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Raouf Dridi

Publications and source records attributed to Raouf Dridi.

24 records · Page 2Linked to original sources
arXiv2016

Very ample line bundles, contextuality and quantum computation

I relate contextuality to line bundles. Line bundles are important in algebraic geometry, they determine through their global sections rational maps to projective spaces. I explain how such maps, if they exist, relate rationally the input and output of measurement based computation (MBQC) and show geometrically that, indeed, contextuality is a necessary resource for the computational advantage in MBQC. I also leverage the definition of MBQC to category theory and present it as a "subfunctor" of the spectral presheaf. In general, the MBQC functor is pointless whereas the computation is trivial.

quant-ph↗
arXiv2015

The many mathematical faces of Mermin's proof of the Kochen-Specker theorem

Mermin's simple "pentagram" proof of the Kochen-Specker theorem is examined from various perspectives. We emphasise the many mathematical structures intimately related to Kochen-Specker proofs, ranging through functional analysis, sheaf theory and topos theory, Coxeter groups and algebraic geometry. Some novel results are presented along the way.

quant-ph↗
arXiv2015

Topos logic in measurement-based quantum computation

We report first steps towards elucidating the relationship between contextuality, measurement-based quantum computation (MBQC) and the non-classical logic of a topos associated with the computation. We show that, in a class of MBQC, classical universality requires non-classical logic, which is 'consumed' during the course of the computation, thereby pinpointing another potential quantum computational resource.

quant-ph↗
arXiv2009

On the geometry of the first and second Painlevé equations

In this paper we \emph{explicitly} compute the transformation that maps the generic second order differential equation $y''= f(x, y, y')$ to the Painlevé first equation $y''=6y^2+x$ (resp. the Painlevé second equation ${y''=2 y^{3}+yx+ α}$). This change of coordinates, which is function of $f$ and its partial derivatives, does not exist for every $f$; it is necessary that the function $f$ satisfies certain conditions that define the equivalence class of the considered Painlevé equation. In this work we won't consider these conditions and the existence issue is solved \emph{on line} as follows: If the input equation is known then it suffices to specialize the change of coordinates on this equation and test by simple substitution if the equivalence holds. The other innovation of this work lies in the exploitation of discrete symmetries for solving the equivalence problem.

math.DG↗
arXiv2008

New classification techniques for ordinary differential equations

The goal of the present paper is to propose an enhanced ordinary differential equations solver by exploitation of the powerful equivalence method of Élie Cartan. This solver returns a target equation equivalent to the equation to be solved and the transformation realizing the equivalence. The target ODE is a member of a dictionary of ODE, that are regarded as well-known, or at least well-studied. The dictionary considered in this article are ODE in a book of Kamke. The major advantage of our solver is that the equivalence transformation is obtained without integrating differential equations. We provide also a theoretical contribution revealing the relationship between the change of coordinates that maps two differential equations and their symmetry pseudo-groups.

math.DG↗