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Автор: Grafiati

Опубліковано: 28 вересня 2024

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1

Matsuta, Takuro, Tohru Koma, and Shu Nakamura. "Improving the Lieb–Robinson Bound for Long-Range Interactions." Annales Henri Poincaré 18, no. 2 (October 20, 2016): 519–28. http://dx.doi.org/10.1007/s00023-016-0526-1.

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2

Woods, M. P., and M. B. Plenio. "Dynamical error bounds for continuum discretisation via Gauss quadrature rules—A Lieb-Robinson bound approach." Journal of Mathematical Physics 57, no. 2 (February 2016): 022105. http://dx.doi.org/10.1063/1.4940436.

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3

Mahoney, Brendan J., and Craig S. Lent. "The Value of the Early-Time Lieb-Robinson Correlation Function for Qubit Arrays." Symmetry 14, no. 11 (October 26, 2022): 2253. http://dx.doi.org/10.3390/sym14112253.

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Анотація:

The Lieb-Robinson correlation function is one way to capture the propagation of quantum entanglement and correlations in many-body systems. We consider arrays of qubits described by the tranverse-field Ising model and examine correlations as the expanding front of entanglement first reaches a particular qubit. Rather than a new bound for the correlation function, we calculate its value, both numerically and analytically. A general analytical result is obtained that enables us to analyze very large arrays of qubits. The velocity of the entanglement front saturates to a constant value, for which an analytic expression is derived. At the leading edge of entanglement, the correlation function is well-described by an exponential reduced by the square root of the distance. This analysis is extended to arbitrary arrays with general coupling and topologies. For regular two and three dimensional qubit arrays with near-neighbor coupling we find the saturation values for the direction-dependent Lieb-Robinson velocity. The symmetry of the underlying 2D or 3D lattice is evident in the shape of surfaces of constant entanglement, even as the correlations front expands over hundreds of qubits.

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4

Strasberg, Philipp, Kavan Modi, and Michalis Skotiniotis. "How long does it take to implement a projective measurement?" European Journal of Physics 43, no. 3 (March 28, 2022): 035404. http://dx.doi.org/10.1088/1361-6404/ac5a7a.

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Abstract According to the Schrödinger equation, a closed quantum system evolves continuously in time. If it is subject to a measurement however, its state changes randomly and discontinuously, which is mathematically described by the projection postulate. But how long does it take for this discontinuous change to occur? Based on simple estimates, whose validity rests solely on the fact that all fundamental forces in nature are finite-ranged, we show that the implementation of a quantum measurement requires a minimum time. This time scales proportionally with the diameter of the quantum mechanical object, on which the measured observable acts non-trivially, with the proportionality constant being around 10−5 s m−1. We confirm our bound by comparison with experimentally reported measurement times for different platforms. We give a pedagogical exposition of our argumentation introducing along the way modern concepts such as ancilla-based measurements, the quantum speed limit, and Lieb–Robinson velocity bounds.

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5

Moosavian, Ali Hamed, Seyed Sajad Kahani, and Salman Beigi. "Limits of Short-Time Evolution of Local Hamiltonians." Quantum 6 (June 27, 2022): 744. http://dx.doi.org/10.22331/q-2022-06-27-744.

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Анотація:

Evolutions of local Hamiltonians in short times are expected to remain local and thus limited. In this paper, we validate this intuition by proving some limitations on short-time evolutions of local time-dependent Hamiltonians. We show that the distribution of the measurement output of short-time (at most logarithmic) evolutions of local Hamiltonians are concentrated and satisfy an isoperimetric inequality. To showcase explicit applications of our results, we study the MAXCUT problem and conclude that quantum annealing needs at least a run-time that scales logarithmically in the problem size to beat classical algorithms on MAXCUT. To establish our results, we also prove a Lieb-Robinson bound that works for time-dependent Hamiltonians which might be of independent interest.

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6

Vershynina, Anna, and Elliott Lieb. "Lieb-Robinson bounds." Scholarpedia 8, no. 9 (2013): 31267. http://dx.doi.org/10.4249/scholarpedia.31267.

