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arXiv:2210.14242 · Physical Review Letters

Scrambling Transition in a Radiative Random Unitary Circuit

Zack Weinstein, Shane P. Kelly, Jamir Marino, Ehud Altman

Published Nov 29, 2023• 48 citations
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Abstract

We study quantum information scrambling in a random unitary circuit that exchanges qubits with an environment at a rate p. As a result, initially localized quantum information not only spreads within the system, but also spills into the environment. Using the out-of-time-order correlator (OTOC) to characterize scrambling, we find a nonequilibrium phase transition in the directed percolation universality class at a critical swap rate p_{c}: for p<p_{c} the ensemble-averaged OTOC exhibits ballistic growth with a tunable light cone velocity, while for p>p_{c} the OTOC fails to percolate within the system and vanishes uniformly within a finite timescale, indicating that all local operators are rapidly swapped into the environment. To elucidate its information-theoretic consequences, we demonstrate that the transition in operator spreading coincides with a transition in an observer's ability to decode the system's initial quantum information from the swapped-out, or "radiated," qubits. We present a simple decoding scheme which recovers the system's initial information with perfect fidelity in the nonpercolating phase, and with continuously decreasing fidelity with decreasing swap rate in the percolating phase. Depending on the initial state of the swapped-in qubits, we further observe a corresponding entanglement transition in the coherent information from the system into the radiated qubits.

