Abstract The Lyapunov exponent of a quantum system has been predicted to be bounded by $$\lambda _L\le 2\pi \,T/\hbar $$ 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.
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.
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].
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.
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.
Abstract We introduce a local operational protocol for measuring quantum information scrambling in a curved spacetime background. In this framework, two Unruh-DeWitt (UDW) detectors, coupled to a massless conformally coupled scalar field in the Hartle–Hawking vacuum of the non-rotating ( 2 + 1 ) -dimensional Banados–Teitelboim–Zanelli (BTZ) black hole, serve as local probes. Evaluating the detector out-of-time-ordered correlator requires a fourth-order Dyson series expansion. Within the scrambling regime, this correlator exhibits exponential growth with Lyapunov exponent, exactly saturating the Maldacena–Shenker–Stanford bound at the global Hawking temperature. Governed by the black hole surface gravity, the Lyapunov exponent is universal and independent of detector positions, energy gaps, and coupling strengths. Furthermore, the scrambling amplitude shows a monotonic dependence on the radial positions of the detectors, vanishing at the event horizon and increasing outward. Local two-level probes thus witness the maximal chaos of the BTZ geometry without reconstructing the global quantum field state.
We implement a chiral spin-chain black hole simulator on IBM superconducting quantum hardware and probe, within a common microscopic framework, both semiclassical horizon physics and interacting quantum scrambling. We first measure the dispersion relation across the exterior, horizon and over-tilted interior regimes, reproducing the predicted evolution of the effective light-cone structure. To probe Hawking thermality, we prepare a localised excitation inside the horizon and monitor its density response at an exterior site, observing the predicted inverse relation between the peak arrival time and the surface gravity, thereby establishing a calibrated dynamical estimator of the Hawking temperature. Beyond the semiclassical regime, we continuously tune the interactions and distinguish non-exponential operator spreading in the free-fermion limit from Lyapunov-like OTOC decay in the strongly interacting chiral regime. These measurements use observable-specific Floquet circuits derived from the same parent chiral model, including its mean-field and coordinate-equivalent XY descriptions, to reduce circuit depth while preserving the physics relevant to each probe. Our results provide a unified programmable platform for studying horizon geometry, Hawking thermality and interacting scrambling on quantum hardware.
The ultimate regularity of quantum mechanics creates a tension with the assumption of classical chaos used in many of our pictures of chemical reaction dynamics. Out-of-time-order correlators (OTOCs) provide a quantum analog to the Lyapunov exponents that characterize classical chaotic motion. Maldacena, Shenker, and Stanford have suggested a fundamental quantum bound for the rate of information scrambling, which resembles a limit suggested by Herzfeld for chemical reaction rates. Here, we use OTOCs to study model reactions based on a double-well reaction coordinate coupled to anharmonic oscillators or to a continuum oscillator bath. Upon cooling, as one enters the tunneling regime where the reaction rate does not strongly depend on temperature, the quantum Lyapunov exponent can approach the scrambling bound and the effective reaction rate obtained from a population correlation function can approach the Herzfeld limit on reaction rates: Tunneling increases scrambling by expanding the state space available to the system. The coupling of a dissipative continuum bath to the reaction coordinate reduces the scrambling rate obtained from the early-time OTOC, thus making the scrambling bound harder to reach, in the same way that friction is known to lower the temperature at which thermally activated barrier crossing goes over to the low-temperature activationless tunneling regime. Thus, chemical reactions entering the tunneling regime can be information scramblers as powerful as the black holes to which the quantum Lyapunov exponent bound has usually been applied.