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7

Doyon, Benjamin. "Hydrodynamic Projections and the Emergence of Linearised Euler Equations in One-Dimensional Isolated Systems." Communications in Mathematical Physics 391, no. 1 (January 27, 2022): 293–356. http://dx.doi.org/10.1007/s00220-022-04310-3.

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AbstractOne of the most profound questions of mathematical physics is that of establishing from first principles the hydrodynamic equations in large, isolated, strongly interacting many-body systems. This involves understanding relaxation at long times under reversible dynamics, determining the space of emergent collective degrees of freedom (the ballistic waves), showing that projection occurs onto them, and establishing their dynamics (the hydrodynamic equations). We make progress in these directions, focussing for simplicity on one-dimensional systems. Under a model-independent definition of the complete space of extensive conserved charges, we show that hydrodynamic projection occurs in Euler-scale two-point correlation functions. A fundamental ingredient is a property of relaxation: we establish ergodicity of correlation functions along almost every direction in space and time. We further show that to every extensive conserved charge with a local density is associated a local current and a continuity equation; and that Euler-scale two-point correlation functions of local conserved densities satisfy a hydrodynamic equation. The results are established rigorously within a general framework based on Hilbert spaces of observables. These spaces occur naturally in the $$C^*$$ C ∗ algebra description of statistical mechanics by the Gelfand–Naimark–Segal construction. Using Araki’s exponential clustering and the Lieb–Robinson bound, we show that the results hold, for instance, in every nonzero-temperature Gibbs state of short-range quantum spin chains. Many techniques we introduce are generalisable to higher dimensions. This provides a precise and universal theory for the emergence of ballistic waves at the Euler scale and how they propagate within homogeneous, stationary states.

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8

Islambekov, Umar, Robert Sims, and Gerald Teschl. "Lieb–Robinson Bounds for the Toda Lattice." Journal of Statistical Physics 148, no. 3 (August 2012): 440–79. http://dx.doi.org/10.1007/s10955-012-0554-2.

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9

NACHTERGAELE, BRUNO, BENJAMIN SCHLEIN, ROBERT SIMS, SHANNON STARR, and VALENTIN ZAGREBNOV. "ON THE EXISTENCE OF THE DYNAMICS FOR ANHARMONIC QUANTUM OSCILLATOR SYSTEMS." Reviews in Mathematical Physics 22, no. 02 (March 2010): 207–31. http://dx.doi.org/10.1142/s0129055x1000393x.

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Анотація:

We construct a W*-dynamical system describing the dynamics of a class of anharmonic quantum oscillator lattice systems in the thermodynamic limit. Our approach is based on recently proved Lieb–Robinson bounds for such systems on finite lattices [19].

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10

Nachtergaele, Bruno, and Robert Sims. "Lieb-Robinson Bounds and the Exponential Clustering Theorem." Communications in Mathematical Physics 265, no. 1 (March 22, 2006): 119–30. http://dx.doi.org/10.1007/s00220-006-1556-1.

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11

Nachtergaele, Bruno, Hillel Raz, Benjamin Schlein, and Robert Sims. "Lieb-Robinson Bounds for Harmonic and Anharmonic Lattice Systems." Communications in Mathematical Physics 286, no. 3 (September 23, 2008): 1073–98. http://dx.doi.org/10.1007/s00220-008-0630-2.

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12

Damanik, David, Marius Lemm, Milivoje Lukic, and William Yessen. "On anomalous Lieb–Robinson bounds for the Fibonacci XY chain." Journal of Spectral Theory 6, no. 3 (2016): 601–28. http://dx.doi.org/10.4171/jst/133.

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13

Sweke, Ryan, Jens Eisert, and Michael Kastner. "Lieb–Robinson bounds for open quantum systems with long-ranged interactions." Journal of Physics A: Mathematical and Theoretical 52, no. 42 (September 24, 2019): 424003. http://dx.doi.org/10.1088/1751-8121/ab3f4a.

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14

Gebert, Martin, and Marius Lemm. "On Polynomial Lieb–Robinson Bounds for the XY Chain in a Decaying Random Field." Journal of Statistical Physics 164, no. 3 (June 10, 2016): 667–79. http://dx.doi.org/10.1007/s10955-016-1558-0.

