4,254 research outputs found

    Coloring non-crossing strings

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    For a family of geometric objects in the plane F={S1,,Sn}\mathcal{F}=\{S_1,\ldots,S_n\}, define χ(F)\chi(\mathcal{F}) as the least integer \ell such that the elements of F\mathcal{F} can be colored with \ell colors, in such a way that any two intersecting objects have distinct colors. When F\mathcal{F} 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 kk pseudo-disks, it can be proven that χ(F)3k/2+o(k)\chi(\mathcal{F})\le 3k/2 + o(k) 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 F\mathcal{F} of pseudo-segments (a.k.a. strings) that do not cross. In other words, any two strings of F\mathcal{F} are only allowed to "touch" each other. Such a family is said to be kk-touching if no point of the plane is contained in more than kk elements of F\mathcal{F}. We give bounds on χ(F)\chi(\mathcal{F}) as a function of kk, and in particular we show that kk-touching segments can be colored with k+5k+5 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

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    The sub-volume scaling of the entanglement entropy with the system's size, nn, 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 log(n)\log\left(n\right) in physically reasonable models such as critical systems. In this paper, we generalize the spin1-1 model of Bravyi et al [3] to all integer spin-ss 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 ncn^{-c}, where using the theory of Brownian excursions, we prove c2c\ge2. 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 nn and that the half-chain entanglement entropy to the leading order scales as n\sqrt{n} (Eq. 16). Geometrically, the ground state is seen as a uniform superposition of all ss-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

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    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

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    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

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    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

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    A dd-box is the cartesian product of dd intervals of R\mathbb{R} and a dd-box representation of a graph GG is a representation of GG as the intersection graph of a set of dd-boxes in Rd\mathbb{R}^d. 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 ff, such that in every graph of genus gg, a set of at most f(g)f(g) vertices can be removed so that the resulting graph has a 5-box representation. We show that such a function ff can be made linear in gg. Finally, we prove that for any proper minor-closed class F\mathcal{F}, there is a constant c(F)c(\mathcal{F}) such that every graph of F\mathcal{F} without cycles of length less than c(F)c(\mathcal{F}) has a 3-box representation, which is best possible.Comment: 16 pages, 6 figures - revised versio
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