743 research outputs found
Boson Sampling as Canonical Transformation: A semiclassical approach in Fock space
We show that a theory of complex scattering between many-body (Fock) states
can be constructed such that its classical limit is a canonical transformation
thus encoding quantum interference in the semiclassical form of the associated
unitary operator. Based on this idea, we study the different coherent effects
expected under different choices of the many-body states and provide different
representations of the associated transition probabilities. In this way, we
derive exact relations and representations of the scattering process that can
be used to attack timely problems related with Boson Sampling.Comment: submitted to Ann. Phy
The Classical Complexity of Boson Sampling
We study the classical complexity of the exact Boson Sampling problem where
the objective is to produce provably correct random samples from a particular
quantum mechanical distribution. The computational framework was proposed by
Aaronson and Arkhipov in 2011 as an attainable demonstration of `quantum
supremacy', that is a practical quantum computing experiment able to produce
output at a speed beyond the reach of classical (that is non-quantum) computer
hardware. Since its introduction Boson Sampling has been the subject of intense
international research in the world of quantum computing. On the face of it,
the problem is challenging for classical computation. Aaronson and Arkhipov
show that exact Boson Sampling is not efficiently solvable by a classical
computer unless and the polynomial hierarchy collapses to
the third level.
The fastest known exact classical algorithm for the standard Boson Sampling
problem takes time to produce samples for a
system with input size and output modes, making it infeasible for
anything but the smallest values of and . We give an algorithm that is
much faster, running in time and
additional space. The algorithm is simple to implement and has low constant
factor overheads. As a consequence our classical algorithm is able to solve the
exact Boson Sampling problem for system sizes far beyond current photonic
quantum computing experimentation, thereby significantly reducing the
likelihood of achieving near-term quantum supremacy in the context of Boson
Sampling.Comment: 15 pages. To appear in SODA '1
Approximating the Permanent with Fractional Belief Propagation
We discuss schemes for exact and approximate computations of permanents, and
compare them with each other. Specifically, we analyze the Belief Propagation
(BP) approach and its Fractional Belief Propagation (FBP) generalization for
computing the permanent of a non-negative matrix. Known bounds and conjectures
are verified in experiments, and some new theoretical relations, bounds and
conjectures are proposed. The Fractional Free Energy (FFE) functional is
parameterized by a scalar parameter , where
corresponds to the BP limit and corresponds to the exclusion
principle (but ignoring perfect matching constraints) Mean-Field (MF) limit.
FFE shows monotonicity and continuity with respect to . For every
non-negative matrix, we define its special value to be the
for which the minimum of the -parameterized FFE functional is
equal to the permanent of the matrix, where the lower and upper bounds of the
-interval corresponds to respective bounds for the permanent. Our
experimental analysis suggests that the distribution of varies for
different ensembles but always lies within the interval.
Moreover, for all ensembles considered the behavior of is highly
distinctive, offering an emprirical practical guidance for estimating
permanents of non-negative matrices via the FFE approach.Comment: 42 pages, 14 figure
Bounds on the permanent and some applications
We give new lower and upper bounds on the permanent of a doubly stochastic
matrix. Combined with previous work, this improves on the deterministic
approximation factor for the permanent.
We also give a combinatorial application of the lower bound, proving S.
Friedland's "Asymptotic Lower Matching Conjecture" for the monomer-dimer
problem
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