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15

Nachtergaele, Bruno, Robert Sims, and Amanda Young. "Quasi-locality bounds for quantum lattice systems. I. Lieb-Robinson bounds, quasi-local maps, and spectral flow automorphisms." Journal of Mathematical Physics 60, no. 6 (June 2019): 061101. http://dx.doi.org/10.1063/1.5095769.

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16

Bachmann, Sven, Wojciech Dybalski, and Pieter Naaijkens. "Lieb–Robinson Bounds, Arveson Spectrum and Haag–Ruelle Scattering Theory for Gapped Quantum Spin Systems." Annales Henri Poincaré 17, no. 7 (October 23, 2015): 1737–91. http://dx.doi.org/10.1007/s00023-015-0440-y.

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17

Bentsen, Gregory, Yingfei Gu, and Andrew Lucas. "Fast scrambling on sparse graphs." Proceedings of the National Academy of Sciences 116, no. 14 (March 21, 2019): 6689–94. http://dx.doi.org/10.1073/pnas.1811033116.

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Анотація:

Given a quantum many-body system with few-body interactions, how rapidly can quantum information be hidden during time evolution? The fast-scrambling conjecture is that the time to thoroughly mix information among N degrees of freedom grows at least logarithmically in N. We derive this inequality for generic quantum systems at infinite temperature, bounding the scrambling time by a finite decay time of local quantum correlations at late times. Using Lieb–Robinson bounds, generalized Sachdev–Ye–Kitaev models, and random unitary circuits, we propose that a logarithmic scrambling time can be achieved in most quantum systems with sparse connectivity. These models also elucidate how quantum chaos is not universally related to scrambling: We construct random few-body circuits with infinite Lyapunov exponent but logarithmic scrambling time. We discuss analogies between quantum models on graphs and quantum black holes and suggest methods to experimentally study scrambling with as many as 100 sparsely connected quantum degrees of freedom.

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18

Gebert, Martin, Bruno Nachtergaele, Jake Reschke, and Robert Sims. "Lieb–Robinson Bounds and Strongly Continuous Dynamics for a Class of Many-Body Fermion Systems in $${\mathbb {R}}^d$$." Annales Henri Poincaré 21, no. 11 (September 24, 2020): 3609–37. http://dx.doi.org/10.1007/s00023-020-00959-5.

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19

Kennett, Malcolm P. "Out-of-Equilibrium Dynamics of the Bose-Hubbard Model." ISRN Condensed Matter Physics 2013 (June 12, 2013): 1–39. http://dx.doi.org/10.1155/2013/393616.

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Анотація:

The Bose-Hubbard model is the simplest model of interacting bosons on a lattice. It has recently been the focus of much attention due to the realization of this model with cold atoms in an optical lattice. The ability to tune parameters in the Hamiltonian as a function of time in cold atom systems has opened up the possibility of studying out-of-equilibrium dynamics, including crossing the quantum critical region of the model in a controlled way. In this paper, I give a brief introduction to the Bose Hubbard model, and its experimental realization and then give an account of theoretical and experimental efforts to understand out-of-equilibrium dynamics in this model, focusing on quantum quenches, both instantaneous and of finite duration. I discuss slow dynamics that have been observed theoretically and experimentally for some quenches from the superfluid phase to the Mott insulating phase and the picture of two timescales, one for fast local equilibration and another for slow global equilibration, that appears to characterize this situation. I also discuss the theoretical and experimental observation of the Lieb-Robinson bounds for a variety of quenches and the Kibble-Zurek mechanism in quenches from the Mott insulator to superfluid. I conclude with a discussion of open questions and future directions.

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20

Hinrichs, Benjamin, Marius Lemm, and Oliver Siebert. "On Lieb–Robinson Bounds for a Class of Continuum Fermions." Annales Henri Poincaré, July 12, 2024. http://dx.doi.org/10.1007/s00023-024-01453-y.

