5,732 research outputs found
Edge-Fault Tolerance of Hypercube-like Networks
This paper considers a kind of generalized measure of fault
tolerance in a hypercube-like graph which contain several well-known
interconnection networks such as hypercubes, varietal hypercubes, twisted
cubes, crossed cubes and M\"obius cubes, and proves for any with by the induction on
and a new technique. This result shows that at least edges of
have to be removed to get a disconnected graph that contains no vertices of
degree less than . Compared with previous results, this result enhances
fault-tolerant ability of the above-mentioned networks theoretically
On Weingarten transformations of hyperbolic nets
Weingarten transformations which, by definition, preserve the asymptotic
lines on smooth surfaces have been studied extensively in classical
differential geometry and also play an important role in connection with the
modern geometric theory of integrable systems. Their natural discrete analogues
have been investigated in great detail in the area of (integrable) discrete
differential geometry and can be traced back at least to the early 1950s. Here,
we propose a canonical analogue of (discrete) Weingarten transformations for
hyperbolic nets, that is, C^1-surfaces which constitute hybrids of smooth and
discrete surfaces "parametrized" in terms of asymptotic coordinates. We prove
the existence of Weingarten pairs and analyse their geometric and algebraic
properties.Comment: 41 pages, 30 figure
Constructing Two Edge-Disjoint Hamiltonian Cycles in Locally Twisted Cubes
The -dimensional hypercube network is one of the most popular
interconnection networks since it has simple structure and is easy to
implement. The -dimensional locally twisted cube, denoted by , an
important variation of the hypercube, has the same number of nodes and the same
number of connections per node as . One advantage of is that the
diameter is only about half of the diameter of . Recently, some
interesting properties of were investigated. In this paper, we
construct two edge-disjoint Hamiltonian cycles in the locally twisted cube
, for any integer . The presence of two edge-disjoint
Hamiltonian cycles provides an advantage when implementing algorithms that
require a ring structure by allowing message traffic to be spread evenly across
the locally twisted cube.Comment: 7 pages, 4 figure
Entanglement Entropy From Tensor Network States for Stabilizer Codes
In this paper, we present the construction of tensor network states (TNS) for
some of the degenerate ground states of 3D stabilizer codes. We then use the
TNS formalism to obtain the entanglement spectrum and entropy of these
ground-states for some special cuts. In particular, we work out the examples of
the 3D toric code, the X-cube model and the Haah code. The latter two models
belong to the category of "fracton" models proposed recently, while the first
one belongs to the conventional topological phases. We mention the cases for
which the entanglement entropy and spectrum can be calculated exactly: for
these, the constructed TNS is the singular value decomposition (SVD) of the
ground states with respect to particular entanglement cuts. Apart from the area
law, the entanglement entropies also have constant and linear corrections for
the fracton models, while the entanglement entropies for the toric code models
only have constant corrections. For the cuts we consider, the entanglement
spectra of these three models are completely flat. We also conjecture that the
negative linear correction to the area law is a signature of extensive ground
state degeneracy. Moreover, the transfer matrices of these TNS can be
constructed. We show that the transfer matrices are projectors whose
eigenvalues are either 1 or 0. The number of nonzero eigenvalues is tightly
related to the ground state degeneracy.Comment: 33+9 pages. 16+3 figure
The crossing number of locally twisted cubes
The {\it crossing number} of a graph is the minimum number of pairwise
intersections of edges in a drawing of . Motivated by the recent work
[Faria, L., Figueiredo, C.M.H. de, Sykora, O., Vrt'o, I.: An improved upper
bound on the crossing number of the hypercube. J. Graph Theory {\bf 59},
145--161 (2008)] which solves the upper bound conjecture on the crossing number
of -dimensional hypercube proposed by Erd\H{o}s and Guy, we give upper and
lower bounds of the crossing number of locally twisted cube, which is one of
variants of hypercube.Comment: 17 pages, 12 figure
Entanglement entropy of (3+1)D topological orders with excitations
Excitations in (3+1)D topologically ordered phases have very rich structures.
(3+1)D topological phases support both point-like and string-like excitations,
and in particular the loop (closed string) excitations may admit knotted and
linked structures. In this work, we ask the question how different types of
topological excitations contribute to the entanglement entropy, or
alternatively, can we use the entanglement entropy to detect the structure of
excitations, and further obtain the information of the underlying topological
orders? We are mainly interested in (3+1)D topological orders that can be
realized in Dijkgraaf-Witten gauge theories, which are labeled by a finite
group and its group 4-cocycle up to group
automorphisms. We find that each topological excitation contributes a universal
constant to the entanglement entropy, where is the quantum
dimension that depends on both the structure of the excitation and the data
. The entanglement entropy of the excitations of the
linked/unlinked topology can capture different information of the DW theory
. In particular, the entanglement entropy introduced by Hopf-link
loop excitations can distinguish certain group 4-cocycles from the
others.Comment: 12 pages, 4 figures; v2: minor changes, published versio
Strong planar subsystem symmetry-protected topological phases and their dual fracton orders
We classify subsystem symmetry-protected topological (SSPT) phases in 3 + 1 dimensions (3 + 1D) protected by planar subsystem symmetries: short-range entangled phases which are dual to long-range entangled Abelian fracton topological orders via a generalized “gauging” duality. We distinguish between weak SSPTs, which can be constructed by stacking 2 + 1D SPTs, and strong SSPTs, which cannot. We identify signatures of strong phases, and show by explicit construction that such phases exist. A classification of strong phases is presented for an arbitrary finite Abelian group. Finally, we show that fracton orders realizable via p-string condensation are dual to weak SSPTs, while those dual to strong SSPTs exhibit statistical interactions prohibiting such a realization
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