1,652 research outputs found
Taming a non-convex landscape with dynamical long-range order: memcomputing Ising benchmarks
Recent work on quantum annealing has emphasized the role of collective
behavior in solving optimization problems. By enabling transitions of clusters
of variables, such solvers are able to navigate their state space and locate
solutions more efficiently despite having only local connections between
elements. However, collective behavior is not exclusive to quantum annealers,
and classical solvers that display collective dynamics should also possess an
advantage in navigating a non-convex landscape. Here, we give evidence that a
benchmark derived from quantum annealing studies is solvable in polynomial time
using digital memcomputing machines, which utilize a collection of dynamical
components with memory to represent the structure of the underlying
optimization problem. To illustrate the role of memory and clarify the
structure of these solvers we propose a simple model of these machines that
demonstrates the emergence of long-range order. This model, when applied to
finding the ground state of the Ising frustrated-loop benchmarks, undergoes a
transient phase of avalanches which can span the entire lattice and
demonstrates a connection between long-range behavior and their probability of
success. These results establish the advantages of computational approaches
based on collective dynamics of continuous dynamical systems
What is the Computational Value of Finite Range Tunneling?
Quantum annealing (QA) has been proposed as a quantum enhanced optimization
heuristic exploiting tunneling. Here, we demonstrate how finite range tunneling
can provide considerable computational advantage. For a crafted problem
designed to have tall and narrow energy barriers separating local minima, the
D-Wave 2X quantum annealer achieves significant runtime advantages relative to
Simulated Annealing (SA). For instances with 945 variables, this results in a
time-to-99%-success-probability that is times faster than SA
running on a single processor core. We also compared physical QA with Quantum
Monte Carlo (QMC), an algorithm that emulates quantum tunneling on classical
processors. We observe a substantial constant overhead against physical QA:
D-Wave 2X again runs up to times faster than an optimized
implementation of QMC on a single core. We note that there exist heuristic
classical algorithms that can solve most instances of Chimera structured
problems in a timescale comparable to the D-Wave 2X. However, we believe that
such solvers will become ineffective for the next generation of annealers
currently being designed. To investigate whether finite range tunneling will
also confer an advantage for problems of practical interest, we conduct
numerical studies on binary optimization problems that cannot yet be
represented on quantum hardware. For random instances of the number
partitioning problem, we find numerically that QMC, as well as other algorithms
designed to simulate QA, scale better than SA. We discuss the implications of
these findings for the design of next generation quantum annealers.Comment: 17 pages, 13 figures. Edited for clarity, in part in response to
comments. Added link to benchmark instance
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
Inference of hidden structures in complex physical systems by multi-scale clustering
We survey the application of a relatively new branch of statistical
physics--"community detection"-- to data mining. In particular, we focus on the
diagnosis of materials and automated image segmentation. Community detection
describes the quest of partitioning a complex system involving many elements
into optimally decoupled subsets or communities of such elements. We review a
multiresolution variant which is used to ascertain structures at different
spatial and temporal scales. Significant patterns are obtained by examining the
correlations between different independent solvers. Similar to other
combinatorial optimization problems in the NP complexity class, community
detection exhibits several phases. Typically, illuminating orders are revealed
by choosing parameters that lead to extremal information theory correlations.Comment: 25 pages, 16 Figures; a review of earlier work
The Quantum Adiabatic Algorithm applied to random optimization problems: the quantum spin glass perspective
Among various algorithms designed to exploit the specific properties of
quantum computers with respect to classical ones, the quantum adiabatic
algorithm is a versatile proposition to find the minimal value of an arbitrary
cost function (ground state energy). Random optimization problems provide a
natural testbed to compare its efficiency with that of classical algorithms.
These problems correspond to mean field spin glasses that have been extensively
studied in the classical case. This paper reviews recent analytical works that
extended these studies to incorporate the effect of quantum fluctuations, and
presents also some original results in this direction.Comment: 151 pages, 21 figure
Cut Size Statistics of Graph Bisection Heuristics
We investigate the statistical properties of cut sizes generated by heuristic
algorithms which solve approximately the graph bisection problem. On an
ensemble of sparse random graphs, we find empirically that the distribution of
the cut sizes found by ``local'' algorithms becomes peaked as the number of
vertices in the graphs becomes large. Evidence is given that this distribution
tends towards a Gaussian whose mean and variance scales linearly with the
number of vertices of the graphs. Given the distribution of cut sizes
associated with each heuristic, we provide a ranking procedure which takes into
account both the quality of the solutions and the speed of the algorithms. This
procedure is demonstrated for a selection of local graph bisection heuristics.Comment: 17 pages, 5 figures, submitted to SIAM Journal on Optimization also
available at http://ipnweb.in2p3.fr/~martin
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