23,725 research outputs found
Classical simulation of measurement-based quantum computation on higher-genus surface-code states
We consider the efficiency of classically simulating measurement-based
quantum computation on surface-code states. We devise a method for calculating
the elements of the probability distribution for the classical output of the
quantum computation. The operational cost of this method is polynomial in the
size of the surface-code state, but in the worst case scales as in the
genus of the surface embedding the code. However, there are states in the
code space for which the simulation becomes efficient. In general, the
simulation cost is exponential in the entanglement contained in a certain
effective state, capturing the encoded state, the encoding and the local
post-measurement states. The same efficiencies hold, with additional
assumptions on the temporal order of measurements and on the tessellations of
the code surfaces, for the harder task of sampling from the distribution of the
computational output.Comment: 21 pages, 13 figure
Graph Theory and Networks in Biology
In this paper, we present a survey of the use of graph theoretical techniques
in Biology. In particular, we discuss recent work on identifying and modelling
the structure of bio-molecular networks, as well as the application of
centrality measures to interaction networks and research on the hierarchical
structure of such networks and network motifs. Work on the link between
structural network properties and dynamics is also described, with emphasis on
synchronization and disease propagation.Comment: 52 pages, 5 figures, Survey Pape
Uniqueness and non-uniqueness in percolation theory
This paper is an up-to-date introduction to the problem of uniqueness versus
non-uniqueness of infinite clusters for percolation on and,
more generally, on transitive graphs. For iid percolation on ,
uniqueness of the infinite cluster is a classical result, while on certain
other transitive graphs uniqueness may fail. Key properties of the graphs in
this context turn out to be amenability and nonamenability. The same problem is
considered for certain dependent percolation models -- most prominently the
Fortuin--Kasteleyn random-cluster model -- and in situations where the standard
connectivity notion is replaced by entanglement or rigidity. So-called
simultaneous uniqueness in couplings of percolation processes is also
considered. Some of the main results are proved in detail, while for others the
proofs are merely sketched, and for yet others they are omitted. Several open
problems are discussed.Comment: Published at http://dx.doi.org/10.1214/154957806000000096 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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