312 research outputs found
On thermalization in the SYK and supersymmetric SYK models
The eigenstate thermalization hypothesis is a compelling conjecture which
strives to explain the apparent thermal behavior of generic observables in
closed quantum systems. Although we are far from a complete analytic
understanding, quantum chaos is often seen as a strong indication that the
ansatz holds true. In this paper, we address the thermalization of energy
eigenstates in the Sachdev-Ye-Kitaev model, a maximally chaotic model of
strongly-interacting Majorana fermions. We numerically investigate eigenstate
thermalization for specific few-body operators in the original SYK model as
well as its supersymmetric extension and find evidence that
these models satisfy ETH. We discuss the implications of ETH for a
gravitational dual and the quantum information-theoretic properties of SYK it
suggests.Comment: Published versio
Chaos, Complexity, and Random Matrices
Chaos and complexity entail an entropic and computational obstruction to
describing a system, and thus are intrinsically difficult to characterize. In
this paper, we consider time evolution by Gaussian Unitary Ensemble (GUE)
Hamiltonians and analytically compute out-of-time-ordered correlation functions
(OTOCs) and frame potentials to quantify scrambling, Haar-randomness, and
circuit complexity. While our random matrix analysis gives a qualitatively
correct prediction of the late-time behavior of chaotic systems, we find
unphysical behavior at early times including an scrambling
time and the apparent breakdown of spatial and temporal locality. The salient
feature of GUE Hamiltonians which gives us computational traction is the
Haar-invariance of the ensemble, meaning that the ensemble-averaged dynamics
look the same in any basis. Motivated by this property of the GUE, we introduce
-invariance as a precise definition of what it means for the dynamics of a
quantum system to be described by random matrix theory. We envision that the
dynamical onset of approximate -invariance will be a useful tool for
capturing the transition from early-time chaos, as seen by OTOCs, to late-time
chaos, as seen by random matrix theory.Comment: 61 pages, 14 figures; v2: references added, typos fixe
Local random quantum circuits form approximate designs on arbitrary architectures
We consider random quantum circuits (RQC) on arbitrary connected graphs whose
edges determine the allowed -qudit interactions. Prior work has established
that such -qudit circuits with local dimension on 1D, complete, and
-dimensional graphs form approximate unitary designs, that is, they generate
unitaries from distributions close to the Haar measure on the unitary group
after polynomially many gates. Here, we extend those results by
proving that RQCs comprised of gates on a wide class of
graphs form approximate unitary -designs. We prove that RQCs on graphs with
spanning trees of bounded degree and height form -designs after
gates, where is the number of edges in the
graph. Furthermore, we identify larger classes of graphs for which RQCs
generate approximate designs in polynomial circuit size. For , we
show that RQCs on graphs of certain maximum degrees form designs after
gates, providing explicit constants. We determine our circuit size
bounds from the spectral gaps of local Hamiltonians. To that end, we extend the
finite-size (or Knabe) method for bounding gaps of frustration-free
Hamiltonians on regular graphs to arbitrary connected graphs. We further
introduce a new method based on the Detectability Lemma for determining the
spectral gaps of Hamiltonians on arbitrary graphs. Our methods have wider
applicability as the first method provides a succinct alternative proof of
[Commun. Math. Phys. 291, 257 (2009)] and the second method proves that RQCs on
any connected architecture form approximate designs in quasi-polynomial circuit
size
Novel approaches to Newtonian noise suppression in interferometric gravitational wave detection
Thesis (S.B.)--Massachusetts Institute of Technology, Dept. of Physics, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 63-65).The Laser Interferometer Gravitational-wave Observatory (LIGO) attempts to detect ripples in the curvature of spacetime using two large scale interferometers. These detectors are several kilometer long Michelson interferometers with Fabry-Perot cavities between two silica test masses in each arm. Given Earth's proximity to various astrophysical phenomena LIGO must be sensitive to relative displacements of 1018 m and thus requires multiple levels of noise reduction to ensure the isolation of the interferometer components from numerous sources of noise. A substantial contributor to the Advanced LIGO noise in the 1-10 Hz range is Newtonian (or gravity gradient) noise which arises from local fluctuations in the Earth's gravitational field. Density fluctuations from seismic activity as well as acoustic and turbulent phenomenon in the Earth's atmosphere both contribute to slight variations in the local value of g. Given the direct coupling of gravitational fields to mass the LIGO test masses cannot be shielded from this noise. In an attempt to characterize and reduce Newtonian noise in interferometric gravitational wave detectors we investigate seismic and atmospheric contributions to the noise and consider the effect of submerging a gravitational wave detector.by Nicholas R. Hunter-Jones.S.B
Chaos and Randomness in Strongly-Interacting Quantum Systems
Quantum chaos entails an entropic and computational obstruction to describing a system and thus is intrinsically difficult to characterize. An understanding of quantum chaos is fundamentally related to the mechanism of thermalization in many-body systems and the quantum nature of black holes. In this thesis we adopt the view that quantum information theory provides a powerful framework in which to elucidate chaos in strongly-interacting quantum systems.
We first push towards a more precise understanding of chaotic dynamics by relating different diagnostics of chaos, studying the time-evolution of random matrix Hamiltonians, and quantifying random matrix behavior in physical systems. We derive relations between out-of-time ordered correlation functions, spectral quantities, and frame potentials to relate the scrambling of quantum information, decay of correlators, and Haar-randomness. We give analytic expressions for these quantities in random matrix theory to explore universal aspects of late-time dynamics. Motivated by our random matrix results, we define k-invariance in order to capture the onset of random matrix behavior in physical systems.
We then refine our diagnostics in order to study chaotic systems with symmetry by considering Haar-randomness with respect to quotients of the unitary group, and in doing so we generalize our quantum information machinery. We further consider extended random matrix ensembles in the context of strongly-interacting quantum systems dual to black holes. Lastly, we study operator growth in classes of random quantum circuits.</p
Chaos and random matrices in supersymmetric SYK
We use random matrix theory to explore late-time chaos in supersymmetric quantum mechanical systems. Motivated by the recent study of supersymmetric SYK models and their random matrix classification, we consider the Wishart-Laguerre unitary ensemble and compute the spectral form factors and frame potentials to quantify chaos and randomness. Compared to the Gaussian ensembles, we observe the absence of a dip regime in the form factor and a slower approach to Haar-random dynamics. We find agreement between our random matrix analysis and predictions from the supersymmetric SYK model, and discuss the implications for supersymmetric chaotic systems
Consistency Conditions for an AdS/MERA Correspondence
The Multi-scale Entanglement Renormalization Ansatz (MERA) is a tensor
network that provides an efficient way of variationally estimating the ground
state of a critical quantum system. The network geometry resembles a
discretization of spatial slices of an AdS spacetime and "geodesics" in the
MERA reproduce the Ryu-Takayanagi formula for the entanglement entropy of a
boundary region in terms of bulk properties. It has therefore been suggested
that there could be an AdS/MERA correspondence, relating states in the Hilbert
space of the boundary quantum system to ones defined on the bulk lattice. Here
we investigate this proposal and derive necessary conditions for it to apply,
using geometric features and entropy inequalities that we expect to hold in the
bulk. We show that, perhaps unsurprisingly, the MERA lattice can only describe
physics on length scales larger than the AdS radius. Further, using the
covariant entropy bound in the bulk, we show that there are no conventional
MERA parameters that completely reproduce bulk physics even on super-AdS
scales. We suggest modifications or generalizations of this kind of tensor
network that may be able to provide a more robust correspondence.Comment: 38 pages, 9 figure
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