetries on the emergence of quantum state designs remain under-explored. We thoroughly investigate this by demonstrating the interplay between symmetries and measurements in constructing approximate state designs. Finally, we outline a few open directions relevant to the current thesis.
Understanding quantum chaos is of profound theoretical interest and carries significant implications for various applications, from condensed matter physics to quantum error correction. Recently, out-of-time ordered correlators (OTOCs) have emerged as a powerful tool to quantify quantum chaos. For a given quantum system, the OTOCs measure incompatibility between an operator evolved in the Heisenberg picture and an unevolved operator. In the first part of this thesis, we employ OTOCs to study the dynamical sensitivity of a perturbed non-Komogorov-Arnold-Moser (non-KAM) system in the quantum limit as the parameter that characterizes the resonance condition is slowly varied. For this purpose, we consider a quantized kicked harmonic oscillator (KHO) model that displays stochastic webs in the phase space. The OTOC analysis is followed by a study of quantum Fisher information (QFI) at the resonances and a comparison with the non-resonance cases. We shall show that scaling of the QFI in time is enhanced at the resonances, making the dynamics of the non-KAM systems good candidates for quantum sensing. In the following chapter, we study the OTOCs in a bipartite system of kicked coupled tops with a special focus on the mixed phase space OTOC dynamics. The last part of the thesis is devoted to the study of the emergence of quantum state designs as a signature of quantum chaos and the role of symmetries in this phenomenon. Recently proposed projected ensemble framework utilizes quantum chaos as a resource to construct approximate higher-order state designs. Despite being ubiquitous, the effects of symmetries on the emergence of quantum state designs remain under-explored. We thoroughly investigate this by demonstrating the interplay between symmetries and measurements in constructing approximate state designs. Finally, we outline a few open directions relevant to the current thesis.
We study operator growth in quantized non-KAM systems using out-of-time-ordered correlators (OTOCs), focusing on the kicked harmonic oscillator as a representative example. Since the classical harmonic oscillator is degenerate, the dynamics fall outside the usual Kolmogorov-Arnold-Moser (KAM) framework, and resonances play a central role in shaping the phase space. We examine the system near resonances, where the ratio between the oscillator and driving frequencies takes integer values. Even though the classical Lyapunov exponent remains small at these points, and hence no conventional chaos, the phase space still undergoes strong structural changes. The OTOCs are particularly sensitive to these resonances, with a quadratic-in-time growth at resonance compared to linear growth away from it. Within a perturbative treatment, we derive closed-form expressions for the OTOCs and uncover a number-theoretic structure emerging in the behavior of OTOCs, governed by the Euler totient function of the frequency ratio. Overall, the results we present in this short note imply that resonant structures can play an important role in controlling information spreading.
Non-KAM (Kolmogorov-Arnold-Moser) systems, when subjected to weak time-dependent perturbations, exhibit an abrupt transition to classical chaos through the breakdown of invariant phase-space tori. We showcase the utilization of non-KAM systems in the quantum regime as quantum sensors, leveraging their sensitivity at \textit{resonances}. Quantum Fisher information (QFI) is a central quantity in quantum parameter estimation theory that measures how much information a quantum state contains about an unknown parameter that is encoded into it. In other words, it quantifies the sensitivity of a quantum state to small changes in that parameter. In this work, through numerical analysis in conjunction with analytical results, we study the performance of the non-KAM systems for quantum sensing applications by computing the QFI. We find that the growth of the QFI is remarkably enhanced when the resonance condition is satisfied. For frequency estimation under Floquet unitary encodings, we derive a transport bound: if the mean excitation number grows as ⟨n^(t)⟩∼tα, the QFI obeys I(t)≲t2α+2. The quantum kicked harmonic oscillator, a paradigmatic non-KAM system, realizes the full hierarchy: localized dynamics (α=0) yield quadratic growth, delocalized diffusion along stochastic webs (α=1) yields quartic growth, and translationally invariant resonances (α=2) saturate the bound with anomalous hexic growth, I(t)∼t6, established analytically at resonance R=2 and numerically at R=4. The enhancement stems from resonance-induced translational symmetry rather than exponential instability, identifying non-KAM resonances as a metrological resource distinct from chaos-assisted and criticality-based sensing.