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AbstractWe consider the quantum dynamics of a many-fermion system in $${{\mathbb {R}}}^d$$ R d with an ultraviolet regularized pair interaction as previously studied in Gebert et al. (Ann Henri Poincaré 21(11):3609–3637, 2020). We provide a Lieb–Robinson bound under substantially relaxed assumptions on the potentials. We also improve the associated one-body Lieb–Robinson bound on $$L^2$$ L 2 -overlaps to an almost ballistic one (i.e., an almost linear light cone) under the same relaxed assumptions. Applications include the existence of the infinite-volume dynamics and clustering of ground states in the presence of a spectral gap. We also develop a fermionic continuum notion of conditional expectation and use it to approximate time-evolved fermionic observables by local ones, which opens the door to other applications of the Lieb–Robinson bounds.

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21

Huang, Zhiqiang, and Xiao-Kan Guo. "Lieb-Robinson bound at finite temperatures." Physical Review E 97, no. 6 (June 18, 2018). http://dx.doi.org/10.1103/physreve.97.062131.

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22

Ranard, Daniel, Michael Walter, and Freek Witteveen. "A Converse to Lieb–Robinson Bounds in One Dimension Using Index Theory." Annales Henri Poincaré, July 26, 2022. http://dx.doi.org/10.1007/s00023-022-01193-x.

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AbstractUnitary dynamics with a strict causal cone (or “light cone”) have been studied extensively, under the name of quantum cellular automata (QCAs). In particular, QCAs in one dimension have been completely classified by an index theory. Physical systems often exhibit only approximate causal cones; Hamiltonian evolutions on the lattice satisfy Lieb–Robinson bounds rather than strict locality. This motivates us to study approximately locality preserving unitaries (ALPUs). We show that the index theory is robust and completely extends to one-dimensional ALPUs. As a consequence, we achieve a converse to the Lieb–Robinson bounds: any ALPU of index zero can be exactly generated by some time-dependent, quasi-local Hamiltonian in constant time. For the special case of finite chains with open boundaries, any unitary satisfying the Lieb–Robinson bound may be generated by such a Hamiltonian. We also discuss some results on the stability of operator algebras which may be of independent interest.

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23

Fu, Hao, Mingqiu Luo, and Peiqing Tong. "Lieb-Robinson bound in one-dimensional inhomogeneous quantum systems." Physica B: Condensed Matter, April 2022, 413958. http://dx.doi.org/10.1016/j.physb.2022.413958.

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24

Fu, Hao, Mingqiu Luo, and Peiqing Tong. "Lieb-Robinson bound in one-dimensional inhomogeneous quantum systems." Physica B: Condensed Matter, April 2022, 413958. http://dx.doi.org/10.1016/j.physb.2022.413958.

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25

Fu, Hao, Peiqing Tong, and Mingqiu Luo. "Lieb-Robinson Bound in One-Dimensional Inhomogeneous Quantum Systems." SSRN Electronic Journal, 2022. http://dx.doi.org/10.2139/ssrn.4053161.

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26

Wang, Zhiyuan, and Kaden R. A. Hazzard. "Tightening the Lieb-Robinson Bound in Locally Interacting Systems." PRX Quantum 1, no. 1 (September 3, 2020). http://dx.doi.org/10.1103/prxquantum.1.010303.

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27

Poulin, David. "Lieb-Robinson Bound and Locality for General Markovian Quantum Dynamics." Physical Review Letters 104, no. 19 (May 11, 2010). http://dx.doi.org/10.1103/physrevlett.104.190401.

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28

Braida, Arthur, Simon Martiel, and Ioan Todinca. "Tight Lieb–Robinson Bound for approximation ratio in quantum annealing." npj Quantum Information 10, no. 1 (April 17, 2024). http://dx.doi.org/10.1038/s41534-024-00832-x.

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AbstractQuantum annealing (QA) holds promise for optimization problems in quantum computing, especially for combinatorial optimization. This analog framework attracts attention for its potential to address complex problems. Its gate-based homologous, QAOA with proven performance, has attracted a lot of attention to the NISQ era. Several numerical benchmarks try to compare these two metaheuristics, however, classical computational power highly limits the performance insights. In this work, we introduce a parametrized version of QA enabling a precise 1-local analysis of the algorithm. We develop a tight Lieb–Robinson bound for regular graphs, achieving the best-known numerical value to analyze QA locally. Studying MaxCut over cubic graph as a benchmark optimization problem, we show that a linear-schedule QA with a 1-local analysis achieves an approximation ratio over 0.7020, outperforming any known 1-local algorithms.

