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Vikram A, Galitski V. Exact Universal Bounds on Quantum Dynamics and Fast Scrambling. PHYSICAL REVIEW LETTERS 2024; 132:040402. [PMID: 38335350 DOI: 10.1103/physrevlett.132.040402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/15/2023] [Revised: 07/30/2023] [Accepted: 12/21/2023] [Indexed: 02/12/2024]
Abstract
Quantum speed limits such as the Mandelstam-Tamm or Margolus-Levitin bounds offer a quantitative formulation of the energy-time uncertainty principle that constrains dynamics over short times. We show that the spectral form factor, a central quantity in quantum chaos, sets a universal state-independent bound on the quantum dynamics of a complete set of initial states over arbitrarily long times, which is tighter than the corresponding state-independent bounds set by known speed limits. This bound further generalizes naturally to the real-time dynamics of time-dependent or dissipative systems where no energy spectrum exists. We use this result to constrain the scrambling of information in interacting many-body systems. For Hamiltonian systems, we show that the fundamental question of the fastest possible scrambling time-without any restrictions on the structure of interactions-maps to a purely mathematical property of the density of states involving the non-negativity of Fourier transforms. We illustrate these bounds in the Sachdev-Ye-Kitaev model, where we show that despite its "maximally chaotic" nature, the sustained scrambling of sufficiently large fermion subsystems via entanglement generation requires an exponentially long time in the subsystem size.
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Affiliation(s)
- Amit Vikram
- Joint Quantum Institute and Department of Physics, University of Maryland, College Park, Maryland 20742, USA
| | - Victor Galitski
- Joint Quantum Institute and Department of Physics, University of Maryland, College Park, Maryland 20742, USA
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2
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Sadhasivam VG, Meuser L, Reichman DR, Althorpe SC. Instantons and the quantum bound to chaos. Proc Natl Acad Sci U S A 2023; 120:e2312378120. [PMID: 38032936 PMCID: PMC10710067 DOI: 10.1073/pnas.2312378120] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/21/2023] [Accepted: 10/26/2023] [Indexed: 12/02/2023] Open
Abstract
The rate at which information scrambles in a quantum system can be quantified using out-of-time-ordered correlators. A remarkable prediction is that the associated Lyapunov exponent [Formula: see text] that quantifies the scrambling rate in chaotic systems obeys a universal bound [Formula: see text]. Previous numerical and analytical studies have indicated that this bound has a quantum-statistical origin. Here, we use path-integral techniques to show that a minimal theory to reproduce this bound involves adding contributions from quantum thermal fluctuations (describing quantum tunneling and zero-point energy) to classical dynamics. By propagating a model quantum-Boltzmann-conserving classical dynamics for a system with a barrier, we show that the bound is controlled by the stability of thermal fluctuations around the barrier instanton (a delocalized structure which dominates the tunneling statistics). This stability requirement appears to be general, implying that there is a close relation between the formation of instantons, or related delocalized structures, and the imposition of the quantum-chaos bound.
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Affiliation(s)
- Vijay Ganesh Sadhasivam
- Yusuf Hamied Department of Chemistry, University of Cambridge, CambridgeCB2 1EW, United Kingdom
| | - Lars Meuser
- Yusuf Hamied Department of Chemistry, University of Cambridge, CambridgeCB2 1EW, United Kingdom
- Department of Chemistry and Applied Biosciences, ETH Zurich (Swiss Federal Institute of Technology), Zürich8093, Switzerland
| | | | - Stuart C. Althorpe
- Yusuf Hamied Department of Chemistry, University of Cambridge, CambridgeCB2 1EW, United Kingdom
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3
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Schuster T, Yao NY. Operator Growth in Open Quantum Systems. PHYSICAL REVIEW LETTERS 2023; 131:160402. [PMID: 37925733 DOI: 10.1103/physrevlett.131.160402] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/28/2022] [Accepted: 09/06/2023] [Indexed: 11/07/2023]
Abstract
The spreading of quantum information in closed systems, often termed scrambling, is a hallmark of many-body quantum dynamics. In open systems, scrambling competes with noise, errors, and decoherence. Here, we provide a universal framework that describes the scrambling of quantum information in open systems: we predict that the effect of open-system dynamics is fundamentally controlled by operator size distributions and independent of the microscopic error mechanism. This framework allows us to demonstrate that open quantum systems exhibit universal classes of information dynamics that fundamentally differ from their unitary counterparts. Implications for the Loschmidt echo, nuclear magnetic resonance experiments, and the classical simulability of open quantum dynamics will be discussed.
