PROBING THE MSS BOUND AND SCRAMBLING DYNAMICS IN A MINIMAL 3-QUBIT SYSTEM ABSTRACT:This study investigates the fundamental limits of quantum chaos—the Maldacena-Shenker-Stanford (MSS) bound—using a minimal 3-qubit non-integrable system[cite: 82, 87]. By prioritizing measurement precision (16,384 shots) over system size on the IBM Quantum Torino processor, I resolve the complete "scrambling life-cycle," including the chaotic burst and the finite-size recurrence of quantum complexity[cite: 89, 91]. The extracted intrinsic Lyapunov exponent (1.30 +/- 0.05) satisfies the MSS bound in the infinite-temperature limit[cite: 90]. These results demonstrate that minimal quantum hardware can probe deep holographic principles and provide a high-precision benchmark for quantum gravity simulations[cite: 92]. KEY SCIENTIFIC ADVANCES:- Intrinsic Lyapunov Extraction: Utilized non-linear data collapse and bootstrap analysis to isolate the scrambling rate from finite-size effects[cite: 138, 140].- Observation of Unitary Recurrence: Resolved the late-time revival of information fidelity at t=3, a hallmark of unitary dynamics in finite Hilbert spaces[cite: 131, 132].- MSS Bound Verification: Defined an effective scrambling temperature scale to confirm adherence to the universal chaos bound[cite: 147, 148]. REPOSITORY CONTENTS:1. Manuscript & SI: Complete text and detailed quantum circuit sequences.2. Raw Experimental Data: Execution counts (shots) from the IBM Quantum Torino processor[cite: 64, 118].3. Analysis Scripts: Python scripts for data collapse fitting and Lyapunov exponent extraction[cite: 64, 139]. NOTE:This work serves as the foundational technical benchmark for larger-scale holographic simulations on NISQ devices[cite: 160].
PROBING THE MSS BOUND AND SCRAMBLING DYNAMICS IN A MINIMAL 3-QUBIT SYSTEM ABSTRACT:This study investigates the fundamental limits of quantum chaos—the Maldacena-Shenker-Stanford (MSS) bound—using a minimal 3-qubit non-integrable system[cite: 82, 87]. By prioritizing measurement precision (16,384 shots) over system size on the IBM Quantum Torino processor, I resolve the complete "scrambling life-cycle," including the chaotic burst and the finite-size recurrence of quantum complexity[cite: 89, 91]. The extracted intrinsic Lyapunov exponent (1.30 +/- 0.05) satisfies the MSS bound in the infinite-temperature limit[cite: 90]. These results demonstrate that minimal quantum hardware can probe deep holographic principles and provide a high-precision benchmark for quantum gravity simulations[cite: 92]. KEY SCIENTIFIC ADVANCES:- Intrinsic Lyapunov Extraction: Utilized non-linear data collapse and bootstrap analysis to isolate the scrambling rate from finite-size effects[cite: 138, 140].- Observation of Unitary Recurrence: Resolved the late-time revival of information fidelity at t=3, a hallmark of unitary dynamics in finite Hilbert spaces[cite: 131, 132].- MSS Bound Verification: Defined an effective scrambling temperature scale to confirm adherence to the universal chaos bound[cite: 147, 148]. REPOSITORY CONTENTS:1. Manuscript & SI: Complete text and detailed quantum circuit sequences.2. Raw Experimental Data: Execution counts (shots) from the IBM Quantum Torino processor[cite: 64, 118].3. Analysis Scripts: Python scripts for data collapse fitting and Lyapunov exponent extraction[cite: 64, 139]. NOTE:This work serves as the foundational technical benchmark for larger-scale holographic simulations on NISQ devices[cite: 160].
This paper examines the mathematical relationship between the Maldacena–Shenker–Stanford (MSS) bound on the growth of chaos and the thin–thick chaos framework developed in recent work on coherent disorder and nonlinear stability. The MSS bound constrains the rate at which a regularized out-of-time-order correlator can deviate from its factorized value, yielding a universal speed limit on information scrambling in thermal quantum systems: the Lyapunov exponent satisfies λL≤2π/βλL≤2π/β, and the scrambling time is t∗=(β/2π)log(1/ε)t∗=(β/2π)log(1/ε), where ε∼1/N2ε∼1/N2 is the microscopic initial amplitude. We introduce a thinness index Θ(t)Θ(t) constructed directly from the correlator deviation and show that, under the analyticity and boundedness hypotheses of the MSS argument, Θ(t)Θ(t) remains parametrically small until times of order the scrambling time. The same analytic strip that yields the Lyapunov bound therefore also controls the rate at which this particular thinness index can collapse. When the additional structural hypotheses of the Trojan Universality Class—spectral gap, two-mode normal form, and near-integrable remainder—are imposed, the prefactor of the exponential growth of Θ(t)Θ(t) is suppressed by a factor η(Δ)≤1η(Δ)≤1, delaying the onset of thickness by a logarithmic shift without altering the scrambling rate itself. The analysis reveals a conceptual distinction between scrambling, the microscopic spread of quantum information, and the transition to thick chaos, the operational crossing of a failure threshold in a chosen projection. For the correlator projection these coincide in time, but structural gaps can decouple them, allowing maximal-rate scrambling to coexist with prolonged operational thinness. Limitations include projection dependence, the restrictive nature of the Trojan criteria, and the absence of a general comparison theorem relating distinct thinness indices. The results are elementary consequences of the existing MSS hypotheses and do not modify the original bound.