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29

Kuwahara, Tomotaka, Tan Van Vu, and Keiji Saito. "Effective light cone and digital quantum simulation of interacting bosons." Nature Communications 15, no. 1 (March 21, 2024). http://dx.doi.org/10.1038/s41467-024-46501-7.

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AbstractThe speed limit of information propagation is one of the most fundamental features in non-equilibrium physics. The region of information propagation by finite-time dynamics is approximately restricted inside the effective light cone that is formulated by the Lieb-Robinson bound. To date, extensive studies have been conducted to identify the shape of effective light cones in most experimentally relevant many-body systems. However, the Lieb-Robinson bound in the interacting boson systems, one of the most ubiquitous quantum systems in nature, has remained a critical open problem for a long time. This study reveals a tight effective light cone to limit the information propagation in interacting bosons, where the shape of the effective light cone depends on the spatial dimension. To achieve it, we prove that the speed for bosons to clump together is finite, which in turn leads to the error guarantee of the boson number truncation at each site. Furthermore, we applied the method to provide a provably efficient algorithm for simulating the interacting boson systems. The results of this study settle the notoriously challenging problem and provide the foundation for elucidating the complexity of many-body boson systems.

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30

Roberts, Daniel A., and Brian Swingle. "Lieb-Robinson Bound and the Butterfly Effect in Quantum Field Theories." Physical Review Letters 117, no. 9 (August 23, 2016). http://dx.doi.org/10.1103/physrevlett.117.091602.

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31

Abeling, Nils O., Lorenzo Cevolani, and Stefan Kehrein. "Analysis of the buildup of spatiotemporal correlations and their bounds outside of the light cone." SciPost Physics 5, no. 5 (November 26, 2018). http://dx.doi.org/10.21468/scipostphys.5.5.052.

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In non-relativistic quantum theories the Lieb-Robinson bound defines an effective light cone with exponentially small tails outside of it. In this work we use it to derive a bound for the correlation function of two local disjoint observables at different times if the initial state has a power-law decay. We show that the exponent of the power-law of the bound is identical to the initial (equilibrium) decay. We explicitly verify this result by studying the full dynamics of the susceptibilities and correlations in the exactly solvable Luttinger model after a sudden quench from the non-interacting to the interacting model.

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32

Jameson, Casey, Bora Basyildiz, Daniel Moore, Kyle Clark, and Zhexuan Gong. "Time optimal quantum state transfer in a fully-connected quantum computer." Quantum Science and Technology, October 26, 2023. http://dx.doi.org/10.1088/2058-9565/ad0770.

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Abstract The speed limit of quantum state transfer (QST) in a system of interacting particles is not only important for quantum information processing, but also directly linked to Lieb-Robinson-type bounds that are crucial for understanding various aspects of quantum many-body physics. For strongly long-range interacting systems such as a fully-connected quantum computer, such a speed limit is still unknown. Here we develop a new Quantum Brachistochrone method that can incorporate inequality constraints on the Hamiltonian. This method allows us to prove an exactly tight bound on the speed of QST on a subclass of Hamiltonians experimentally realizable by a fully-connected quantum computer.

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33

Gebert, Martin, Alvin Moon, and Bruno Nachtergaele. "A Lieb–Robinson bound for quantum spin chains with strong on-site impurities." Reviews in Mathematical Physics, January 15, 2022. http://dx.doi.org/10.1142/s0129055x22500076.

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We consider a quantum spin chain with nearest neighbor interactions and sparsely distributed on-site impurities. We prove commutator bounds for its Heisenberg dynamics which incorporate the coupling strengths of the impurities. The impurities are assumed to satisfy a minimum spacing, and each impurity has a non-degenerate spectrum. Our results are proven in a broadly applicable setting, both in finite volume and in thermodynamic limit. We apply our results to improve Lieb–Robinson bounds for the Heisenberg spin chain with a random, sparse transverse field drawn from a heavy-tailed distribution.

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34

Kuwahara, Tomotaka, and Keiji Saito. "Lieb-Robinson Bound and Almost-Linear Light Cone in Interacting Boson Systems." Physical Review Letters 127, no. 7 (August 13, 2021). http://dx.doi.org/10.1103/physrevlett.127.070403.