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Affiliation(s)
- Thomas Schuster
- Department of Physics, University of California, Berkeley, California 94720, USA
- Walter Burke Institute for Theoretical Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
| | - Norman Y Yao
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
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4
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Tunable Geometries in Sparse Clifford Circuits. Symmetry (Basel) 2022. [DOI: 10.3390/sym14040666] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/04/2023] Open
Abstract
We investigate the emergence of different effective geometries in stochastic Clifford circuits with sparse coupling. By changing the probability distribution for choosing two-site gates as a function of distance, we generate sparse interactions that either decay or grow with distance as a function of a single tunable parameter. Tuning this parameter reveals three distinct regimes of geometry for the spreading of correlations and growth of entanglement in the system. We observe linear geometry for short-range interactions, treelike geometry on a sparse coupling graph for long-range interactions, and an intermediate fast scrambling regime at the crossover point between the linear and treelike geometries. This transition in geometry is revealed in calculations of the subsystem entanglement entropy and tripartite mutual information. We also study emergent lightcones that govern these effective geometries by teleporting a single qubit of information from an input qubit to an output qubit. These tools help to analyze distinct geometries arising in dynamics and correlation spreading in quantum many-body systems.
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Hashizume T, Bentsen GS, Weber S, Daley AJ. Deterministic Fast Scrambling with Neutral Atom Arrays. PHYSICAL REVIEW LETTERS 2021; 126:200603. [PMID: 34110181 DOI: 10.1103/physrevlett.126.200603] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/26/2021] [Accepted: 04/15/2021] [Indexed: 06/12/2023]
Abstract
Fast scramblers are dynamical quantum systems that produce many-body entanglement on a timescale that grows logarithmically with the system size N. We propose and investigate a family of deterministic, fast scrambling quantum circuits realizable in near-term experiments with arrays of neutral atoms. We show that three experimental tools-nearest-neighbor Rydberg interactions, global single-qubit rotations, and shuffling operations facilitated by an auxiliary tweezer array-are sufficient to generate nonlocal interaction graphs capable of scrambling quantum information using only O(logN) parallel applications of nearest-neighbor gates. These tools enable direct experimental access to fast scrambling dynamics in a highly controlled and programmable way and can be harnessed to produce highly entangled states with varied applications.
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Affiliation(s)
- Tomohiro Hashizume
- Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, United Kingdom
| | - Gregory S Bentsen
- Martin A. Fisher School of Physics, Brandeis University, Waltham, Massachusetts 02465, USA
| | - Sebastian Weber
- Institute for Theoretical Physics III and Center for Integrated Quantum Science and Technology, University of Stuttgart, 70550 Stuttgart, Germany
| | - Andrew J Daley
- Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, United Kingdom
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Kumar S, Tripathi V. Signature of universal fast scrambling in the transient response of a driven mott insulator system. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2021; 33:244003. [PMID: 33827053 DOI: 10.1088/1361-648x/abf592] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/29/2020] [Accepted: 04/07/2021] [Indexed: 06/12/2023]
Abstract
The scrambling rateλs, a measure of the early growth of decoherence in an interacting quantum system, has been conjectured to have a universal saturation bound,λs⩽ 2πkBT/ℏ, whereTis the temperature. This decoherence arises from the spread of quantum information over a large number of untracked degrees of freedom. The commonly studied indicator of scrambling is the out of time-ordered correlator (OTOC) of noncommuting quantum operators, in-turn related to generalized uncertainty relations, and reminiscent of the Lyapunov exponent of classically chaotic systems. From a practical measurement point of view, other quantities besides OTOCs, that are also sensitive to these generalized uncertainty relations, may capture the scrambling behavior. Here, using a large-NKeldysh field theory approach, we show that the nonequilibrium current response of a Mott insulator system consisting of a mesoscopic quantum dot array, when subjected to an electric field quench, reveals this phenomenon on account of number-phase uncertainty. Both ac and dc field quenches are considered. The passage from the initial Mott insulator phase with well-defined charge excitations, to the final nonequilibrium steady current state, is revealed in the transient current response that has Bloch-like oscillations. We find that the amplitude of these oscillations decreases at the universal rate, 2πkBT/ℏ, associated with fast scramblers. Our Mott insulator model provides a new example of a fast scrambler in addition to the known ones such as extremal black holes and the Sachdev-Ye-Kitaev (SYK) model.
