807 research outputs found
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
Symmetry Breaking Constraints: Recent Results
Symmetry is an important problem in many combinatorial problems. One way of
dealing with symmetry is to add constraints that eliminate symmetric solutions.
We survey recent results in this area, focusing especially on two common and
useful cases: symmetry breaking constraints for row and column symmetry, and
symmetry breaking constraints for eliminating value symmetryComment: To appear in Proceedings of Twenty-Sixth Conference on Artificial
Intelligence (AAAI-12
Generation of All Possible Multiselections from a Multiset
The concept of a [k1, k2,..., kK]-selection applied on a multiset is introduced and an algorithm is outlined to generate all [k1, k2,..., kK]-selections from a given multiset. Key words: Multiselection; Mutiset; Contingency matrix; Combinatorie
Subset-lex: did we miss an order?
We generalize a well-known algorithm for the generation of all subsets of a
set in lexicographic order with respect to the sets as lists of elements
(subset-lex order). We obtain algorithms for various combinatorial objects such
as the subsets of a multiset, compositions and partitions represented as lists
of parts, and for certain restricted growth strings. The algorithms are often
loopless and require at most one extra variable for the computation of the next
object. The performance of the algorithms is very competitive even when not
loopless. A Gray code corresponding to the subset-lex order and a Gray code for
compositions that was found during this work are described.Comment: Two obvious errors corrected (indicated by "Correction:" in the LaTeX
source
Encodings and Arithmetic Operations in P Systems
Following, we present in this paper various number encodings and operations over multisets. We obtain the most compact encoding and several other interesting
encodings and study their properties using elements of combinatorics over multisets. We
also construct P systems that implement their associated operations. We quantify the effect of adding order to a multiset thus obtaining a string, as going from encoding lengths
of the number n in base b and time complexities of operations of the order b
p
n to lengths
and complexities of order logbn
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