639 research outputs found
On prisms, M\"obius ladders and the cycle space of dense graphs
For a graph X, let f_0(X) denote its number of vertices, d(X) its minimum
degree and Z_1(X;Z/2) its cycle space in the standard graph-theoretical sense
(i.e. 1-dimensional cycle group in the sense of simplicial homology theory with
Z/2-coefficients). Call a graph Hamilton-generated if and only if the set of
all Hamilton circuits is a Z/2-generating system for Z_1(X;Z/2). The main
purpose of this paper is to prove the following: for every s > 0 there exists
n_0 such that for every graph X with f_0(X) >= n_0 vertices, (1) if d(X) >=
(1/2 + s) f_0(X) and f_0(X) is odd, then X is Hamilton-generated, (2) if d(X)
>= (1/2 + s) f_0(X) and f_0(X) is even, then the set of all Hamilton circuits
of X generates a codimension-one subspace of Z_1(X;Z/2), and the set of all
circuits of X having length either f_0(X)-1 or f_0(X) generates all of
Z_1(X;Z/2), (3) if d(X) >= (1/4 + s) f_0(X) and X is square bipartite, then X
is Hamilton-generated. All these degree-conditions are essentially
best-possible. The implications in (1) and (2) give an asymptotic affirmative
answer to a special case of an open conjecture which according to [European J.
Combin. 4 (1983), no. 3, p. 246] originates with A. Bondy.Comment: 33 pages; 5 figure
3nj Morphogenesis and Semiclassical Disentangling
Recoupling coefficients (3nj symbols) are unitary transformations between
binary coupled eigenstates of N=(n+1) mutually commuting SU(2) angular momentum
operators. They have been used in a variety of applications in spectroscopy,
quantum chemistry and nuclear physics and quite recently also in quantum
gravity and quantum computing. These coefficients, naturally associated to
cubic Yutsis graphs, share a number of intriguing combinatorial, algebraic, and
analytical features that make them fashinating objects to be studied on their
own. In this paper we develop a bottom--up, systematic procedure for the
generation of 3nj from 3(n-1)j diagrams by resorting to diagrammatical and
algebraic methods. We provide also a novel approach to the problem of
classifying various regimes of semiclassical expansions of 3nj coefficients
(asymptotic disentangling of 3nj diagrams) for n > 2 by means of combinatorial,
analytical and numerical tools
Traversing Every Edge in Each Direction Once, But Not at Once: Cubic (Polyhedral) Graphs
A {\em retracting-free bidirectional circuit} in a graph is a closed walk which traverses every edge exactly once in each direction and such that no edge is succeeded by the same edge in the opposite direction. Such a circuit revisits each vertex only in a number of steps. Studying the class of all graphs admitting at least one retracting-free bidirectional circuit was proposed by Ore (1951) and is by now of practical use to nanotechnology. The latter needs in various molecular polyhedra that are constructed from a single chain molecule in the retracting-free way. Some earlier results for simple graphs, obtained by Thomassen and, then, by other authors, are specially refined by us for a cubic graph . Most of such refinements depend only on the number of vertices of
Dynamically manipulating topological physics and edge modes in a single degenerate optical cavity
We propose a scheme to simulate topological physics within a single
degenerate cavity, whose modes are mapped to lattice sites. A crucial
ingredient of the scheme is to construct a sharp boundary so that the open
boundary condition can be implemented for this effective lattice system. In
doing so, the topological properties of the system can manifest themselves on
the edge states, which can be probed from the spectrum of an output cavity
field. We demonstrate this with two examples: a static Su-Schrieffer-Heeger
chain and a periodically driven Floquet topological insulator. Our work opens
up new avenues to explore exotic photonic topological phases inside a single
optical cavity.Comment: 6 pages, 5 figure
Experimental measurement-based quantum computing beyond the cluster-state model
The paradigm of measurement-based quantum computation opens new experimental
avenues to realize a quantum computer and deepens our understanding of quantum
physics. Measurement-based quantum computation starts from a highly entangled
universal resource state. For years, clusters states have been the only known
universal resources. Surprisingly, a novel framework namely quantum computation
in correlation space has opened new routes to implement measurement-based
quantum computation based on quantum states possessing entanglement properties
different from cluster states. Here we report an experimental demonstration of
every building block of such a model. With a four-qubit and a six-qubit state
as distinct from cluster states, we have realized a universal set of
single-qubit rotations, two-qubit entangling gates and further Deutsch's
algorithm. Besides being of fundamental interest, our experiment proves
in-principle the feasibility of universal measurement-based quantum computation
without using cluster states, which represents a new approach towards the
realization of a quantum computer.Comment: 26 pages, final version, comments welcom
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