57,103 research outputs found
Calculation of Densities of States and Spectral Functions by Chebyshev Recursion and Maximum Entropy
We present an efficient algorithm for calculating spectral properties of
large sparse Hamiltonian matrices such as densities of states and spectral
functions. The combination of Chebyshev recursion and maximum entropy achieves
high energy resolution without significant roundoff error, machine precision or
numerical instability limitations. If controlled statistical or systematic
errors are acceptable, cpu and memory requirements scale linearly in the number
of states. The inference of spectral properties from moments is much better
conditioned for Chebyshev moments than for power moments. We adapt concepts
from the kernel polynomial approximation, a linear Chebyshev approximation with
optimized Gibbs damping, to control the accuracy of Fourier integrals of
positive non-analytic functions. We compare the performance of kernel
polynomial and maximum entropy algorithms for an electronic structure example.Comment: 8 pages RevTex, 3 postscript figure
Regimes of classical simulability for noisy Gaussian boson sampling
As a promising candidate for exhibiting quantum computational supremacy,
Gaussian Boson Sampling (GBS) is designed to exploit the ease of experimental
preparation of Gaussian states. However, sufficiently large and inevitable
experimental noise might render GBS classically simulable. In this work, we
formalize this intuition by establishing a sufficient condition for approximate
polynomial-time classical simulation of noisy GBS --- in the form of an
inequality between the input squeezing parameter, the overall transmission rate
and the quality of photon detectors. Our result serves as a non-classicality
test that must be passed by any quantum computationalsupremacy demonstration
based on GBS. We show that, for most linear-optical architectures, where photon
loss increases exponentially with the circuit depth, noisy GBS loses its
quantum advantage in the asymptotic limit. Our results thus delineate
intermediate-sized regimes where GBS devices might considerably outperform
classical computers for modest noise levels. Finally, we find that increasing
the amount of input squeezing is helpful to evade our classical simulation
algorithm, which suggests a potential route to mitigate photon loss.Comment: 13 pages, 4 figures, final version accepted for publication in
Physical Review Letter
Quantifying Privacy: A Novel Entropy-Based Measure of Disclosure Risk
It is well recognised that data mining and statistical analysis pose a
serious treat to privacy. This is true for financial, medical, criminal and
marketing research. Numerous techniques have been proposed to protect privacy,
including restriction and data modification. Recently proposed privacy models
such as differential privacy and k-anonymity received a lot of attention and
for the latter there are now several improvements of the original scheme, each
removing some security shortcomings of the previous one. However, the challenge
lies in evaluating and comparing privacy provided by various techniques. In
this paper we propose a novel entropy based security measure that can be
applied to any generalisation, restriction or data modification technique. We
use our measure to empirically evaluate and compare a few popular methods,
namely query restriction, sampling and noise addition.Comment: 20 pages, 4 figure
Entropy of unimodular Lattice Triangulations
Triangulations are important objects of study in combinatorics, finite
element simulations and quantum gravity, where its entropy is crucial for many
physical properties. Due to their inherent complex topological structure even
the number of possible triangulations is unknown for large systems. We present
a novel algorithm for an approximate enumeration which is based on calculations
of the density of states using the Wang-Landau flat histogram sampling. For
triangulations on two-dimensional integer lattices we achive excellent
agreement with known exact numbers of small triangulations as well as an
improvement of analytical calculated asymptotics. The entropy density is
consistent with rigorous upper and lower bounds. The presented
numerical scheme can easily be applied to other counting and optimization
problems.Comment: 6 pages, 7 figure
The number of matchings in random graphs
We study matchings on sparse random graphs by means of the cavity method. We
first show how the method reproduces several known results about maximum and
perfect matchings in regular and Erdos-Renyi random graphs. Our main new result
is the computation of the entropy, i.e. the leading order of the logarithm of
the number of solutions, of matchings with a given size. We derive both an
algorithm to compute this entropy for an arbitrary graph with a girth that
diverges in the large size limit, and an analytic result for the entropy in
regular and Erdos-Renyi random graph ensembles.Comment: 17 pages, 6 figures, to be published in Journal of Statistical
Mechanic
Observations Outside the Light-Cone: Algorithms for Non-Equilibrium and Thermal States
We apply algorithms based on Lieb-Robinson bounds to simulate time-dependent
and thermal quantities in quantum systems. For time-dependent systems, we
modify a previous mapping to quantum circuits to significantly reduce the
computer resources required. This modification is based on a principle of
"observing" the system outside the light-cone. We apply this method to study
spin relaxation in systems started out of equilibrium with initial conditions
that give rise to very rapid entanglement growth. We also show that it is
possible to approximate time evolution under a local Hamiltonian by a quantum
circuit whose light-cone naturally matches the Lieb-Robinson velocity.
Asymptotically, these modified methods allow a doubling of the system size that
one can obtain compared to direct simulation. We then consider a different
problem of thermal properties of disordered spin chains and use quantum belief
propagation to average over different configurations. We test this algorithm on
one dimensional systems with mixed ferromagnetic and anti-ferromagnetic bonds,
where we can compare to quantum Monte Carlo, and then we apply it to the study
of disordered, frustrated spin systems.Comment: 19 pages, 12 figure
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