53,406 research outputs found
Optimal Summation and Integration by Deterministic, Randomized, and Quantum Algorithms
We survey old and new results about optimal algorithms for summation of
finite sequences and for integration of functions from Hoelder or Sobolev
spaces. First we discuss optimal deterministic and randomized algorithms. Then
we add a new aspect, which has not been covered before on conferences about
(quasi-) Monte Carlo methods: quantum computation. We give a short introduction
into this setting and present recent results of the authors on optimal quantum
algorithms for summation and integration. We discuss comparisons between the
three settings. The most interesting case for Monte Carlo and quantum
integration is that of moderate smoothness k and large dimension d which, in
fact, occurs in a number of important applied problems. In that case the
deterministic exponent is negligible, so the n^{-1/2} Monte Carlo and the
n^{-1} quantum speedup essentially constitute the entire convergence rate. We
observe that -- there is an exponential speed-up of quantum algorithms over
deterministic (classical) algorithms, if k/d tends to zero; -- there is a
(roughly) quadratic speed-up of quantum algorithms over randomized classical
algorithms, if k/d is small.Comment: 13 pages, contribution to the 4th International Conference on Monte
Carlo and Quasi-Monte Carlo Methods, Hong Kong 200
Turbulent mixing simulation via a quantum algorithm
Probability density function (PDF) methods have been very useful in describing many physical aspects of turbulent mixing. In applications of these methods, modeled PDF transport equations are commonly simulated via classical Monte Carlo techniques, which provide estimates of moments of the PDF at arbitrary accuracy. In this work, recently developed techniques in quantum computing and quantum enhanced measurements (quantum metrology) are used to construct a quantum algorithm that accelerates the computation of such estimates. This quantum algorithm provides a quadratic speedup over classical Monte Carlo methods in terms of the number of repetitions needed to achieve the desired precision. This paper illustrates the power of this algorithm by considering a binary scalar mixing process modeled by means of the coalescence/dispersion (C/D) closure. The equation is first simulated using classical Monte Carlo methods, where error estimates for the computation of central moments are provided. Then the quantum algorithm for this problem is simulated by sampling from the same probability distribution as that of the output of a quantum computer, and it is shown that significantly fewer resources are required to achieve the same precision. The results demonstrate potential applications of future quantum computers for simulation of turbulent mixing, and large classes of related problems
Quantum Complexity of Integration
It is known that quantum computers yield a speed-up for certain discrete
problems. Here we want to know whether quantum computers are useful for
continuous problems. We study the computation of the integral of functions from
the classical Hoelder classes with d variables. The optimal orders for the
complexity of deterministic and (general) randomized methods are known. We
obtain the respective optimal orders for quantum algorithms and also for
restricted Monte Carlo (only coin tossing instead of general random numbers).
To summarize the results one can say that (1) there is an exponential speed-up
of quantum algorithms over deterministic (classical) algorithms, if the
smoothness is small; (2) there is a (roughly) quadratic speed-up of quantum
algorithms over randomized classical methods, if the smoothness is small.Comment: 13 pages, some minor correction
On the efficient Monte Carlo implementation of path integrals
We demonstrate that the Levy-Ciesielski implementation of Lie-Trotter
products enjoys several properties that make it extremely suitable for
path-integral Monte Carlo simulations: fast computation of paths, fast Monte
Carlo sampling, and the ability to use different numbers of time slices for the
different degrees of freedom, commensurate with the quantum effects. It is
demonstrated that a Monte Carlo simulation for which particles or small groups
of variables are updated in a sequential fashion has a statistical efficiency
that is always comparable to or better than that of an all-particle or
all-variable update sampler. The sequential sampler results in significant
computational savings if updating a variable costs only a fraction of the cost
for updating all variables simultaneously or if the variables are independent.
In the Levy-Ciesielski representation, the path variables are grouped in a
small number of layers, with the variables from the same layer being
statistically independent. The superior performance of the fast sampling
algorithm is shown to be a consequence of these observations. Both mathematical
arguments and numerical simulations are employed in order to quantify the
computational advantages of the sequential sampler, the Levy-Ciesielski
implementation of path integrals, and the fast sampling algorithm.Comment: 14 pages, 3 figures; submitted to Phys. Rev.
Bridging lattice-scale physics and continuum field theory with quantum Monte Carlo simulations
We discuss designer Hamiltonians---lattice models tailored to be free from
sign problems ("de-signed") when simulated with quantum Monte Carlo methods but
which still host complex many-body states and quantum phase transitions of
interest in condensed matter physics. We focus on quantum spin systems in which
competing interactions lead to non-magnetic ground states. These states and the
associated quantum phase transitions can be studied in great detail, enabling
direct access to universal properties and connections with low-energy effective
quantum field theories. As specific examples, we discuss the transition from a
Neel antiferromagnet to either a uniform quantum paramagnet or a spontaneously
symmetry-broken valence-bond solid in SU(2) and SU(N) invariant spin models. We
also discuss anisotropic (XXZ) systems harboring topological Z2 spin liquids
and the XY* transition. We briefly review recent progress on quantum Monte
Carlo algorithms, including ground state projection in the valence-bond basis
and direct computation of the Renyi variants of the entanglement entropy.Comment: 23 pages, 10 figure
Stochastic series expansion method for quantum Ising models with arbitrary interactions
A quantum Monte Carlo algorithm for the transverse Ising model with arbitrary
short- or long-range interactions is presented. The algorithm is based on
sampling the diagonal matrix elements of the power series expansion of the
density matrix (stochastic series expansion), and avoids the interaction
summations necessary in conventional methods. In the case of long-range
interactions, the scaling of the computation time with the system size N is
therefore reduced from N^2 to Nln(N). The method is tested on a one-dimensional
ferromagnet in a transverse field, with interactions decaying as 1/r^2.Comment: 9 pages, 5 figure
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