618,454 research outputs found
Complexity of Graph State Preparation
The graph state formalism is a useful abstraction of entanglement. It is used
in some multipartite purification schemes and it adequately represents
universal resources for measurement-only quantum computation. We focus in this
paper on the complexity of graph state preparation. We consider the number of
ancillary qubits, the size of the primitive operators, and the duration of
preparation. For each lexicographic order over these parameters we give upper
and lower bounds for the complexity of graph state preparation. The first part
motivates our work and introduces basic notions and notations for the study of
graph states. Then we study some graph properties of graph states,
characterizing their minimal degree by local unitary transformations, we
propose an algorithm to reduce the degree of a graph state, and show the
relationship with Sutner sigma-game.
These properties are used in the last part, where algorithms and lower bounds
for each lexicographic order over the considered parameters are presented.Comment: 17 page
Neural complexity: a graph theoretic interpretation
One of the central challenges facing modern neuroscience is to explain the ability of the nervous system to coherently integrate information across distinct functional modules in the absence of a central executive. To this end Tononi et al. [Proc. Nat. Acad. Sci. USA 91, 5033 (1994)] proposed a measure of neural complexity that purports to capture this property based on mutual information between complementary subsets of a system. Neural complexity, so defined, is one of a family of information theoretic metrics developed to measure the balance between the segregation and integration of a system's dynamics. One key question arising for such measures involves understanding how they are influenced by network topology. Sporns et al. [Cereb. Cortex 10, 127 (2000)] employed numerical models in order to determine the dependence of neural complexity on the topological features of a network. However, a complete picture has yet to be established. While De Lucia et al. [Phys. Rev. E 71, 016114 (2005)] made the first attempts at an analytical account of this relationship, their work utilized a formulation of neural complexity that, we argue, did not reflect the intuitions of the original work. In this paper we start by describing weighted connection matrices formed by applying a random continuous weight distribution to binary adjacency matrices. This allows us to derive an approximation for neural complexity in terms of the moments of the weight distribution and elementary graph motifs. In particular we explicitly establish a dependency of neural complexity on cyclic graph motifs
Graph sharing games: complexity and connectivity
We study the following combinatorial game played by two players, Alice and
Bob, which generalizes the Pizza game considered by Brown, Winkler and others.
Given a connected graph G with nonnegative weights assigned to its vertices,
the players alternately take one vertex of G in each turn. The first turn is
Alice's. The vertices are to be taken according to one (or both) of the
following two rules: (T) the subgraph of G induced by the taken vertices is
connected during the whole game, (R) the subgraph of G induced by the remaining
vertices is connected during the whole game. We show that if rules (T) and/or
(R) are required then for every epsilon > 0 and for every positive integer k
there is a k-connected graph G for which Bob has a strategy to obtain
(1-epsilon) of the total weight of the vertices. This contrasts with the
original Pizza game played on a cycle, where Alice is known to have a strategy
to obtain 4/9 of the total weight.
We show that the problem of deciding whether Alice has a winning strategy
(i.e., a strategy to obtain more than half of the total weight) is
PSPACE-complete if condition (R) or both conditions (T) and (R) are required.
We also consider a game played on connected graphs (without weights) where the
first player who violates condition (T) or (R) loses the game. We show that
deciding who has the winning strategy is PSPACE-complete.Comment: 22 pages, 11 figures; updated references, minor stylistical change
Parameterized Complexity of Graph Constraint Logic
Graph constraint logic is a framework introduced by Hearn and Demaine, which
provides several problems that are often a convenient starting point for
reductions. We study the parameterized complexity of Constraint Graph
Satisfiability and both bounded and unbounded versions of Nondeterministic
Constraint Logic (NCL) with respect to solution length, treewidth and maximum
degree of the underlying constraint graph as parameters. As a main result we
show that restricted NCL remains PSPACE-complete on graphs of bounded
bandwidth, strengthening Hearn and Demaine's framework. This allows us to
improve upon existing results obtained by reduction from NCL. We show that
reconfiguration versions of several classical graph problems (including
independent set, feedback vertex set and dominating set) are PSPACE-complete on
planar graphs of bounded bandwidth and that Rush Hour, generalized to boards, is PSPACE-complete even when is at most a constant
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