On the Pauli graphs of N-qudits

A comprehensive graph theoretical and finite geometrical study of the commutation relations between the generalized Pauli operators of N-qudits is performed in which vertices/points correspond to the operators and edges/lines join commuting pairs of them. As per two-qubits, all basic properties and partitionings of the corresponding Pauli graph are embodied in the geometry of the generalized quadrangle of order two. Here, one identifies the operators with the points of the quadrangle and groups of maximally commuting subsets of the operators with the lines of the quadrangle. The three basic partitionings are (a) a pencil of lines and a cube, (b) a Mermin's array and a bipartite-part and (c) a maximum independent set and the Petersen graph. These factorizations stem naturally from the existence of three distinct geometric hyperplanes of the quadrangle, namely a set of points collinear with a given point, a grid and an ovoid, which answer to three distinguished subsets of the Pauli graph, namely a set of six operators commuting with a given one, a Mermin's square, and set of five mutually non-commuting operators, respectively. The generalized Pauli graph for multiple qubits is found to follow from symplectic polar spaces of order two, where maximal totally isotropic subspaces stand for maximal subsets of mutually commuting operators. The substructure of the (strongly regular) N-qubit Pauli graph is shown to be pseudo-geometric, i. e., isomorphic to a graph of a partial geometry. Finally, the (not strongly regular) Pauli graph of a two-qutrit system is introduced; here it turns out more convenient to deal with its dual in order to see all the parallels with the two-qubit case and its surmised relation with the generalized quadrangle Q(4, 3), the dual ofW(3).Comment: 17 pages. Expanded section on two-qutrits, Quantum Information and Computation (2007) accept\'

A restriction estimate using polynomial partitioning

If $S$ is a smooth compact surface in $\mathbb{R}^3$ with strictly positive second fundamental form, and $E_S$ is the corresponding extension operator, then we prove that for all $p > 3.25$, $\| E_S f\|_{L^p(\mathbb{R}^3)} \le C(p,S) \| f \|_{L^\infty(S)}$. The proof uses polynomial partitioning arguments from incidence geometry.Comment: 42 pages. Minor revisions. Accepted for publication in JAM

Output Sensitive Algorithms for Approximate Incidences and Their Applications

An epsilon-approximate incidence between a point and some geometric object (line, circle, plane, sphere) occurs when the point and the object lie at distance at most epsilon from each other. Given a set of points and a set of objects, computing the approximate incidences between them is a major step in many database and web-based applications in computer vision and graphics, including robust model fitting, approximate point pattern matching, and estimating the fundamental matrix in epipolar (stereo) geometry. In a typical approximate incidence problem of this sort, we are given a set P of m points in two or three dimensions, a set S of n objects (lines, circles, planes, spheres), and an error parameter epsilon>0, and our goal is to report all pairs (p,s) in P times S that lie at distance at most epsilon from one another. We present efficient output-sensitive approximation algorithms for quite a few cases, including points and lines or circles in the plane, and points and planes, spheres, lines, or circles in three dimensions. Several of these cases arise in the applications mentioned above. Our algorithms report all pairs at distance 1. Our algorithms are based on simple primal and dual grid decompositions and are easy to implement. We note though that (a) the use of duality, which leads to significant improvements in the overhead cost of the algorithms, appears to be novel for this kind of problems; (b) the correct choice of duality in some of these problems is fairly intricate and requires some care; and (c) the correctness and performance analysis of the algorithms (especially in the more advanced versions) is fairly non-trivial. We analyze our algorithms and prove guaranteed upper bounds on their running time and on the "distortion" parameter alpha. We also briefly describe the motivating applications, and show how they can effectively exploit our solutions. The superior theoretical bounds on the performance of our algorithms, and their simplicity, make them indeed ideal tools for these applications. In a series of preliminary experimentations (not included in this abstract), we substantiate this feeling, and show that our algorithms lead in practice to significant improved performance of the aforementioned applications

Correlated ab-initio calculations for ground-state properties of II-VI semiconductors

Correlated ab-initio ground-state calculations, using relativistic energy-consistent pseudopotentials, are performed for six II-VI semiconductors. Valence ($ns,np$) correlations are evaluated using the coupled cluster approach with single and double excitations. An incremental scheme is applied based on correlation contributions of localized bond orbitals and of pairs and triples of such bonds. In view of the high polarity of the bonds in II-VI compounds, we examine both, ionic and covalent embedding schemes for the calculation of individual bond increments. Also, a partitioning of the correlation energy according to local ionic increments is tested. Core-valence ($nsp,(n-1)d$) correlation effects are taken into account via a core-polarization potential. Combining the results at the correlated level with corresponding Hartree-Fock data we recover about 94% of the experimental cohesive energies; lattice constants are accurate to \sim 1%; bulk moduli are on average 10% too large compared with experiment.Comment: 10 pages, twocolumn, RevTex, 3 figures, accepted Phys. Rev.

Bisector energy and few distinct distances

We introduce the bisector energy of an $n$-point set $P$ in $\mathbb{R}^2$, defined as the number of quadruples $(a,b,c,d)$ from $P$ such that $a$ and $b$ determine the same perpendicular bisector as $c$ and $d$. If no line or circle contains $M(n)$ points of $P$, then we prove that the bisector energy is $O(M(n)^{\frac{2}{5}}n^{\frac{12}{5}+\epsilon} + M(n)n^2).$. We also prove the lower bound $\Omega(M(n)n^2)$, which matches our upper bound when $M(n)$ is large. We use our upper bound on the bisector energy to obtain two rather different results: (i) If $P$ determines $O(n/\sqrt{\log n})$ distinct distances, then for any $0<\alpha\le 1/4$, either there exists a line or circle that contains $n^\alpha$ points of $P$, or there exist $\Omega(n^{8/5-12\alpha/5-\epsilon})$ distinct lines that contain $\Omega(\sqrt{\log n})$ points of $P$. This result provides new information on a conjecture of Erd\H{o}s regarding the structure of point sets with few distinct distances. (ii) If no line or circle contains $M(n)$ points of $P$, then the number of distinct perpendicular bisectors determined by $P$ is $\Omega(\min\{M(n)^{-2/5}n^{8/5-\epsilon}, M(n)^{-1} n^2\})$. This appears to be the first higher-dimensional example in a framework for studying the expansion properties of polynomials and rational functions over $\mathbb{R}$, initiated by Elekes and R\'onyai.Comment: 18 pages, 2 figure

On the restriction problem for discrete paraboloid in lower dimension

We apply geometric incidence estimates in positive characteristic to prove the optimal $L^2 \to L^3$ Fourier extension estimate for the paraboloid in the four-dimensional vector space over a prime residue field. In three dimensions, when $-1$ is not a square, we prove an $L^2 \to L^{\frac{32}{9} }$ extension estimate, improving the previously known exponent $\frac{68}{19}.$Comment: Final versio

Partial ovoids and partial spreads in symplectic and orthogonal polar spaces

We present improved lower bounds on the sizes of small maximal partial ovoids and small maximal partial spreads in the classical symplectic and orthogonal polar spaces, and improved upper bounds on the sizes of large maximal partial ovoids and large maximal partial spreads in the classical symplectic and orthogonal polar spaces. An overview of the status regarding these results is given in tables. The similar results for the hermitian classical polar spaces are presented in [J. De Beule, A. Klein, K. Metsch, L. Storme, Partial ovoids and partial spreads in hermitian polar spaces, Des. Codes Cryptogr. (in press)]