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Claire I. Levaillant

Publications and source records attributed to Claire I. Levaillant.

16 recordsLinked to original sources
arXiv2026

Quantum multiple gray scale images encryption scheme in the bit plane representation model

After introducing a bit-plane quantum representation for a multi-image, we present a novel way to encrypt/decrypt multiple images using a quantum computer. Our encryption scheme is based on a two-stage scrambling of the images and of the bit planes on one hand and of the pixel positions on the other hand, each time using quantum baker maps. The resulting quantum multi-image is then diffused with controlled CNOT gates using a sine chaotification of a two-dimensional Hénon map as well as Chebyshev polynomials. The decryption is processed by operating all the inverse quantum gates in the reverse order.

quant-ph↗
arXiv2024

Multiple harmonic sums $\mathcal{H}_{\lbrace s\rbrace^{2l}=1;p-1}$ modulo $p^4$ and applications

Wilson's theorem for the factorial got generalized to the moduli $p^2$ in 1900 and $p^3$ in 2000 by J.W.L. Glaisher and Z-H. Sun respectively. This paper which studies more generally the multiple harmonic sums $\mathcal{H}_{\lbrace s\rbrace^{2l}=1;p-1},2\leq 2l\leq p-1$ modulo $p^4$ in association with the Stirling numbers $\left[\begin{array}{l}\;\;\;p\\2s-1\end{array}\right], 2\leq 2s\leq p-1$ modulo $p^4$ is concerned with establishing a generalization of Wilson, Glaisher and Sun's results to the modulus $p^4$. We also break p-residues of convolutions of three divided Bernoulli numbers of respective orders $p-1$, $p-3$ and $p-5$ into smaller pieces and generalize some results of Sun for some of the generalized harmonic numbers of order $p-1$ modulo $p^4$.

math.NT↗
arXiv2021

A congruence concerning a convolution involving weighted Bernoulli numbers

Given a prime $p\geq 5$, we reduce modulo p a convolution of order p-1 of powers of two weighted Bernoulli numbers with Bernoulli numbers in terms of harmonic numbers and generalized harmonic numbers. Our proof is based on studying the p-adic expansion of the number of permutations of Sym(p-2) with an even number of ascents, up to the modulus $p^2$.

math.NT↗
arXiv2021

A note on the Voronoi congruences and the residue of the Fermat quotient

Given an odd prime p, we prove a congruence on the p-residue of the Fermat quotient q_p(a) in base a with 0<a<p, which arises from a generalization of the Voronoi congruences and from some other congruences on sums and weighted sums of divided Bernoulli numbers. As an application in the base 2 case, we retrieve a congruence for the first generalized harmonic number of order (p-1)/2, a generalization originally due to Z-H. Sun of a classical congruence known since long for the harmonic number of order (p-1)/2 as a special case of the Lerch formula. We find a sharpening of the Voronoi congruences which is different from the one of W. Johnson and which is more computationally efficient. We prove an additional related congruence, which specialized to base 2 allows to retrieve several congruences that were originally due to E. Lehmer.

math.NT↗
arXiv2020

Congruences related to Miki's identity

Given an odd prime p, we present three independent ways of relating modulo p certain truncated convolutions of divided Bernoulli numbers to certain full convolutions of divided Bernoulli numbers.

math.CO↗
arXiv2015

Interferometry versus projective measurement of anyons

The distinct methods for measuring topological charge in a non-abelian anyonic system have been discussed in the literature: projective measurement of a single point-like quasiparticle and interferometric measurement of the total topological charge of a group of quasiparticles. Projective measurement by definition is only applied near a point and will project to a topological charge sector near that point. Thus, if it is to be applied to a \emph{group} of anyons to project to a \emph{total} charge, then the anyons must first be fused one by one to obtain a single anyon carrying the collective charge. We show that interferometric measurement is strictly stronger: Any protocol involving projective measurement can be simulated at low overhead by another protocol involving only interferometric measurement.

quant-ph↗
arXiv2014

Making a circulant 2-qubit entangling gate

We present a way to physically realize a circulant 2-qubit entangling gate in the Kauffman-Jones version of SU(2) Chern-Simons theory at level 4. Our approach uses qubit and qutrit ancillas, braids, fusions and interferometric measurements. Our qubit is formed by four anyons of topological charges 1221. Among other 2-qubit entangling gates we generate in the present paper, we produce in particular the circulant gate CEG = 1/4 I + I sqrt(3)/4 J - 3/4 J^2 + I sqrt(3)/4 J^3, where J denotes the permutation matrix associated with the cycle (1432) and I denotes the identity matrix.

