21,424 research outputs found
Forgetting the starting distribution in finite interacting tempering
Markov chain Monte Carlo (MCMC) methods are frequently used to approximately
simulate high-dimensional, multimodal probability distributions. In adaptive
MCMC methods, the transition kernel is changed "on the fly" in the hope to
speed up convergence. We study interacting tempering, an adaptive MCMC
algorithm based on interacting Markov chains, that can be seen as a simplified
version of the equi-energy sampler. Using a coupling argument, we show that
under easy to verify assumptions on the target distribution (on a finite
space), the interacting tempering process rapidly forgets its starting
distribution. The result applies, among others, to exponential random graph
models, the Ising and Potts models (in mean field or on a bounded degree
graph), as well as (Edwards-Anderson) Ising spin glasses. As a cautionary note,
we also exhibit an example of a target distribution for which the interacting
tempering process rapidly forgets its starting distribution, but takes an
exponential number of steps (in the dimension of the state space) to converge
to its limiting distribution. As a consequence, we argue that convergence
diagnostics that are based on demonstrating that the process has forgotten its
starting distribution might be of limited use for adaptive MCMC algorithms like
interacting tempering
Network Community Detection On Small Quantum Computers
In recent years a number of quantum computing devices with small numbers of
qubits became available. We present a hybrid quantum local search (QLS)
approach that combines a classical machine and a small quantum device to solve
problems of practical size. The proposed approach is applied to the network
community detection problem. QLS is hardware-agnostic and easily extendable to
new quantum computing devices as they become available. We demonstrate it to
solve the 2-community detection problem on graphs of size up to 410 vertices
using the 16-qubit IBM quantum computer and D-Wave 2000Q, and compare their
performance with the optimal solutions. Our results demonstrate that QLS
perform similarly in terms of quality of the solution and the number of
iterations to convergence on both types of quantum computers and it is capable
of achieving results comparable to state-of-the-art solvers in terms of quality
of the solution including reaching the optimal solutions
Demonstration of a scaling advantage for a quantum annealer over simulated annealing
The observation of an unequivocal quantum speedup remains an elusive
objective for quantum computing. The D-Wave quantum annealing processors have
been at the forefront of experimental attempts to address this goal, given
their relatively large numbers of qubits and programmability. A complete
determination of the optimal time-to-solution (TTS) using these processors has
not been possible to date, preventing definitive conclusions about the presence
of a scaling advantage. The main technical obstacle has been the inability to
verify an optimal annealing time within the available range. Here we overcome
this obstacle and present a class of problem instances for which we observe an
optimal annealing time using a D-Wave 2000Q processor over a range spanning up
to more than qubits. This allows us to perform an optimal TTS
benchmarking analysis and perform a comparison to several classical algorithms,
including simulated annealing, spin-vector Monte Carlo, and discrete-time
simulated quantum annealing. We establish the first example of a scaling
advantage for an experimental quantum annealer over classical simulated
annealing: we find that the D-Wave device exhibits certifiably better scaling
than simulated annealing, with confidence, over the range of problem
sizes that we can test. However, we do not find evidence for a quantum speedup:
simulated quantum annealing exhibits the best scaling by a significant margin.
Our construction of instance classes with verifiably optimal annealing times
opens up the possibility of generating many new such classes, paving the way
for further definitive assessments of scaling advantages using current and
future quantum annealing devices.Comment: 26 pages, 22 figures. v2: Updated benchmarking results with
additional analysis. v3: Updated to published versio
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