831 research outputs found
Derivatives and inequalities for order parameters in the Ising spin glass
Identities and inequalities are proved for the order parameters, correlation
functions and their derivatives of the Ising spin glass. The results serve as
additional evidence that the ferromagnetic phase is composed of two regions,
one with strong ferromagnetic ordering and the other with the effects of
disorder dominant. The Nishimori line marks a crossover between these two
regions.Comment: 10 pages; 3 figures; new inequalities added, title slightly change
Quantum annealing with antiferromagnetic fluctuations
We introduce antiferromagnetic quantum fluctuations into quantum annealing in
addition to the conventional transverse-field term. We apply this method to the
infinite-range ferromagnetic p-spin model, for which the conventional quantum
annealing has been shown to have difficulties to find the ground state
efficiently due to a first-order transition. We study the phase diagram of this
system both analytically and numerically. Using the static approximation, we
find that there exists a quantum path to reach the final ground state from the
trivial initial state that avoids first-order transitions for intermediate
values of p. We also study numerically the energy gap between the ground state
and the first excited state and find evidence for intermediate values of p that
the time complexity scales polynomially with the system size at a second-order
transition point along the quantum path that avoids first-order transitions.
These results suggest that quantum annealing would be able to solve this
problem with intermediate values of p efficiently in contrast to the case with
only simple transverse-field fluctuations.Comment: 19 pages, 11 figures; Added references; To be published in Physical
Review
Location of the Multicritical Point for the Ising Spin Glass on the Triangular and Hexagonal Lattices
A conjecture is given for the exact location of the multicritical point in
the phase diagram of the +/- J Ising model on the triangular lattice. The
result p_c=0.8358058 agrees well with a recent numerical estimate. From this
value, it is possible to derive a comparable conjecture for the exact location
of the multicritical point for the hexagonal lattice, p_c=0.9327041, again in
excellent agreement with a numerical study. The method is a variant of duality
transformation to relate the triangular lattice directly with its dual
triangular lattice without recourse to the hexagonal lattice, in conjunction
with the replica method.Comment: 9 pages, 1 figure; Minor corrections in notatio
Multicritical Points of Potts Spin Glasses on the Triangular Lattice
We predict the locations of several multicritical points of the Potts spin
glass model on the triangular lattice. In particular, continuous multicritical
lines, which consist of multicritical points, are obtained for two types of
two-state Potts (i.e., Ising) spin glasses with two- and three-body
interactions on the triangular lattice. These results provide us with numerous
examples to further verify the validity of the conjecture, which has succeeded
in deriving highly precise locations of multicritical points for several spin
glass models. The technique, called the direct triangular duality, a variant of
the ordinary duality transformation, directly relates the triangular lattice
with its dual triangular lattice in conjunction with the replica method.Comment: 18 pages, 2, figure
Quantum Annealing in the Transverse Ising Model
We introduce quantum fluctuations into the simulated annealing process of
optimization problems, aiming at faster convergence to the optimal state.
Quantum fluctuations cause transitions between states and thus play the same
role as thermal fluctuations in the conventional approach. The idea is tested
by the transverse Ising model, in which the transverse field is a function of
time similar to the temperature in the conventional method. The goal is to find
the ground state of the diagonal part of the Hamiltonian with high accuracy as
quickly as possible. We have solved the time-dependent Schr\"odinger equation
numerically for small size systems with various exchange interactions.
Comparison with the results of the corresponding classical (thermal) method
reveals that the quantum annealing leads to the ground state with much larger
probability in almost all cases if we use the same annealing schedule.Comment: 15 pages, RevTeX, 8 figure
Duality in finite-dimensional spin glasses
We present an analysis leading to a conjecture on the exact location of the
multicritical point in the phase diagram of spin glasses in finite dimensions.
The conjecture, in satisfactory agreement with a number of numerical results,
was previously derived using an ansatz emerging from duality and the replica
method. In the present paper we carefully examine the ansatz and reduce it to a
hypothesis on analyticity of a function appearing in the duality relation. Thus
the problem is now clearer than before from a mathematical point of view: The
ansatz, somewhat arbitrarily introduced previously, has now been shown to be
closely related to the analyticity of a well-defined function.Comment: 12 pages, 3 figures; A reference added; to appear in J. Stat. Phy
Ensemble learning of linear perceptron; Online learning theory
Within the framework of on-line learning, we study the generalization error
of an ensemble learning machine learning from a linear teacher perceptron. The
generalization error achieved by an ensemble of linear perceptrons having
homogeneous or inhomogeneous initial weight vectors is precisely calculated at
the thermodynamic limit of a large number of input elements and shows rich
behavior. Our main findings are as follows. For learning with homogeneous
initial weight vectors, the generalization error using an infinite number of
linear student perceptrons is equal to only half that of a single linear
perceptron, and converges with that of the infinite case with O(1/K) for a
finite number of K linear perceptrons. For learning with inhomogeneous initial
weight vectors, it is advantageous to use an approach of weighted averaging
over the output of the linear perceptrons, and we show the conditions under
which the optimal weights are constant during the learning process. The optimal
weights depend on only correlation of the initial weight vectors.Comment: 14 pages, 3 figures, submitted to Physical Review
Statistical mechanics of image restoration and error-correcting codes
We develop a statistical-mechanical formulation for image restoration and
error-correcting codes. These problems are shown to be equivalent to the Ising
spin glass with ferromagnetic bias under random external fields. We prove that
the quality of restoration/decoding is maximized at a specific set of parameter
values determined by the source and channel properties. For image restoration
in mean-field system a line of optimal performance is shown to exist in the
parameter space. These results are illustrated by solving exactly the
infinite-range model. The solutions enable us to determine how precisely one
should estimate unknown parameters. Monte Carlo simulations are carried out to
see how far the conclusions from the infinite-range model are applicable to the
more realistic two-dimensional case in image restoration.Comment: 20 pages, 9 figures, ReVTe
Competition between ferro-retrieval and anti-ferro orders in a Hopfield-like network model for plant intelligence
We introduce a simple cellular-network model to explain the capacity of the
plants as memory devices. Following earlier observations (Bose \cite{Bose} and
others), we regard the plant as a network in which each of the elements (plant
cells) are connected via negative (inhibitory) interactions. To investigate the
performance of the network, we construct a model following that of Hopfield,
whose energy function possesses both Hebbian spin glass and anti-ferromagnetic
terms. With the assistance of the replica method, we find that the memory state
of the network decreases enormously due to the effect of the anti-ferromagnetic
order induced by the inhibitory connections. We conclude that the ability of
the plant as a memory device is rather weak.Comment: To be pulished in Physica A (Proc. STATPHYS-KOLKATA V), 9 pages, 6
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