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35

Else, Dominic V., Francisco Machado, Chetan Nayak, and Norman Y. Yao. "Improved Lieb-Robinson bound for many-body Hamiltonians with power-law interactions." Physical Review A 101, no. 2 (February 26, 2020). http://dx.doi.org/10.1103/physreva.101.022333.

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36

Nickelsen, Daniel, and Michael Kastner. "Classical Lieb-Robinson Bound for Estimating Equilibration Timescales of Isolated Quantum Systems." Physical Review Letters 122, no. 18 (May 10, 2019). http://dx.doi.org/10.1103/physrevlett.122.180602.

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37

Gong, Zongping, Tommaso Guaita, and J. Ignacio Cirac. "Long-Range Free Fermions: Lieb-Robinson Bound, Clustering Properties, and Topological Phases." Physical Review Letters 130, no. 7 (February 17, 2023). http://dx.doi.org/10.1103/physrevlett.130.070401.

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38

Shiraishi, Naoto, and Hiroyasu Tajima. "Efficiency versus speed in quantum heat engines: Rigorous constraint from Lieb-Robinson bound." Physical Review E 96, no. 2 (August 16, 2017). http://dx.doi.org/10.1103/physreve.96.022138.

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39

Chen, Xiao, Yingfei Gu, and Andrew Lucas. "Many-body quantum dynamics slows down at low density." SciPost Physics 9, no. 5 (November 12, 2020). http://dx.doi.org/10.21468/scipostphys.9.5.071.

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Анотація:

We study quantum many-body systems with a global U(1) conservation law, focusing on a theory of N interacting fermions with charge conservation, or N interacting spins with one conserved component of total spin. We define an effective operator size at finite chemical potential through suitably regularized out-of-time-ordered correlation functions. The growth rate of this density-dependent operator size vanishes algebraically with charge density; hence we obtain new bounds on Lyapunov exponents and butterfly velocities in charged systems at a given density, which are parametrically stronger than any Lieb-Robinson bound. We argue that the density dependence of our bound on the Lyapunov exponent is saturated in the charged Sachdev-Ye-Kitaev model. We also study random automaton quantum circuits and Brownian Sachdev-Ye-Kitaev models, each of which exhibit a different density dependence for the Lyapunov exponent, and explain the discrepancy. We propose that our results are a cartoon for understanding Planckian-limited energy-conserving dynamics at finite temperature.

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40

Gong, Zongping, and Ryusuke Hamazaki. "Bounds in nonequilibrium quantum dynamics." International Journal of Modern Physics B, September 26, 2022. http://dx.doi.org/10.1142/s0217979222300079.

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Анотація:

We review various bounds concerning out-of-equilibrium dynamics in few-level and many-body quantum systems. We primarily focus on closed quantum systems but will also mention some related results for open quantum systems and classical stochastic systems. We start from the speed limits, the universal bounds on the speeds of (either quantum or classical) dynamical evolutions. We then turn to review the bounds that address how good and how long would a quantum system equilibrate or thermalize. Afterward, we focus on the stringent constraint set by locality in many-body systems, rigorously formalized as the Lieb–Robinson bound. We also review the bounds related to the dynamics of entanglement, a genuine quantum property. Apart from some other miscellaneous topics, several notable error bounds for approximated quantum dynamics are discussed. While far from comprehensive, this topical review covers a considerable amount of recent progress and thus could hopefully serve as a convenient starting point and up-to-date guidance for interested readers.

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41

Ageev, Dmitry S., Andrey A. Bagrov, Aleksandr I. Belokon, Askar Iliasov, Vasilii V. Pushkarev, and Femke Verheijen. "Local quenches in fracton field theory: Lieb-Robinson bound, noncausal dynamics and fractal excitation patterns." Physical Review D 110, no. 6 (September 13, 2024). http://dx.doi.org/10.1103/physrevd.110.065011.

