33,408 research outputs found
Quantum Annealing and Analog Quantum Computation
We review here the recent success in quantum annealing, i.e., optimization of
the cost or energy functions of complex systems utilizing quantum fluctuations.
The concept is introduced in successive steps through the studies of mapping of
such computationally hard problems to the classical spin glass problems. The
quantum spin glass problems arise with the introduction of quantum
fluctuations, and the annealing behavior of the systems as these fluctuations
are reduced slowly to zero. This provides a general framework for realizing
analog quantum computation.Comment: 22 pages, 7 figs (color online); new References Added. Reviews of
Modern Physics (in press
Glassy Phase of Optimal Quantum Control
We study the problem of preparing a quantum many-body system from an initial
to a target state by optimizing the fidelity over the family of bang-bang
protocols. We present compelling numerical evidence for a universal
spin-glass-like transition controlled by the protocol time duration. The glassy
critical point is marked by a proliferation of protocols with close-to-optimal
fidelity and with a true optimum that appears exponentially difficult to
locate. Using a machine learning (ML) inspired framework based on the manifold
learning algorithm t-SNE, we are able to visualize the geometry of the
high-dimensional control landscape in an effective low-dimensional
representation. Across the transition, the control landscape features an
exponential number of clusters separated by extensive barriers, which bears a
strong resemblance with replica symmetry breaking in spin glasses and random
satisfiability problems. We further show that the quantum control landscape
maps onto a disorder-free classical Ising model with frustrated nonlocal,
multibody interactions. Our work highlights an intricate but unexpected
connection between optimal quantum control and spin glass physics, and shows
how tools from ML can be used to visualize and understand glassy optimization
landscapes.Comment: Modified figures in appendix and main text (color schemes). Corrected
references. Added figures in SI and pseudo-cod
Genetic embedded matching approach to ground states in continuous-spin systems
Due to an extremely rugged structure of the free energy landscape, the
determination of spin-glass ground states is among the hardest known
optimization problems, found to be NP-hard in the most general case. Owing to
the specific structure of local (free) energy minima, general-purpose
optimization strategies perform relatively poorly on these problems, and a
number of specially tailored optimization techniques have been developed in
particular for the Ising spin glass and similar discrete systems. Here, an
efficient optimization heuristic for the much less discussed case of continuous
spins is introduced, based on the combination of an embedding of Ising spins
into the continuous rotators and an appropriate variant of a genetic algorithm.
Statistical techniques for insuring high reliability in finding (numerically)
exact ground states are discussed, and the method is benchmarked against the
simulated annealing approach.Comment: 17 pages, 12 figures, 1 tabl
Hessian spectrum at the global minimum of high-dimensional random landscapes
Using the replica method we calculate the mean spectral density of the
Hessian matrix at the global minimum of a random dimensional
isotropic, translationally invariant Gaussian random landscape confined by a
parabolic potential with fixed curvature . Simple landscapes with
generically a single minimum are typical for , and we show that
the Hessian at the global minimum is always {\it gapped}, with the low spectral
edge being strictly positive. When approaching from above the transitional
point separating simple landscapes from 'glassy' ones, with
exponentially abundant minima, the spectral gap vanishes as .
For the Hessian spectrum is qualitatively different for 'moderately
complex' and 'genuinely complex' landscapes. The former are typical for
short-range correlated random potentials and correspond to 1-step
replica-symmetry breaking mechanism. Their Hessian spectra turn out to be again
gapped, with the gap vanishing on approaching from below with a larger
critical exponent, as . At the same time in the 'most complex'
landscapes with long-ranged power-law correlations the replica symmetry is
completely broken. We show that in that case the Hessian remains gapless for
all values of , indicating the presence of 'marginally stable'
spatial directions. Finally, the potentials with {\it logarithmic} correlations
share both 1RSB nature and gapless spectrum. The spectral density of the
Hessian always takes the semi-circular form, up to a shift and an amplitude
that we explicitly calculate.Comment: 28 pages, 1 figure; a brief summary of main results is added to the
introductio
First excitations in two- and three-dimensional random-field Ising systems
We present results on the first excited states for the random-field Ising
model. These are based on an exact algorithm, with which we study the
excitation energies and the excitation sizes for two- and three-dimensional
random-field Ising systems with a Gaussian distribution of the random fields.
Our algorithm is based on an approach of Frontera and Vives which, in some
cases, does not yield the true first excited states. Using the corrected
algorithm, we find that the order-disorder phase transition for three
dimensions is visible via crossings of the excitations-energy curves for
different system sizes, while in two-dimensions these crossings converge to
zero disorder. Furthermore, we obtain in three dimensions a fractal dimension
of the excitations cluster of d_s=2.42(2). We also provide analytical droplet
arguments to understand the behavior of the excitation energies for small and
large disorder as well as close to the critical point.Comment: 17 pages, 12 figure
Zero-temperature phase of the XY spin glass in two dimensions: Genetic embedded matching heuristic
For many real spin-glass materials, the Edwards-Anderson model with
continuous-symmetry spins is more realistic than the rather better understood
Ising variant. In principle, the nature of an occurring spin-glass phase in
such systems might be inferred from an analysis of the zero-temperature
properties. Unfortunately, with few exceptions, the problem of finding
ground-state configurations is a non-polynomial problem computationally, such
that efficient approximation algorithms are called for. Here, we employ the
recently developed genetic embedded matching (GEM) heuristic to investigate the
nature of the zero-temperature phase of the bimodal XY spin glass in two
dimensions. We analyze bulk properties such as the asymptotic ground-state
energy and the phase diagram of disorder strength vs. disorder concentration.
For the case of a symmetric distribution of ferromagnetic and antiferromagnetic
bonds, we find that the ground state of the model is unique up to a global O(2)
rotation of the spins. In particular, there are no extensive degeneracies in
this model. The main focus of this work is on an investigation of the
excitation spectrum as probed by changing the boundary conditions. Using
appropriate finite-size scaling techniques, we consistently determine the
stiffness of spin and chiral domain walls and the corresponding fractal
dimensions. Most noteworthy, we find that the spin and chiral channels are
characterized by two distinct stiffness exponents and, consequently, the system
displays spin-chirality decoupling at large length scales. Results for the
overlap distribution do not support the possibility of a multitude of
thermodynamic pure states.Comment: 18 pages, RevTex 4, moderately revised version as publishe
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