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arXiv:2608.25511

Superextensive learning in quantum reservoirs at the onset of information scrambling

Jonas Freiheit, Francesco Campaioli

quant-phcond-mat.dis-nnSubmitted Aug 26, 2026• Announced Aug 27, 2026• 0 citations
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Abstract

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.

Comments
15 pages, 8 figures

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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=20N=20N=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 ∼Nα\sim N^α∼Nα in the reservoir size, superextensively (α>1)(α>1)(α>1) at the onset of scrambling and only sublinearly (α<1)(α<1)(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.
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The performance of a quantum reservoir computer in temporal processing tasks depends on how its driven quantum substrate retains information about past inputs, distributes it across physical degrees of freedom, and loses it through environmental dissipation and measurement feedback. We formulate these processes using a classical-quantum state obtained by restricting a reservoir process tensor to classical input encoding and single-time readout. Conditional subsystem Holevo quantities describe information about selected input histories and bound its accessibility to measurements on subsystems of the reservoir. In a six-qubit all-to-all transverse-field Ising reservoir, we find that the total stored information changes relatively little across Hamiltonian parameters, while its spatial distribution and temporal decay vary strongly. Effective diagnostics of these two behaviours identify different regions of high information-processing capacity for linear and higher-degree temporal tasks. Measurement-induced dephasing can improve noiseless task performance when it increases forgetting rates without strongly reducing information delocalisation. The framework separates storage from subsystem accessibility and provides a common description of information flow in driven quantum learning systems.
Machine learning has shown significant breakthroughs in quantum science, where in particular deep neural networks exhibited remarkable power in modeling quantum many-body systems. Here, we explore how the capacity of data-driven deep neural networks in learning the dynamics of physical observables is correlated with the scrambling of quantum information. We train a neural network to find a mapping from the parameters of a model to the evolution of observables in random quantum circuits for various regimes of quantum scrambling and test its generalization and extrapolation capabilities in applying it to unseen circuits. Our results show that a particular type of recurrent neural network is extremely powerful in generalizing its predictions within the system size and time window that it has been trained on for both, localized and scrambled regimes. These include regimes where classical learning approaches are known to fail in sampling from a representation of the full wave function. Moreover, the considered neural network succeeds in extrapolating its predictions beyond the time window and system size that it has been trained on for models that show localization, but not in scrambled regimes.
How fast quantum information scrambles such that it becomes inaccessible by local probes turns out to be central to various fields. Motivated by recent works on spin systems with nonlocal interactions, we study information scrambling in different variants of the Ising model. Our work reveals that nonlocal interactions can induce operator dynamics not precisely captured by out-of-time-order correlators (OTOCs). In particular, the operator size exhibits a slowdown in systems with generic power-law interactions despite a highly nonlinear light cone. A recently proposed microscopic model for fast scrambling does not show this slowdown, which uncovers a distinct analogy between a local operator under unitary evolution and the entanglement entropy following a quantum quench. Our work gives new insights on scrambling properties of systems in reach of current quantum simulation platforms and complements results on possibly observing features of quantum gravity in the laboratory.