DOI
10.1103/physrevlett.131.220404

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We study quantum information scrambling in a random unitary circuit that exchanges qubits with an environment at a rate p. As a result, initially localized quantum information not only spreads within the system, but also spills into the environment. Using the out-of-time-order correlator (OTOC) to characterize scrambling, we find a nonequilibrium phase transition in the directed percolation universality class at a critical swap rate p_{c}: for p<p_{c} the ensemble-averaged OTOC exhibits ballistic growth with a tunable light cone velocity, while for p>p_{c} the OTOC fails to percolate within the system and vanishes uniformly within a finite timescale, indicating that all local operators are rapidly swapped into the environment. To elucidate its information-theoretic consequences, we demonstrate that the transition in operator spreading coincides with a transition in an observer's ability to decode the system's initial quantum information from the swapped-out, or "radiated," qubits. We present a simple decoding scheme which recovers the system's initial information with perfect fidelity in the nonpercolating phase, and with continuously decreasing fidelity with decreasing swap rate in the percolating phase. Depending on the initial state of the swapped-in qubits, we further observe a corresponding entanglement transition in the coherent information from the system into the radiated qubits.
A closed quantum system never forgets its initial state, but the encoded information can get scrambled and become inaccessible without measuring a large fraction of all the system degrees of freedom. This scrambling can be diagnosed by studying the spatial spreading of initially local operators under the Heisenberg time evolution, and the decay of the out-of-time-ordered correlators (OTOC). What insights can OTOCs provide to understand the dynamics of quantum many-body systems? What are the characteristic behaviors of OTOCs during the time evolution? How is information scrambling affected by the dissipation in open quantum many-body systems? We first study slow scrambling in many-body localized systems via calculating various correlators, two-point retarded correlators and OTOCs. Comparing with retarded correlators, OTOCs provide more information about the dynamics. We find that disorder slows and partially halts the onset of information scrambling. Instead of ballistic spreading, propagation of information forms a logarithmic light cone. Next, we study the finite-size scaling of OTOCs at late times in generic thermalizing quantum many-body systems. When energy is conserved, the late-time saturation value of the OTOC of generic traceless local operators scales as an inverse polynomial in the system size. This is in contrast to the inverse exponential scaling expected for chaotic dynamics without energy conservation. We also study information scrambling in open quantum many-body systems. We define a dissipative version of OTOC and study its behaviors in a prototypical chaotic quantum chain with dissipation. We find that dissipation leads to not only the overall decay of the scrambled information due to leaking, but also structural changes so that the information light cone can only reach a finite distance even when the effect of overall decay is removed. Finally, we construct a family of local Hamiltonians for understanding the asymmetric information scrambling. Our models live on a one-dimensional lattice and exhibit asymmetric butterfly light cone between the left and right spatial directions.
Abstract The study of information scrambling has profoundly deepened our understanding of many-body quantum systems. Much recent research has been devote to understanding the interplay between scrambling and decoherence in open systems. Continuing in this vein, we investigate scrambling in a noisy encoding-decoding circuit model. Specifically, we consider an LLL-qubit circuit consisting of a Haar-random unitary, followed by noise acting on a subset of qubits, and then by the inverse unitary. Scrambling is measured using the bipartite algebraic out-of-time-order correlator (A\mathcal{A}A-OTOC), which allows us to track information spread between extensively sized subsystems. We derive an analytic expression for the A\mathcal{A}A-OTOC that depends on system size and noise strength. In the thermodynamic limit, this system displays a \textit{butterfly effect} in which infinitesimal noise induces macroscopic information scrambling. We also perform numerical simulations while relaxing the condition of Haar-randomness, which preliminarily suggest that this effect may manifest in a larger set of circuits.
Thermalization of chaotic quantum many-body systems under unitary time evolution is related to the growth in complexity of initially simple Heisenberg operators.Operator growth is a manifestation of information scrambling and can be diagnosed by out-of-time-order correlators (OTOCs).However, the behavior of OTOCs of local operators in generic chaotic local Hamiltonians remains poorly understood, with some semiclassical and large-N models exhibiting exponential growth of OTOCs and a sharp chaos wave front and other random circuit models showing a diffusively broadened wave front.In this paper, we propose a unified physical picture for scrambling in chaotic local Hamiltonians.We construct a random time-dependent Hamiltonian model featuring a large-N limit where the OTOC obeys a Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) type equation and exhibits exponential growth and a sharp wave front.We show that quantum fluctuations manifest as noise (distinct from the randomness of the couplings in the underlying Hamiltonian) in the FKPP equation and that the noise-averaged OTOC exhibits a crossover to a diffusively broadened wave front.At small N, we demonstrate that operator growth dynamics, averaged over the random couplings, can be efficiently simulated for all time using matrix product state techniques.To show that time-dependent randomness is not essential to our conclusions, we push our previous matrix product operator methods to very large size and show that data for a timeindependent Hamiltonian model are also consistent with a diffusively broadened wave front.
Scrambling, the delocalization of initially localized quantum information, is commonly characterized by the out-of-time ordered correlator (OTOC). Employing the OTOC-Renyi-2 entropy theorem we derive a quantum speed limit for the OTOC, which sets an lower bound for the rate with which information can be scrambled. This bound becomes particularly tractable by describing the scrambling of information in a closed quantum system as an effective decoherence process of an open system interacting with an environment. We prove that decay of the OTOC can be bounded by the strength of the system-environment coupling and two-point environmental correlation functions. We validate our analytic bound numerically using the non-integrable transverse field Ising model. Our results provide a universal and model-agnostic quantitative framework for understanding the dynamical limits of information spreading across quantum many-body physics, condensed matter, and engineered quantum platforms.
We study information scrambling (a spread of initially localized quantum information into the system's many degree of freedom) in discrete-time quantum walks. We consider out-of-time-ordered correlators (OTOC) and K complexity as a probe of information scrambling. The OTOC for local spin operators in all directions has a light-cone structure which is ``shell-like.'' As the wavefront passes, the OTOC approaches to zero in the long-time limit, showing no signature of scrambling. The introduction of spatial or temporal disorder changes the shape of the light-cone akin to localization of wavefuction. We formulate the K complexity in system with discrete-time evolution, and show that it grows linearly in discrete-time quantum walk. The presence of disorder modifies this growth to sublinear. Our study present interesting case to explore many-body phenomenon in a discrete-time quantum walk using scrambling.
Current quantum hardware is limited by noise and decoherence, which restrict the depth of unitary circuits that can be implemented with high fidelity. We investigate how the compressibility of time-evolution operators depends on the dynamical regime of the underlying many-body system. As a testbed, we study a one-dimensional Floquet random circuit with a tunable competition between interactions and on-site disorder. Using tensor-network simulations, we characterize operator growth through the operator-entanglement entropy of the Floquet unitary as well as of out-of-time-ordered correlators (OTOCs). We find rapid operator scrambling at weak disorder, while strong disorder leads to slow OTOC-front propagation and logarithmic or near-logarithmic operator-entanglement growth over the accessible time window. We then optimize shallow brickwall circuits to approximate the Floquet evolution and show that strong-disorder circuits can be compressed to substantially smaller depths than weak-disorder circuits at fixed logarithmic fidelity density. These results suggest that localized or slowly scrambling dynamics provide a favorable regime for compressed quantum simulation on noisy devices.