Non-Kolmogorov-Arnold-Moser (KAM) systems, when perturbed by weak time-dependent fields, offer a fast route to classical chaos through an abrupt breaking of invariant phase-space tori. In this work, we employ out-of-time-order correlators (OTOCs) to study the dynamical sensitivity of a perturbed non-KAM system in the quantum limit as the parameter that characterizes the resonance condition is slowly varied. For this purpose, we consider a quantized kicked harmonic oscillator (KHO) model, which displays stochastic webs resembling Arnold's diffusion that facilitate large-scale diffusion in the phase space. Although the Lyapunov exponent of the KHO at resonances remains close to zero in the weak perturbative regime, making the system weakly chaotic in the conventional sense, the classical phase space undergoes significant structural changes. Motivated by this, we study the OTOCs when the system is in resonance and contrast the results with the nonresonant case. At resonances, we observe that the long-time dynamics of the OTOCs are sensitive to these structural changes, where they grow quadratically as opposed to linear or stagnant growth at nonresonances. On the other hand, our findings suggest that the short-time dynamics remain relatively more stable and show the exponential growth found in the literature for unstable fixed points. The numerical results are backed by analytical expressions derived for a few special cases. We will then extend our findings concerning the nonresonant cases to a broad class of near-integrable KAM systems.
The symmetry of chaotic systems plays a pivotal role in determining the universality class of spectral statistics and dynamical behaviors, which can be described within the framework of random matrix theory. Understanding the influence of system symmetry on these behaviors is crucial for characterizing universal properties in quantum chaotic systems. In this work, we explore the universality of out-of-time-ordered correlator (OTOC) dynamics in quantum chaotic systems, focusing on the kicked rotor and the kicked Harper model. By modulating the periodically kicked potential, we control system symmetry to examine its impact on OTOC dynamics and level spacing distributions. Our results show that ensemble-averaged OTOC dynamics exhibit distinct universal behaviors depending on system symmetry, enabling classification through random matrix theory. These distinctions become evident after the localization time in localized regimes and emerge at specific time scales corresponding to the translational period of the Floquet operator in momentum space under quantum resonance conditions. Our findings provide a rigorous understanding of the relationship between symmetry and quantum chaotic dynamics, contributing to a deeper comprehension of universal behaviors in these systems.
While classical chaos has been successfully characterized with consistent theories and intuitive techniques, such as with the use of Lyapunov exponents, quantum chaos is still poorly understood, as well as its relation with multi-partite entanglement and information scrambling. We consider a benchmark system, the kicked top model, which displays chaotic behaviour in the classical version, and proceed to characterize the quantum case with a thorough diagnosis of the growth of chaos and entanglement in time. As a novel tool for the characterization of quantum chaos, we introduce for this scope the quasi-probability distribution behind the out-of-time-ordered correlator (OTOC). We calculate the cumulative nonclassicality of this distribution, which has already been shown to outperform the simple use of OTOC as a probe to distinguish between integrable and nonintegrable Hamiltonians. To provide a thorough comparative analysis, we contrast the behavior of the nonclassicality with entanglement measures, such as the tripartite mutual information of the Hamiltonian as well as the entanglement entropy. We find that systems whose initial states would lie in the "sea of chaos" in the classical kicked-top model, exhibit, as they evolve in time, characteristics associated with chaotic behavior and entanglement production in closed quantum systems. We corroborate this indication by capturing it with this novel OTOC-based measure.
Quantum computing has introduced new paradigms in how information is stored, processed, and transmitted. Unlike classical computation, where data follows deterministic pathways, quantum systems display complex interactions driven by superposition and entanglement. One of the most powerful tools for studying these interactions is the Out-of-Time-Ordered Correlator (OTOC), a mathematical framework used to quantify how information disperses and becomes scrambled in quantum systems. This paper explores information scrambling dynamics in simulated quantum environments using OTOC-based analysis. Rather than relying on a specific quantum processor, this study uses computational simulations to emulate noisy and noise-mitigated conditions, analyzing how OTOC behavior varies across these regimes. Through numerical modeling and circuit-level quantum simulations, the study demonstrates how information spreads, how chaos develops, and how error mitigation influences the stability of information propagation. The findings provide theoretical foundations for designing stable, error-tolerant quantum algorithms and highlight the intersection between quantum physics, computer science, and information theory.