Abstract Chaos, in quantum systems, can be diagnosed by certain out-of-time-order correlators (OTOCs) that obey the chaos bound of Maldacena, Shenker, and Stanford (MSS). We begin by deriving a dispersion relation for this class of OTOCs, implying that they must satisfy many more constraints beyond the MSS bound. Motivated by this observation, we perform a systematic analysis obtaining an infinite set of constraints on the OTOC. This infinite set includes the MSS bound as the leading constraint. In addition, it also contains subleading bounds that are highly constraining, especially when the MSS bound is saturated by the leading term. These new bounds, among other things, imply that the MSS bound cannot be exactly saturated over any duration of time, however short. Furthermore, we derive a sharp bound on the Lyapunov exponent λ2 ≤ β6π 6 π β of the subleading correction to maximal chaos.
Abstract The Lyapunov exponent of a quantum system has been predicted to be bounded by λL≤2πT/ℏ λ L ≤ 2 π T / ħ , where T is its temperature, as established by Maldacena, Shenker, and Stanford (MSS). This bound plays an important role in studying very diverse topics of physics, ranging from the dynamics of interacting many-body systems to the black hole information problem, and it is saturated when the system under consideration is the exact holographic dual of a black hole. Based on the fact that an inverted harmonic oscillator (IHO) may exhibit the behavior of thermal energy emission, in close analogy to the Hawking radiation emitted by black holes, we propose using a trapped ion as an implementation of the IHO to verify, in a concrete analogue-gravity system, whether the MSS bound can be identically saturated. To this end, we provide prescriptions for experimentally observing the scattering process at the IHO potential, which yields an analogue of Hawking radiation, as well as for how to measure the corresponding out-of-time-ordered correlation function (OTOC), diagnosing quantum chaos, in this thermally excited semiclassical system. We theoretically show, for an experimentally realizable analogue-gravity setup, that the effective Hawking temperature of the trapped-ion-IHO indeed matches the upper MSS bound for the speed of scrambling.
Out-of-Time-Ordered Commutators (OTOCs), representing a key diagnostic for scrambling as a facet of short-time quantum chaos, have attracted wide-ranging interest, from many-body physics to quantum gravity. By means of a suitable form of the Wigner-Moyal expansion, and invoking ensemble equivalence in statistical physics, we provide a consistent approach to the growth rate of the OTOC for many-body systems with chaotic classical limit where both the classical Lyapunov exponent and the quantum nature of the density of states enter. Applying this construction to quantized high-dimensional hyperbolic motion, i.e., a quantum chaotic system that exhibits gravity-like correlation functions in the late-time regime, we compute the OTOC growth rate Λ as a function of the number of degrees of freedom, f, and inverse temperature, β.
We show that the scaled growth rate, Λ/f, can be described by a universal function of fβ and displays a cross-over from classical to quantum behavior as we increase f and/or lower the temperature. In the deep quantum regime of infinite f, we find maximally fast scrambling in the sense of the Maldacena-Shenker-Stanford bound on chaos. This elucidates the non-perturbative mechanism underlying the saturation of the bound via quantum contributions to the mean density of states, and it provides further support for this dynamical system as a dual to two-dimensional quantum gravity. In this way, we present first evidence of maximally fast scrambling in a quantum chaotic system with a well-defined classical Hamiltonian limit, without invoking any external mechanism such as (disorder) averaging.
We study numerically the onset of chaos and thermalization in the Banks-Fischler-Shenker-Susskind (BFSS) matrix model with and without fermions, considering Lyapunov exponents, entanglement generation, and quasinormal ringing.We approximate the real-time dynamics in terms of the most general Gaussian density matrices with parameters which obey self-consistent equations of motion, thus extending the applicability of real-time simulations beyond the classical limit.Initial values of these Gaussian density matrices are optimized to be as close as possible to the thermal equilibrium state of the system.Thus attempting to bridge between the low-energy regime with a calculable holographic description and the classical regime at high energies, we find that quantum corrections to classical dynamics tend to decrease the Lyapunov exponents, which is essential for consistency with the Maldacena-Shenker-Stanford bound at low temperatures.The entanglement entropy is found to exhibit an expected "scrambling" behavior-rapid initial growth followed by saturation.At least at high temperatures the entanglement saturation time appears to be governed by classical Lyapunov exponents.Decay of quasinormal modes is found to be characterized by the shortest timescale of all.We also find that while the bosonic matrix model becomes nonchaotic in the low-temperature regime, for the full BFSS model with fermions the leading Lyapunov exponent, entanglement saturation time, and decay rate of quasinormal modes all remain finite and nonzero down to the lowest temperatures.
Out-of-Time-Ordered Commutators (OTOCs), representing a key diagnostic for scrambling as a facet of short-time quantum chaos, have attracted wide-ranging interest, from many-body physics to quantum gravity. By means of a suitable form of the Wigner-Moyal expansion, and invoking ensemble equivalence in statistical physics, we provide a consistent approach to the growth rate of the OTOC for many-body systems with chaotic classical limit where both the classical Lyapunov exponent and the quantum nature of the density of states enter. Applying this construction to quantized high-dimensional hyperbolic motion, i.e., a quantum chaotic system that exhibits gravity-like correlation functions in the late-time regime, we compute the OTOC growth rate Λ as a function of the number of degrees of freedom, f, and inverse temperature, β. We show that the scaled growth rate, Λ/f, can be described by a universal function of fβ and displays a cross-over from classical to quantum behavior as we increase f and/or lower the temperature. In the deep quantum regime of infinite f, we find maximally fast scrambling in the sense of the Maldacena-Shenker-Stanford bound on chaos. This elucidates the non-perturbative mechanism underlying the saturation of the bound via quantum contributions to the mean density of states, and it provides further support for this dynamical system as a dual to two-dimensional quantum gravity. In this way, we present first evidence of maximally fast scrambling in a quantum chaotic system with a well-defined classical Hamiltonian limit, without invoking any external mechanism such as (disorder) averaging.