4,254 research outputs found
Coloring non-crossing strings
For a family of geometric objects in the plane
, define as the least
integer such that the elements of can be colored with
colors, in such a way that any two intersecting objects have distinct
colors. When is a set of pseudo-disks that may only intersect on
their boundaries, and such that any point of the plane is contained in at most
pseudo-disks, it can be proven that
since the problem is equivalent to cyclic coloring of plane graphs. In this
paper, we study the same problem when pseudo-disks are replaced by a family
of pseudo-segments (a.k.a. strings) that do not cross. In other
words, any two strings of are only allowed to "touch" each other.
Such a family is said to be -touching if no point of the plane is contained
in more than elements of . We give bounds on
as a function of , and in particular we show that
-touching segments can be colored with colors. This partially answers
a question of Hlin\v{e}n\'y (1998) on the chromatic number of contact systems
of strings.Comment: 19 pages. A preliminary version of this work appeared in the
proceedings of EuroComb'09 under the title "Coloring a set of touching
strings
Power law violation of the area law in quantum spin chains
The sub-volume scaling of the entanglement entropy with the system's size,
, has been a subject of vigorous study in the last decade [1]. The area law
provably holds for gapped one dimensional systems [2] and it was believed to be
violated by at most a factor of in physically reasonable
models such as critical systems.
In this paper, we generalize the spin model of Bravyi et al [3] to all
integer spin- chains, whereby we introduce a class of exactly solvable
models that are physical and exhibit signatures of criticality, yet violate the
area law by a power law. The proposed Hamiltonian is local and translationally
invariant in the bulk. We prove that it is frustration free and has a unique
ground state. Moreover, we prove that the energy gap scales as , where
using the theory of Brownian excursions, we prove . This rules out the
possibility of these models being described by a conformal field theory. We
analytically show that the Schmidt rank grows exponentially with and that
the half-chain entanglement entropy to the leading order scales as
(Eq. 16). Geometrically, the ground state is seen as a uniform superposition of
all colored Motzkin walks. Lastly, we introduce an external field which
allows us to remove the boundary terms yet retain the desired properties of the
model. Our techniques for obtaining the asymptotic form of the entanglement
entropy, the gap upper bound and the self-contained expositions of the
combinatorial techniques, more akin to lattice paths, may be of independent
interest.Comment: v3: 10+33 pages. In the PNAS publication, the abstract was rewritten
and title changed to "Supercritical entanglement in local systems:
Counterexample to the area law for quantum matter". The content is same
otherwise. v2: a section was added with an external field to include a model
with no boundary terms (open and closed chain). Asymptotic technique is
improved. v1:37 pages, 10 figures. Proc. Natl. Acad. Sci. USA, (Nov. 2016
Fault-tolerant error correction with the gauge color code
The constituent parts of a quantum computer are inherently vulnerable to
errors. To this end we have developed quantum error-correcting codes to protect
quantum information from noise. However, discovering codes that are capable of
a universal set of computational operations with the minimal cost in quantum
resources remains an important and ongoing challenge. One proposal of
significant recent interest is the gauge color code. Notably, this code may
offer a reduced resource cost over other well-studied fault-tolerant
architectures using a new method, known as gauge fixing, for performing the
non-Clifford logical operations that are essential for universal quantum
computation. Here we examine the gauge color code when it is subject to noise.
Specifically we make use of single-shot error correction to develop a simple
decoding algorithm for the gauge color code, and we numerically analyse its
performance. Remarkably, we find threshold error rates comparable to those of
other leading proposals. Our results thus provide encouraging preliminary data
of a comparative study between the gauge color code and other promising
computational architectures.Comment: v1 - 5+4 pages, 11 figures, comments welcome; v2 - minor revisions,
new supplemental including a discussion on correlated errors and details on
threshold calculations; v3 - Author accepted manuscript. Accepted on
21/06/16. Deposited on 29/07/16. 9+5 pages, 17 figures, new version includes
resource scaling analysis in below threshold regime, see eqn. (4) and methods
sectio
Quadratic and Near-Quadratic Lower Bounds for the CONGEST Model
We present the first super-linear lower bounds for natural graph problems in the CONGEST model, answering a long-standing open question.
Specifically, we show that any exact computation of a minimum vertex cover or a maximum independent set requires a near-quadratic number of rounds in the CONGEST model, as well as any algorithm for computing the chromatic number of the graph. We further show that such strong lower bounds are not limited to NP-hard problems, by showing two simple graph problems in P which require a quadratic and near-quadratic number of rounds.
Finally, we address the problem of computing an exact solution to weighted all-pairs-shortest-paths (APSP), which arguably may be considered as a candidate for having a super-linear lower bound. We show a simple linear lower bound for this problem, which implies a separation between the weighted and unweighted cases, since the latter is known to have a sub-linear complexity. We also formally prove that the standard Alice-Bob framework is incapable of providing a super-linear lower bound for exact weighted APSP, whose complexity remains an intriguing open question
Generating series and asymptotics of classical spin networks
We study classical spin networks with group SU(2). In the first part, using
gaussian integrals, we compute their generating series in the case where the
networks are equipped with holonomies; this generalizes Westbury's formula. In
the second part, we use an integral formula for the square of the spin network
and perform stationary phase approximation under some non-degeneracy
hypothesis. This gives a precise asymptotic behavior when the labels are
rescaled by a constant going to infinity.Comment: 33 pages, 3 figures; in version 2 added one reference and a comment
on the hypotheses of Theorem 1.
Box representations of embedded graphs
A -box is the cartesian product of intervals of and a
-box representation of a graph is a representation of as the
intersection graph of a set of -boxes in . It was proved by
Thomassen in 1986 that every planar graph has a 3-box representation. In this
paper we prove that every graph embedded in a fixed orientable surface, without
short non-contractible cycles, has a 5-box representation. This directly
implies that there is a function , such that in every graph of genus , a
set of at most vertices can be removed so that the resulting graph has a
5-box representation. We show that such a function can be made linear in
. Finally, we prove that for any proper minor-closed class ,
there is a constant such that every graph of
without cycles of length less than has a 3-box representation,
which is best possible.Comment: 16 pages, 6 figures - revised versio
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