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Affiliation(s)
- Sanjeev Kumar
- Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Navy Nagar, Mumbai 400005, India
| | - Vikram Tripathi
- Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Navy Nagar, Mumbai 400005, India
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Carrega M, Kim J, Rosa D. Unveiling Operator Growth Using Spin Correlation Functions. ENTROPY (BASEL, SWITZERLAND) 2021; 23:587. [PMID: 34068630 PMCID: PMC8151211 DOI: 10.3390/e23050587] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 04/09/2021] [Revised: 05/01/2021] [Accepted: 05/06/2021] [Indexed: 11/16/2022]
Abstract
In this paper, we study non-equilibrium dynamics induced by a sudden quench of strongly correlated Hamiltonians with all-to-all interactions. By relying on a Sachdev-Ye-Kitaev (SYK)-based quench protocol, we show that the time evolution of simple spin-spin correlation functions is highly sensitive to the degree of k-locality of the corresponding operators, once an appropriate set of fundamental fields is identified. By tracking the time-evolution of specific spin-spin correlation functions and their decay, we argue that it is possible to distinguish between operator-hopping and operator growth dynamics; the latter being a hallmark of quantum chaos in many-body quantum systems. Such an observation, in turn, could constitute a promising tool to probe the emergence of chaotic behavior, rather accessible in state-of-the-art quench setups.
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Affiliation(s)
- Matteo Carrega
- NEST, Istituto Nanoscienze—CNR and Scuola Normale Superiore, I-56127 Pisa, Italy;
| | - Joonho Kim
- Institute for Advanced Study, Princeton, NJ 08540, USA;
| | - Dario Rosa
- School of Physics, Korea Institute for Advanced Study, 85 Hoegiro Dongdaemun-gu, Seoul 02455, Korea
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Expo-ro 55, Yuseong-gu, Daejeon 34126, Korea
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Kuwahara T, Saito K. Absence of Fast Scrambling in Thermodynamically Stable Long-Range Interacting Systems. PHYSICAL REVIEW LETTERS 2021; 126:030604. [PMID: 33543944 DOI: 10.1103/physrevlett.126.030604] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/09/2020] [Accepted: 12/22/2020] [Indexed: 06/12/2023]
Abstract
In this study, we investigate out-of-time-order correlators (OTOCs) in systems with power-law decaying interactions such as R^{-α}, where R is the distance. In such systems, the fast scrambling of quantum information or the exponential growth of information propagation can potentially occur according to the decay rate α. In this regard, a crucial open challenge is to identify the optimal condition for α such that fast scrambling cannot occur. In this study, we disprove fast scrambling in generic long-range interacting systems with α>D (D: spatial dimension), where the total energy is extensive in terms of system size and the thermodynamic limit is well defined. We rigorously demonstrate that the OTOC shows a polynomial growth over time as long as α>D and the necessary scrambling time over a distance R is larger than t≳R^{[(2α-2D)/(2α-D+1)]}.
