8 research outputs found
Counting Euler Tours in Undirected Bounded Treewidth Graphs
We show that counting Euler tours in undirected bounded tree-width graphs is
tractable even in parallel - by proving a upper bound. This is in
stark contrast to #P-completeness of the same problem in general graphs.
Our main technical contribution is to show how (an instance of) dynamic
programming on bounded \emph{clique-width} graphs can be performed efficiently
in parallel. Thus we show that the sequential result of Espelage, Gurski and
Wanke for efficiently computing Hamiltonian paths in bounded clique-width
graphs can be adapted in the parallel setting to count the number of
Hamiltonian paths which in turn is a tool for counting the number of Euler
tours in bounded tree-width graphs. Our technique also yields parallel
algorithms for counting longest paths and bipartite perfect matchings in
bounded-clique width graphs.
While establishing that counting Euler tours in bounded tree-width graphs can
be computed by non-uniform monotone arithmetic circuits of polynomial degree
(which characterize ) is relatively easy, establishing a uniform
bound needs a careful use of polynomial interpolation.Comment: 17 pages; There was an error in the proof of the GapL upper bound
claimed in the previous version which has been subsequently remove
Counting single-qubit Clifford equivalent graph states is #P-Complete
Graph states, which include for example Bell states, GHZ states and cluster
states, form a well-known class of quantum states with applications ranging
from quantum networks to error-correction. Deciding whether two graph states
are equivalent up to single-qubit Clifford operations is known to be decidable
in polynomial time and have been studied both in the context of producing
certain required states in a quantum network but also in relation to stabilizer
codes. The reason for the latter this is that single-qubit Clifford equivalent
graph states exactly corresponds to equivalent stabilizer codes. We here
consider the computational complexity of, given a graph state |G>, counting the
number of graph states, single-qubit Clifford equivalent to |G>. We show that
this problem is #P-Complete. To prove our main result we make use of the notion
of isotropic systems in graph theory. We review the definition of isotropic
systems and point out their strong relation to graph states. We believe that
these isotropic systems can be useful beyond the results presented in this
paper.Comment: 10 pages, no figure
Counting single-qubit Clifford equivalent graph states is #ℙ-complete
Graph states, which include for example Bell states, GHZ states and cluster states, form a well-known class of quantum states with applications ranging from quantum networks to error-correction. Deciding whether two graph states are equivalent up to single-qubit Clifford operations is known to be decidable in polynomial time and have been studied both in the context of producing certain required states in a quantum network but also in relation to stabilizer codes. The reason for the latter this is that single-qubit Clifford equivalent graph states exactly corresponds to equivalent stabilizer codes. We here consider the computational complexity of, given a graph state |G>, counting the number of graph states, single-qubit Clifford equivalent to |G>. We show that this problem is #P-Complete. To prove our main result we make use of the notion of isotropic systems in graph theory. We review the definition of isotropic systems and point out their strong relation to graph states. We believe that these isotropic systems can be useful beyond the results presented in this paper