1,564 research outputs found
Quantum annealing with Jarzynski equality
We show a practical application of the Jarzynski equality in quantum
computation. Its implementation may open a way to solve combinatorial
optimization problems, minimization of a real single-valued function, cost
function, with many arguments. We consider to incorpolate the Jarzynski
equality into quantum annealing, which is one of the generic algorithms to
solve the combinatorial optimization problem. The ordinary quantum annealing
suffers from non-adiabatic transitions whose rate is characterized by the
minimum energy gap of the quantum system under
consideration. The quantum sweep speed is therefore restricted to be extremely
slow for the achievement to obtain a solution without relevant errors. However,
in our strategy shown in the present study, we find that such a difficulty
would not matter.Comment: 4 pages, to appear in Phys. Rev. Let
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
Edge Elimination in TSP Instances
The Traveling Salesman Problem is one of the best studied NP-hard problems in
combinatorial optimization. Powerful methods have been developed over the last
60 years to find optimum solutions to large TSP instances. The largest TSP
instance so far that has been solved optimally has 85,900 vertices. Its
solution required more than 136 years of total CPU time using the
branch-and-cut based Concorde TSP code [1]. In this paper we present graph
theoretic results that allow to prove that some edges of a TSP instance cannot
occur in any optimum TSP tour. Based on these results we propose a
combinatorial algorithm to identify such edges. The runtime of the main part of
our algorithm is for an n-vertex TSP instance. By combining our
approach with the Concorde TSP solver we are able to solve a large TSPLIB
instance more than 11 times faster than Concorde alone
First order phase transition in the Quantum Adiabatic Algorithm
We simulate the quantum adiabatic algorithm (QAA) for the exact cover problem
for sizes up to N=256 using quantum Monte Carlo simulations incorporating
parallel tempering. At large N we find that some instances have a discontinuous
(first order) quantum phase transition during the evolution of the QAA. This
fraction increases with increasing N and may tend to 1 for N -> infinity.Comment: 5 pages, 3 figures. Replaced with published version; two figures
slightly changed and some small changes to the tex
On the Complexity of Local Search for Weighted Standard Set Problems
In this paper, we study the complexity of computing locally optimal solutions
for weighted versions of standard set problems such as SetCover, SetPacking,
and many more. For our investigation, we use the framework of PLS, as defined
in Johnson et al., [JPY88]. We show that for most of these problems, computing
a locally optimal solution is already PLS-complete for a simple neighborhood of
size one. For the local search versions of weighted SetPacking and SetCover, we
derive tight bounds for a simple neighborhood of size two. To the best of our
knowledge, these are one of the very few PLS results about local search for
weighted standard set problems
RNA secondary structure design
We consider the inverse-folding problem for RNA secondary structures: for a
given (pseudo-knot-free) secondary structure find a sequence that has that
structure as its ground state. If such a sequence exists, the structure is
called designable. We implemented a branch-and-bound algorithm that is able to
do an exhaustive search within the sequence space, i.e., gives an exact answer
whether such a sequence exists. The bound required by the branch-and-bound
algorithm are calculated by a dynamic programming algorithm. We consider
different alphabet sizes and an ensemble of random structures, which we want to
design. We find that for two letters almost none of these structures are
designable. The designability improves for the three-letter case, but still a
significant fraction of structures is undesignable. This changes when we look
at the natural four-letter case with two pairs of complementary bases:
undesignable structures are the exception, although they still exist. Finally,
we also study the relation between designability and the algorithmic complexity
of the branch-and-bound algorithm. Within the ensemble of structures, a high
average degree of undesignability is correlated to a long time to prove that a
given structure is (un-)designable. In the four-letter case, where the
designability is high everywhere, the algorithmic complexity is highest in the
region of naturally occurring RNA.Comment: 11 pages, 10 figure
Excitation Gap from Optimized Correlation Functions in Quantum Monte Carlo Simulations
We give a prescription for finding optimized correlation functions for the
extraction of the gap to the first excited state within quantum Monte Carlo
simulations. We demonstrate that optimized correlation functions provide a more
accurate reading of the gap when compared to other `non-optimized' correlation
functions and are generally characterized by considerably larger
signal-to-noise ratios. We also analyze the cost of the procedure and show that
it is not computationally demanding. We illustrate the effectiveness of the
proposed procedure by analyzing several exemplary many-body systems of
interacting spin-1/2 particles.Comment: 11 pages, 5 figure
Optimal Vertex Cover for the Small-World Hanoi Networks
The vertex-cover problem on the Hanoi networks HN3 and HN5 is analyzed with
an exact renormalization group and parallel-tempering Monte Carlo simulations.
The grand canonical partition function of the equivalent hard-core repulsive
lattice-gas problem is recast first as an Ising-like canonical partition
function, which allows for a closed set of renormalization group equations. The
flow of these equations is analyzed for the limit of infinite chemical
potential, at which the vertex-cover problem is attained. The relevant fixed
point and its neighborhood are analyzed, and non-trivial results are obtained
both, for the coverage as well as for the ground state entropy density, which
indicates the complex structure of the solution space. Using special
hierarchy-dependent operators in the renormalization group and Monte-Carlo
simulations, structural details of optimal configurations are revealed. These
studies indicate that the optimal coverages (or packings) are not related by a
simple symmetry. Using a clustering analysis of the solutions obtained in the
Monte Carlo simulations, a complex solution space structure is revealed for
each system size. Nevertheless, in the thermodynamic limit, the solution
landscape is dominated by one huge set of very similar solutions.Comment: RevTex, 24 pages; many corrections in text and figures; final
version; for related information, see
http://www.physics.emory.edu/faculty/boettcher
Reduction of Two-Dimensional Dilute Ising Spin Glasses
The recently proposed reduction method is applied to the Edwards-Anderson
model on bond-diluted square lattices. This allows, in combination with a
graph-theoretical matching algorithm, to calculate numerically exact ground
states of large systems. Low-temperature domain-wall excitations are studied to
determine the stiffness exponent y_2. A value of y_2=-0.281(3) is found,
consistent with previous results obtained on undiluted lattices. This
comparison demonstrates the validity of the reduction method for bond-diluted
spin systems and provides strong support for similar studies proclaiming
accurate results for stiffness exponents in dimensions d=3,...,7.Comment: 7 pages, RevTex4, 6 ps-figures included, for related information, see
http://www.physics.emory.edu/faculty/boettcher
Coloring random graphs
We study the graph coloring problem over random graphs of finite average
connectivity . Given a number of available colors, we find that graphs
with low connectivity admit almost always a proper coloring whereas graphs with
high connectivity are uncolorable. Depending on , we find the precise value
of the critical average connectivity . Moreover, we show that below
there exist a clustering phase in which ground states
spontaneously divide into an exponential number of clusters and where the
proliferation of metastable states is responsible for the onset of complexity
in local search algorithms.Comment: 4 pages, 1 figure, version to app. in PR
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