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Affiliation(s)
- Tomotaka Kuwahara
- Mathematical Science Team, RIKEN Center for Advanced Intelligence Project (AIP), 1-4-1 Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan
- Interdisciplinary Theoretical & Mathematical Sciences Program (iTHEMS) RIKEN 2-1, Hirosawa, Wako, Saitama 351-0198, Japan
| | - Keiji Saito
- Department of Physics, Keio University, Yokohama 223-8522, Japan
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Guo AY, Tran MC, Childs AM, Gorshkov AV, Gong ZX. Signaling and scrambling with strongly long-range interactions. PHYSICAL REVIEW. A 2020; 102:10.1103/PhysRevA.102.010401. [PMID: 33367192 PMCID: PMC7754795 DOI: 10.1103/physreva.102.010401] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
Strongly long-range interacting quantum systems-those with interactions decaying as a power law 1/r α in the distance r on a D-dimensional lattice for α ⩽ D-have received significant interest in recent years. They are present in leading experimental platforms for quantum computation and simulation, as well as in theoretical models of quantum-information scrambling and fast entanglement creation. Since no notion of locality is expected in such systems, a general understanding of their dynamics is lacking. In a step towards rectifying this problem, we prove two Lieb-Robinson-type bounds that constrain the time for signaling and scrambling in strongly long-range interacting systems, for which no tight bounds were previously known. Our first bound applies to systems mappable to free-particle Hamiltonians with long-range hopping, and is saturable for α ⩽ D/2. Our second bound pertains to generic long-range interacting spin Hamiltonians and gives a tight lower bound for the signaling time to extensive subsets of the system for all α< D. This many-site signaling time lower bounds the scrambling time in strongly long-range interacting systems.
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Affiliation(s)
- Andrew Y. Guo
- Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA
- Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
| | - Minh C. Tran
- Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA
- Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
- Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA
| | - Andrew M. Childs
- Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA
- Department of Computer Science, University of Maryland, College Park, Maryland 20742, USA
- Institute for Advanced Computer Studies, University of Maryland, College Park, Maryland 20742, USA
| | - Alexey V. Gorshkov
- Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA
- Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
| | - Zhe-Xuan Gong
- Department of Physics, Colorado School of Mines, Golden, Colorado 80401, USA
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Bentsen G, Hashizume T, Buyskikh AS, Davis EJ, Daley AJ, Gubser SS, Schleier-Smith M. Treelike Interactions and Fast Scrambling with Cold Atoms. PHYSICAL REVIEW LETTERS 2019; 123:130601. [PMID: 31697527 DOI: 10.1103/physrevlett.123.130601] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/28/2019] [Indexed: 06/10/2023]
Abstract
We propose an experimentally realizable quantum spin model that exhibits fast scrambling, based on nonlocal interactions that couple sites whose separation is a power of 2. By controlling the relative strengths of deterministic, nonrandom couplings, we can continuously tune from the linear geometry of a nearest-neighbor spin chain to an ultrametric geometry in which the effective distance between spins is governed by their positions on a tree graph. The transition in geometry can be observed in quench dynamics, and is furthermore manifest in calculations of the entanglement entropy. Between the linear and treelike regimes, we find a peak in entanglement and exponentially fast spreading of quantum information across the system. Our proposed implementation, harnessing photon-mediated interactions among cold atoms in an optical cavity, offers a test case for experimentally observing the emergent geometry of a quantum many-body system.
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Affiliation(s)
- Gregory Bentsen
- Department of Physics, Stanford University, Stanford, California 94305, USA
- SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
| | - Tomohiro Hashizume
- Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, United Kingdom
| | - Anton S Buyskikh
- Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, United Kingdom
| | - Emily J Davis
- Department of Physics, Stanford University, Stanford, California 94305, USA
| | - Andrew J Daley
- Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, United Kingdom
| | - Steven S Gubser
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - Monika Schleier-Smith
- Department of Physics, Stanford University, Stanford, California 94305, USA
- SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
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Lucas A. Operator Size at Finite Temperature and Planckian Bounds on Quantum Dynamics. PHYSICAL REVIEW LETTERS 2019; 122:216601. [PMID: 31283307 DOI: 10.1103/physrevlett.122.216601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/29/2018] [Indexed: 06/09/2023]
Abstract
It has long been believed that dissipative timescales τ obey a "Planckian" bound τ≳(ℏ/k_{B}T) in strongly coupled quantum systems. Despite much circumstantial evidence, however, there is no known τ for which this bound is universal. Here we define operator size at a finite temperature, and conjecture such a τ: the timescale over which small operators become large. All known many-body theories are consistent with this conjecture. This proposed bound explains why previously conjectured Planckian bounds do not always apply to weakly coupled theories, and how Planckian timescales can be relevant to both transport and chaos.
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Affiliation(s)
- Andrew Lucas
- Department of Physics, Stanford University, Stanford California 94305, USA
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