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We study the out-of-equilibrium dynamics induced by a local perturbation in fracton field theory. For the Z4- and Z8-symmetric free fractonic theories, we compute the time dynamics of several observables such as the two-point Green’s function, ⟨ϕ2⟩ condensate, energy density, and the dipole momentum. The time-dependent considerations highlight that the free fractonic theory breaks causality and exhibits instantaneous signal propagation, even if an additional relativistic term is included to enforce a speed limit in the system. We show that it is related to the fact that the Lieb-Robinson bound does not hold in the continuum limit of the fracton field theory, and the effective bounded speed of light does not emerge. For the theory in finite volume, we show that the fracton wave front acquires fractal shape with nontrivial Hausdorff dimension and argue that this phenomenon cannot be explained by a simple self-interference effect. Published by the American Physical Society 2024

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42

Braida, Arthur, Simon Martiel, and Ioan Todinca. "On constant-time quantum annealing and guaranteed approximations for graph optimization problems." Quantum Science and Technology, September 1, 2022. http://dx.doi.org/10.1088/2058-9565/ac8e91.

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Abstract Quantum Annealing (QA) is a computational framework where a quantum system’s continuous evolution is used to ﬁnd the global minimum of an objective function over an unstructured search space. It can be seen as a general metaheuristic for optimization problems, including NP-hard ones if we allow an exponentially large running time. While QA is widely studied from a heuristic point of view, little is known about theoretical guarantees on the quality of the solutions obtained in polynomial time. In this paper, we use a technique borrowed from theoretical physics, the Lieb-Robinson (LR) bound, and develop new tools proving that short, constant-time quantum annealing guarantees constant factor approximations ratios for some optimization problems when restricted to bounded degree graphs. Informally, on bounded degree graphs the LR bound allows us to retrieve a (relaxed) locality argument, through which the approximation ratio can be deduced by studying subgraphs of bounded radius. We illustrate our tools on problems MaxCut and Maximum Independent Set for cubic graphs, providing explicit approximation ratios and the runtimes needed to obtain them. Our results are of similar ﬂavor to the well-known ones obtained in the diﬀerent but related QAOA (quantum optimization algorithms) framework.

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43

Ampelogiannis, Dimitrios, and Benjamin Doyon. "Long-Time Dynamics in Quantum Spin Lattices: Ergodicity and Hydrodynamic Projections at All Frequencies and Wavelengths." Annales Henri Poincaré, May 5, 2023. http://dx.doi.org/10.1007/s00023-023-01304-2.

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AbstractObtaining rigorous and general results about the non-equilibrium dynamics of extended many-body systems is a difficult task. In quantum lattice models with short-range interactions, the Lieb–Robinson bound tells us that the spatial extent of operators grows at most linearly in time. But what happens within this light-cone? We discuss rigorous results on ergodicity and the emergence of the Euler hydrodynamic scale in correlation functions, which establish fundamental principles at the root of non-equilibrium physics. One key idea of the present work is that general structures of Euler hydrodynamics, obtained under ballistic scaling, follow independently from the details of the microscopic dynamics, and in particular do not necessitate chaos; they are consequences of “extensivity”. Another crucial observation is that these apply at arbitrary frequencies and wavelengths. That is, long-time, persistent oscillations of correlation functions over ballistic regions of spacetime, which may be of microscopic frequencies and wavelengths, are predicted by a general Euler-hydrodynamic theory that takes the same form as that for smoothed-out correlation functions. This involves a natural extension of notions of conserved quantities and hydrodynamic projection and shows that the Euler hydrodynamic paradigm covers the full frequency-wavelength plane.

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44

Ponnaganti, Ravi Teja, Matthieu Mambrini, and Didier Poilblanc. "Tensor network variational optimizations for real-time dynamics: Application to the time-evolution of spin liquids." SciPost Physics 15, no. 4 (October 12, 2023). http://dx.doi.org/10.21468/scipostphys.15.4.158.

