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arXiv:2301.07722 · Physical review. B./Physical review. B

Scrambling in quantum cellular automata

Brian Kent, Sarah Racz, Sanjit Shashi

Published Apr 21, 2023• 11 citations
arXivHTMLPDF

Abstract

Scrambling is the delocalization of quantum information over a many-body system and underlies all quantum-chaotic dynamics. We employ discrete quantum cellular automata as classically simulable toy models of scrambling. We observe that these automata break ergodicity, i.e., they exhibit quantum scarring. We also find that the time scale of scrambling rises with the local Hilbert-space dimension and obeys a specific combinatorial pattern. We then show that scarring is mostly suppressed in a semiclassical limit, demonstrating that semiclassical-chaotic systems are more ergodic.

DOI
10.1103/physrevb.107.144306

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Scrambling is the delocalization of quantum information over a many-body system and underlies all quantum-chaotic dynamics. We employ discrete quantum cellular automata as classically simulable toy models of scrambling. We observe that these automata break ergodicity, i.e., they exhibit quantum scarring. We also find that the time scale of scrambling rises with the local Hilbert-space dimension and obeys a specific combinatorial pattern. We then show that scarring is mostly suppressed in a semiclassical limit, demonstrating that semiclassical-chaotic systems are more ergodic.
We introduce the concept of ergodicity and explore its deviation caused by quantum scars in an isolated quantum system, employing a pedagogical approach based on a toy model. Quantum scars, originally identified as traces of classically unstable orbits in certain wavefunctions of chaotic systems, have recently regained interest for their role in non-ergodic dynamics, as they retain memory of their initial states. We elucidate these features of quantum scars within the same framework of this toy model. The integrable part of the model consists of two large spins, with a classical counterpart, which we combine with a random matrix to induce ergodic behavior. Scarred states can be selectively generated from the integrable spin Hamiltonian by protecting them from the ergodic states using a projector method. Deformed projectors mimic the ‘quantum leakage’ of scarred states, enabling tunable mixing with ergodic states and thereby controlling the degree of scarring. In this simple model, we investigate various properties of quantum scarring and shed light on different aspects of many-body quantum scars observed in more complex quantum systems. Notably, the underlying classicality can be revealed through the entanglement spectrum and the dynamics of ‘out-of-time-ordered correlators’.
Ergodicity breaking is observed in the blockade regime of Rydberg atom arrays, in the form of low entanglement eigenstates known as scars, which fail to thermalize. The signature of these states persists in periodically driven systems, where they coexist with an extensive number of chaotic states. Here we investigate a quantum cellular automaton based on the classical rule that updates a site if its two neighbors are in the lower state. We show that the breaking of ergodicity extends to chaotic states. The dynamical breaking of ergodicity is controlled by chiral quasiparticle excitations which propagate entanglement. Evidence of nonlocal entanglement is found, showing that these nonthermal chaotic states may be useful to quantum computation.
Quantum chaotic interacting N-particle systems are assumed to show fast and irreversible spreading of quantum information on short (Ehrenfest) time scales ∼logN. Here, we show that, near criticality, certain many-body systems exhibit fast initial scrambling, followed subsequently by oscillatory behavior between reentrant localization and delocalization of information in Hilbert space. We consider both integrable and nonintegrable quantum critical bosonic systems with attractive contact interaction that exhibit locally unstable dynamics in the corresponding many-body phase space of the large-N limit. Semiclassical quantization of the latter accounts for many-body correlations in excellent agreement with simulations. Most notably, it predicts an asymptotically constant local level spacing ℏ/τ, again given by τ∼logN. This unique timescale governs the long-time behavior of out-of-time-order correlators that feature quasiperiodic recurrences indicating reversibility.
In quantum systems with a classical limit, advanced semiclassical methods provide the crucial link between phase-space structures, reflecting the distinction between chaotic, mixed or integrable classical dynamics, and the corresponding quantum properties. Well established techniques dealing with ergodic wave interference in the usual semiclassical limit ℏ→0\hbar \to 0ℏ→0, where the classical limit is given by Hamiltonian mechanics of particles, constitute a now standard part of the toolkit of theoretical physics. During the last years, these ideas have been extended into the field theoretical domain of systems composed of NNN indistinguishable particles, aka quantum fields, displaying a different type of semiclassical limit ℏeff=1/N→0\hbar_{\rm eff}=1/N \to 0ℏeff​=1/N→0 and accounting for genuine many-body quantum interference. The foundational concept behind this idea of many-body interference, the many-body version of the van Vleck-Gutzwillers semiclassical propagator, is explained in detail. Based on this the corresponding semiclassical many-body theory is reviewed. It provides a unified framework for understanding a variety of quantum chaotic phenomena addressed, including random-matrix spectral correlations in many-body systems, the universal morphology of many-body eigenstates, interference effects kin to mesoscopic weak localization, and the key to the scrambling of many-body correlations characterized by out-of-time-order correlators.
Quantum many-body scars are atypical, non-thermal eigenstates that constitute a mechanism of weak ergodicity breaking, yet their stability in realistic, open quantum systems remains an outstanding question. We investigate the interplay between stability and quantum information dynamics within the PXP model subjected to local dissipation. Employing a non-Hermitian formalism, we analytically demonstrate and numerically corroborate that scar eigenstates possess intrinsic decay rates parametrically smaller than those of the surrounding chaotic bulk, revealing a pronounced stability hierarchy. This spectral robustness is a direct manifestation of the approximate algebraic structure underpinning the scars. This resilience has direct dynamical consequences. We find a suppression of information scrambling, quantified by the out-of-time-order correlator (OTOC), for dynamics initiated from scar-like states. Analysis of entanglement dynamics reveals that quantum correlations in scar states persist for longer timescales under weak dissipation. These results demonstrate that algebraic confinement reduces the susceptibility of scar states to both dissipation and chaotic scrambling; the signature of this resilience is a persistent suppression of the scrambling rate and the presence of coherent oscillations, not a uniformly longer signal lifetime.
We systematically investigate scrambling (or delocalizing) processes of quantum information encoded in quantum many-body systems by using numerical exact diagonalization. As a measure of scrambling, we adopt the tripartite mutual information (TMI) that becomes negative when quantum information is delocalized. We clarify that scrambling is an independent property of the integrability of Hamiltonians; TMI can be negative or positive for both integrable and nonintegrable systems. This implies that scrambling is a separate concept from conventional quantum chaos characterized by nonintegrability. Specifically, we argue that there are a few exceptional initial states that do not exhibit scrambling, and show that such exceptional initial states have small effective dimensions. Furthermore, we calculate TMI in the Sachdev-Ye-Kitaev (SYK) model, a fermionic toy model of quantum gravity. We find that disorder does not make scrambling slower but makes it smoother in the SYK model, in contrast to many-body localization in spin chains.