4,144 research outputs found
Minimum cycle and homology bases of surface embedded graphs
We study the problems of finding a minimum cycle basis (a minimum weight set
of cycles that form a basis for the cycle space) and a minimum homology basis
(a minimum weight set of cycles that generates the -dimensional
()-homology classes) of an undirected graph embedded on a
surface. The problems are closely related, because the minimum cycle basis of a
graph contains its minimum homology basis, and the minimum homology basis of
the -skeleton of any graph is exactly its minimum cycle basis.
For the minimum cycle basis problem, we give a deterministic
-time algorithm for graphs embedded on an orientable
surface of genus . The best known existing algorithms for surface embedded
graphs are those for general graphs: an time Monte Carlo
algorithm and a deterministic time algorithm. For the
minimum homology basis problem, we give a deterministic -time algorithm for graphs embedded on an orientable or non-orientable
surface of genus with boundary components, assuming shortest paths are
unique, improving on existing algorithms for many values of and . The
assumption of unique shortest paths can be avoided with high probability using
randomization or deterministically by increasing the running time of the
homology basis algorithm by a factor of .Comment: A preliminary version of this work was presented at the 32nd Annual
International Symposium on Computational Geometr
All-Pairs Minimum Cuts in Near-Linear Time for Surface-Embedded Graphs
For an undirected -vertex graph with non-negative edge-weights, we
consider the following type of query: given two vertices and in ,
what is the weight of a minimum -cut in ? We solve this problem in
preprocessing time for graphs of bounded genus, giving the first
sub-quadratic time algorithm for this class of graphs. Our result also improves
by a logarithmic factor a previous algorithm by Borradaile, Sankowski and
Wulff-Nilsen (FOCS 2010) that applied only to planar graphs. Our algorithm
constructs a Gomory-Hu tree for the given graph, providing a data structure
with space that can answer minimum-cut queries in constant time. The
dependence on the genus of the input graph in our preprocessing time is
Homological Error Correction: Classical and Quantum Codes
We prove several theorems characterizing the existence of homological error
correction codes both classically and quantumly. Not every classical code is
homological, but we find a family of classical homological codes saturating the
Hamming bound. In the quantum case, we show that for non-orientable surfaces it
is impossible to construct homological codes based on qudits of dimension
, while for orientable surfaces with boundaries it is possible to
construct them for arbitrary dimension . We give a method to obtain planar
homological codes based on the construction of quantum codes on compact
surfaces without boundaries. We show how the original Shor's 9-qubit code can
be visualized as a homological quantum code. We study the problem of
constructing quantum codes with optimal encoding rate. In the particular case
of toric codes we construct an optimal family and give an explicit proof of its
optimality. For homological quantum codes on surfaces of arbitrary genus we
also construct a family of codes asymptotically attaining the maximum possible
encoding rate. We provide the tools of homology group theory for graphs
embedded on surfaces in a self-contained manner.Comment: Revtex4 fil
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
A Turaev surface approach to Khovanov homology
We introduce Khovanov homology for ribbon graphs and show that the Khovanov
homology of a certain ribbon graph embedded on the Turaev surface of a link is
isomorphic to the Khovanov homology of the link (after a grading shift). We
also present a spanning quasi-tree model for the Khovanov homology of a ribbon
graph.Comment: 30 pages, 18 figures, added sections on virtual links and
Reidemeister move
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