8,726 research outputs found
Self-Learning Monte Carlo Method: Continuous-Time Algorithm
The recently-introduced self-learning Monte Carlo method is a general-purpose
numerical method that speeds up Monte Carlo simulations by training an
effective model to propose uncorrelated configurations in the Markov chain. We
implement this method in the framework of continuous time Monte Carlo method
with auxiliary field in quantum impurity models. We introduce and train a
diagram generating function (DGF) to model the probability distribution of
auxiliary field configurations in continuous imaginary time, at all orders of
diagrammatic expansion. By using DGF to propose global moves in configuration
space, we show that the self-learning continuous-time Monte Carlo method can
significantly reduce the computational complexity of the simulation.Comment: 6 pages, 5 figures + 2 page supplemental materials, to be published
in Phys. Rev. B Rapid communication sectio
Quantum algorithm for approximating partition functions
We present a quantum algorithm based on classical fully polynomial randomized approximation schemes (FPRASs) for estimating partition functions that combine simulated annealing with the Monte Carlo Markov chain method and use nonadaptive cooling schedules. We achieve a twofold polynomial improvement in time complexity: a quadratic reduction with respect to the spectral gap of the underlying Markov chains and a quadratic reduction with respect to the parameter characterizing the desired accuracy of the estimate output by the FPRAS. Both reductions are intimately related and cannot be achieved separately. First, we use Grover\u27s fixed-point search, quantum walks, and phase estimation to efficiently prepare approximate coherent encodings of stationary distributions of the Markov chains. The speed up we obtain in this way is due to the quadratic relation between the spectral and phase gaps of classical and quantum walks. The second speed up with respect to accuracy comes from generalized quantum counting used instead of classical sampling to estimate expected values of quantum observables
Quantum Speed-up for Approximating Partition Functions
We achieve a quantum speed-up of fully polynomial randomized approximation
schemes (FPRAS) for estimating partition functions that combine simulated
annealing with the Monte-Carlo Markov Chain method and use non-adaptive cooling
schedules. The improvement in time complexity is twofold: a quadratic reduction
with respect to the spectral gap of the underlying Markov chains and a
quadratic reduction with respect to the parameter characterizing the desired
accuracy of the estimate output by the FPRAS. Both reductions are intimately
related and cannot be achieved separately.
First, we use Grover's fixed point search, quantum walks and phase estimation
to efficiently prepare approximate coherent encodings of stationary
distributions of the Markov chains. The speed-up we obtain in this way is due
to the quadratic relation between the spectral and phase gaps of classical and
quantum walks. Second, we generalize the method of quantum counting, showing
how to estimate expected values of quantum observables. Using this method
instead of classical sampling, we obtain the speed-up with respect to accuracy.Comment: 17 pages; v3: corrected typos, added a reference about efficient
implementations of quantum walk
Some Results on the Complexity of Numerical Integration
This is a survey (21 pages, 124 references) written for the MCQMC 2014
conference in Leuven, April 2014. We start with the seminal paper of Bakhvalov
(1959) and end with new results on the curse of dimension and on the complexity
of oscillatory integrals. Some small errors of earlier versions are corrected
Monte Carlo techniques for real-time quantum dynamics
The stochastic-gauge representation is a method of mapping the equation of
motion for the quantum mechanical density operator onto a set of equivalent
stochastic differential equations. One of the stochastic variables is termed
the "weight", and its magnitude is related to the importance of the stochastic
trajectory. We investigate the use of Monte Carlo algorithms to improve the
sampling of the weighted trajectories and thus reduce sampling error in a
simulation of quantum dynamics. The method can be applied to calculations in
real time, as well as imaginary time for which Monte Carlo algorithms are
more-commonly used. The method is applicable when the weight is guaranteed to
be real, and we demonstrate how to ensure this is the case. Examples are given
for the anharmonic oscillator, where large improvements over stochastic
sampling are observed.Comment: 28 pages, submitted to J. Comp. Phy
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