The idea that information processing is optimised near the boundary between order and chaos has emerged as a recurring principle across neuroscience, complex systems, and machine learning. Here we test this hypothesis in quantum many-body systems, numerically simulating two-dimensional Ising networks of up to $N=20$ spins, operated as quantum reservoirs for time-series forecasting. Using out-of-time-order correlators (OTOCs), we locate the onset of information scrambling as the input strength is swept, separating regimes where information is frozen and scrambled across the whole reservoir state. We show that prediction precision peaks at the onset of scrambling, along with the number of computational-basis states that the reservoir actively populates. We then show that prediction precision grows as a power law $\sim N^α$ in the reservoir size, superextensively $(α>1)$ at the onset of scrambling and only sublinearly $(α<1)$ in either neighbouring regime. Finally, we show that scrambling enhances the nonlinear components of the reservoir memory while reducing its linear capacity. At the onset, the total memory capacity grows superextensively, provided that the necessary ``forgetting'' mechanism is supplied by a collective relaxation channel. These results consolidate the role of information scrambling in learning systems, turning it from an operating point into a scaling law for the performance of quantum reservoirs.The idea that information processing is optimised near the boundary between order and chaos has emerged as a recurring principle across neuroscience, complex systems, and machine learning. Here we test this hypothesis in quantum many-body systems, numerically simulating two-dimensional Ising networks of up to $N=20$ spins, operated as quantum reservoirs for time-series forecasting. Using out-of-time-order correlators (OTOCs), we locate the onset of information scrambling as the input strength is swept, separating regimes where information is frozen and scrambled across the whole reservoir state. We show that prediction precision peaks at the onset of scrambling, along with the number of computational-basis states that the reservoir actively populates. We then show that prediction precision grows as a power law $\sim N^α$ in the reservoir size, superextensively $(α>1)$ at the onset of scrambling and only sublinearly $(α<1)$ in either neighbouring regime. Finally, we show that scrambling enhances the nonlinear components of the reservoir memory while reducing its linear capacity. At the onset, the total memory capacity grows superextensively, provided that the necessary ``forgetting'' mechanism is supplied by a collective relaxation channel. These results consolidate the role of information scrambling in learning systems, turning it from an operating point into a scaling law for the performance of quantum reservoirs.

quant-pharXiv:2608.25511Aug 27, 20260 citations

Onset of chaos and relaxation in isolated systems of interacting spins: Energy shell approach

Léa Ferreira dos Santos, F. Borgonovi, Felix M. Izrailev

We study the onset of chaos and statistical relaxation in two isolated dynamical quantum systems of interacting spins 1/2, one of which is integrable and the other chaotic. Our approach to identifying the emergence of chaos is based on the level of delocalization of the eigenstates with respect to the energy shell, the latter being determined by the interaction strength between particles or quasiparticles. We also discuss how the onset of chaos may be anticipated by a careful analysis of the Hamiltonian matrices, even before diagonalization. We find that despite differences between the two models, their relaxation processes following a quench are very similar and can be described analytically with a theory previously developed for systems with two-body random interactions. Our results imply that global features of statistical relaxation depend on the degree of spread of the eigenstates within the energy shell and may happen to both integrable and nonintegrable systems.We study the onset of chaos and statistical relaxation in two isolated dynamical quantum systems of interacting spins 1/2, one of which is integrable and the other chaotic. Our approach to identifying the emergence of chaos is based on the level of delocalization of the eigenstates with respect to the energy shell, the latter being determined by the interaction strength between particles or quasiparticles. We also discuss how the onset of chaos may be anticipated by a careful analysis of the Hamiltonian matrices, even before diagonalization. We find that despite differences between the two models, their relaxation processes following a quench are very similar and can be described analytically with a theory previously developed for systems with two-body random interactions. Our results imply that global features of statistical relaxation depend on the degree of spread of the eigenstates within the energy shell and may happen to both integrable and nonintegrable systems.

arXiv:1201.0186Physical Review EMar 20, 2012131 citations

Chaos in quantum many-body systems

Gary E. Mitchell

Bulletin of the American Physical SocietyOct 28, 20061 citation

Dynamical Chaos and Many-Body Quantum Systems

Pierre Gaspard

Springer eBooksJan 1, 199210 citations

Scrambling and quantum chaos indicators from long-time properties of operator distributions

Sivaprasad Omanakuttan, Karthik Chinni, Philip Daniel Blocher +1 more

Scrambling is a key concept in the analysis of nonequilibrium properties of quantum many-body systems. Most studies focus on its characterization via out-of-time-ordered correlation (OTOC) functions, particularly through the early-time decay of the OTOC. However, scrambling is a complex process which involves operator spreading and operator entanglement, and a full characterization requires one to access more refined information on the operator dynamics at several timescales. In this work we analyze operator scrambling by expanding the target operator in a complete basis and studying the structure of the expansion coefficients treated as a coarse-grained probability distribution in the space of operators. We study different features of this distribution, such as its mean, variance, and participation ratio, for the Ising model with longitudinal and transverse fields, kicked collective spin models, and random circuit models. We show that the long-time properties of the operator distribution display common features across these cases and discuss how these properties can be used as a proxy for the onset of quantum chaos. Finally, we discuss the connection with OTOCs and analyze the cost of probing the operator distribution experimentally using these correlation functions.Scrambling is a key concept in the analysis of nonequilibrium properties of quantum many-body systems. Most studies focus on its characterization via out-of-time-ordered correlation (OTOC) functions, particularly through the early-time decay of the OTOC. However, scrambling is a complex process which involves operator spreading and operator entanglement, and a full characterization requires one to access more refined information on the operator dynamics at several timescales. In this work we analyze operator scrambling by expanding the target operator in a complete basis and studying the structure of the expansion coefficients treated as a coarse-grained probability distribution in the space of operators. We study different features of this distribution, such as its mean, variance, and participation ratio, for the Ising model with longitudinal and transverse fields, kicked collective spin models, and random circuit models. We show that the long-time properties of the operator distribution display common features across these cases and discuss how these properties can be used as a proxy for the onset of quantum chaos. Finally, we discuss the connection with OTOCs and analyze the cost of probing the operator distribution experimentally using these correlation functions.

arXiv:2211.15872Physical Review AMar 21, 202324 citations

N=20

spins, operated as quantum reservoirs for time-series forecasting. Using out-of-time-order correlators (OTOCs), we locate the onset of information scrambling as the input strength is swept, separating regimes where information is frozen and scrambled across the whole reservoir state. We show that prediction precision peaks at the onset of scrambling, along with the number of computational-basis states that the reservoir actively populates. We then show that prediction precision grows as a power law

\sim N^α

in the reservoir size, superextensively

(α>1)

at the onset of scrambling and only sublinearly

(α<1)

in either neighbouring regime. Finally, we show that scrambling enhances the nonlinear components of the reservoir memory while reducing its linear capacity. At the onset, the total memory capacity grows superextensively, provided that the necessary ``forgetting'' mechanism is supplied by a collective relaxation channel. These results consolidate the role of information scrambling in learning systems, turning it from an operating point into a scaling law for the performance of quantum reservoirs.