13 research outputs found
A faster hafnian formula for complex matrices and its benchmarking on a supercomputer
We introduce new and simple algorithms for the calculation of the number of
perfect matchings of complex weighted, undirected graphs with and without
loops. Our compact formulas for the hafnian and loop hafnian of
complex matrices run in time, are embarrassingly
parallelizable and, to the best of our knowledge, are the fastest exact
algorithms to compute these quantities. Despite our highly optimized algorithm,
numerical benchmarks on the Titan supercomputer with matrices up to size indicate that one would require the 288000 CPUs of this machine for
about a month and a half to compute the hafnian of a matrix.Comment: 11 pages, 7 figures. The source code of the library is available at
https://github.com/XanaduAI/hafnian . Accepted for publication in Journal of
Experimental Algorithmic
Gaussian Boson Sampling using threshold detectors
We study what is arguably the most experimentally appealing Boson Sampling
architecture: Gaussian states sampled with threshold detectors. We show that in
this setting, the probability of observing a given outcome is related to a
matrix function that we name the Torontonian, which plays an analogous role to
the permanent or the Hafnian in other models. We also prove that, provided that
the probability of observing two or more photons in a single output mode is
sufficiently small, our model remains intractable to simulate classically under
standard complexity-theoretic conjectures. Finally, we leverage the
mathematical simplicity of the model to introduce a physically motivated, exact
sampling algorithm for all Boson Sampling models that employ Gaussian states
and threshold detectors.Comment: 5+5 pages, 2 figures. Closer to published versio
Franck-Condon factors by counting perfect matchings of graphs with loops
We show that the Franck-Condon Factor (FCF) associated to a transition
between initial and final vibrational states in two different potential energy
surfaces, having and vibrational quanta, respectively, is equivalent to
calculating the number of perfect matchings of a weighted graph with loops that
has vertices. This last quantity is the loop hafnian of the
(symmetric) adjacency matrix of the graph which can be calculated in steps. In the limit of small numbers of vibrational quanta per normal
mode our loop hafnian formula significantly improves the speed at which FCFs
can be calculated. Our results more generally apply to the calculation of the
matrix elements of a bosonic Gaussian unitary between two multimode Fock states
having and photons in total and provide a useful link between certain
calculations of quantum chemistry, quantum optics and graph theory.Comment: 13+3 pages, 4 figures. Source code available at
https://github.com/XanaduAI/fockgaussia