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Within the Projected Entangled Pair State (PEPS) tensor network formalism, a simple update (SU) method has been used to investigate the time evolution of a two-dimensional U(1) critical spin-1/2 spin liquid under Hamiltonian quench [Phys. Rev. B 106, 195132 (2022)]. Here we introduce two different variational frameworks to describe the time dynamics of SU(2)-symmetric translationally-invariant PEPS, aiming to improve the accuracy. In one approach, after using a Trotter-Suzuki decomposition of the time evolution operator in term of two-site elementary gates, one considers a single bond embedded in an environment approximated by a Corner Transfer Matrix Renormalization Group (CTMRG). A variational update of the two tensors on the bond is performed under the application of the elementary gate and then, after symmetrization of the site tensors, the environment is updated. In the second approach, a cluster optimization is performed on a finite (periodic) cluster, maximizing the overlap of the exact time-evolved state with a symmetric finite-size PEPS ansatz. Observables are then computed on the infinite lattice contracting the infinite-PEPS (iPEPS) by CTMRG. We show that the variational schemes outperform the SU method and remain accurate over a significant time interval before hitting the entanglement barrier. Studying the spectrum of the transfer matrix, we find that the asymptotic correlations are very well preserved under time evolution, including the critical nature of the singlet correlations, as expected from the Lieb-Robinson (LR) bound theorem. Consistently, the system (asymptotic) boundary is found to be described by the same Conformal Field Theory of central charge c = 1c=1 during time evolution. We also compute the time-evolution of the short distance spin-spin correlations and estimate the LR velocity.

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45

Ampelogiannis, Dimitrios, and Benjamin Doyon. "Almost Everywhere Ergodicity in Quantum Lattice Models." Communications in Mathematical Physics, October 30, 2023. http://dx.doi.org/10.1007/s00220-023-04849-9.

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AbstractWe rigorously examine, in generality, the ergodic properties of quantum lattice models with short range interactions, in the $$C^*$$ C ∗ algebra formulation of statistical mechanics. Ergodicity results, in the context of group actions on $$C^*$$ C ∗ algebras, assume that the algebra is asymptotically abelian, which is not generically the case for time evolution. The Lieb-Robinson bound tells us that, in a precise sense, the spatial extent of any time-evolved local operator grows linearly with time. This means that the algebra of observables is asymptotically abelian in a space-like region, and implies a form of ergodicity outside the light-cone. But what happens within it? We show that the long-time limit of the n-th moment of a ray-averaged observable, along space-time rays of almost every speed, converges to the n-th power of its expectation in the state (i.e. its ensemble average). Thus ray averages do not fluctuate in the long time limit. This is a statement of ergodicity, and holds in any state that is invariant under space-time translations and that satisfies weak clustering properties in space. The ray averages can be performed in a way that accounts for oscillations, showing that ray-averaged observables cannot sustainably oscillate in the long time limit. We also show that in the GNS representation of the algebra of observables, for any KMS state with the above properties, the long-time limit of the ray average of any observable converges (in the strong operator topology) to the ensemble average times the identity, again along space-time rays of almost every speed. This is a strong version of ergodicity, and indicates that, as operators, observables get “thinner” almost everywhere within the light-cone. A similar statement holds under oscillatory averaging.

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46

Wilming, Henrik, and Albert H. Werner. "Lieb-Robinson bounds imply locality of interactions." Physical Review B 105, no. 12 (March 2, 2022). http://dx.doi.org/10.1103/physrevb.105.125101.

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47

Damanik, David, Marius Lemm, Milivoje Lukic, and William Yessen. "New Anomalous Lieb-Robinson Bounds in QuasiperiodicXYChains." Physical Review Letters 113, no. 12 (September 18, 2014). http://dx.doi.org/10.1103/physrevlett.113.127202.

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48

Prémont-Schwarz, Isabeau, Alioscia Hamma, Israel Klich, and Fotini Markopoulou-Kalamara. "Lieb-Robinson bounds for commutator-bounded operators." Physical Review A 81, no. 4 (April 27, 2010). http://dx.doi.org/10.1103/physreva.81.040102.

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49

Baldwin, Christopher L., Adam Ehrenberg, Andrew Y. Guo, and Alexey V. Gorshkov. "Disordered Lieb-Robinson Bounds in One Dimension." PRX Quantum 4, no. 2 (June 22, 2023). http://dx.doi.org/10.1103/prxquantum.4.020349.

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50

Faupin, Jérémy, Marius Lemm, and Israel Michael Sigal. "On Lieb–Robinson Bounds for the Bose–Hubbard Model." Communications in Mathematical Physics, June 29, 2022. http://dx.doi.org/10.1007/s00220-022-04416-8.

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