quant-ph↗
arXiv2013

On some projective unitary qutrit gates

As part of a protocol, we braid in a certain way six anyons of topological charges $222211$ in the Kauffman-Jones version of $SU(2)$ Chern-Simons theory at level $4$. The gate we obtain is a braid for the usual qutrit $2222$ but with respect to a different basis. With respect to that basis, the Freedman group of \cite{LEV} is identical to the $D$-group $D(18,1,1;2,1,1)$. We give a physical interpretation for each Blichfeld generator of the group $D(18,1,1;2,1,1)$. Inspired by these new techniques for the qutrit, we are able to make new ancillas, namely $\frac{1}{\sqrt{2}}(|1>\,+|3>)$ and $\frac{1}{\sqrt{2}}(|1>\,-|3>)$, for the qubit $1221$.

math.QA↗
arXiv2013

The Freedman group: a physical interpretation for the SU(3)-subgroup D(18,1,1;2,1,1) of order 648

We study a subgroup $Fr(162\times 4)$ of SU(3) of order 648 which is an extension of $D(9,1,1;2,1,1)$ and whose generators arise from anyonic systems. We show that this group is isomorphic to a semi-direct product $(\mathbb{Z}/18\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z})\rtimes S_3$ with respect to conjugation and we give a presentation of the group. We show that the group $D(18,1,1;2,1,1)$ from the series $(D)$ in the existing classification for finite SU(3)-subgroups is also isomorphic to a semi-direct product $(\mathbb{Z}/18\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z})\rtimes S_3$, also with respect to conjugation. We show that the two groups $Fr(162\times 4)$ and $D(18,1,1;2,1,1)$ are isomorphic and we provide an isomorphism between both groups. We prove that $Fr(162\times 4)$ is not isomorphic to the exceptional SU(3) subgroup $Σ(216\times 3)$ of the same order 648. We further prove that the only SU(3) finite subgroups from the 1916 classification by Blichfeldt or its extended version which $Fr(162\times 4)$ may be isomorphic to belong to the $(D)$-series. Finally, we show that $Fr(162\times 4)$ and $D(18,1,1;2,1,1)$ are both conjugate under an orthogonal matrix which we provide.

math.QA↗
arXiv2011

Tangles of type $E_n$ and a reducibility criterion for the Cohen-Wales representation of the Artin group of type $E_6$

We introduce tangles of type $E_n$ and construct a representation of the Birman-Murakami-Wenzl algebra (BMW algebra) of type $E_6$. As a representation of the Artin group of type $E_6$, this representation is equivalent to the faithful Cohen-Wales representation of type $E_6$ that was used to show the linearity of the Artin group of type $E_6$. We find a reducibility criterion for this representation and complex values of the parameters for which the algebra is not semisimple.

math.GR↗
arXiv2011

Classification of the invariant subspaces of the Cohen-Wales representation of the Artin group of type $D_n$

Recently, Cohen and Wales built a faithful linear representation of the Artin group of type $D_n$, hence showing the linearity of this group. It was later discovered that this representation is reducible for some complex values of its two parameters. It was also shown that when the representation is reducible, the action on a proper invariant subspace is a Hecke algebra action of type $D_n$. The goal of this paper is to classify these proper invariant subspaces in terms of Specht modules indexed by double partitions of the integer $n$. This work is the continuation of arXiv:1103.5673

math.RT↗
arXiv2011

Reducibility of the Cohen-Wales representation of the Artin group of type $D_n$

Using knot theory, we construct a linear representation of the CGW algebra of type $D_n$. This representation has degree $n^2-n$, the number of positive roots of a root system of type $D_n$. We show that the representation is generically irreducible, but that when the parameters of the algebra are related in a certain way, it becomes reducible. As a representation of the Artin group of type $D_n$, this representation is equivalent to the faithful linear representation of Cohen-Wales. We give a reducibility criterion for this representation as well as a conjecture on the semisimplicity of the CGW algebra of type $D_n$. Our proof is computer-assisted using Mathematica.

math.GR↗
arXiv2010

Classification of the invariant subspaces of the Lawrence-Krammer representation

The Lawrence-Krammer representation was used in $2000$ to show the linearity of the braid group. The problem had remained open for many years. The fact that the Lawrence-Krammer representation of the braid group is reducible for some complex values of its two parameters is now known, as well as the complete description of these values under some restrictions on one of the parameters. It is also known that when the representation is reducible, the action on a proper invariant subspace is an Iwahori-Hecke algebra action. In this paper, we prove a theorem of classification for the invariant subspaces of the Lawrence-Krammer space. We classify the proper invariant subspaces in terms of Specht modules. We fully describe them in terms of dimension and spanning vectors in the Lawrence-Krammer space.

math.RT↗
arXiv2009

Parameters for which the Lawrence-Krammer representation is reducible

We show that the Lawrence-Krammer representation based on two parameters that was used by Bigelow and independently Krammer to show the linearity of the braid group is generically irreducible, but that when its parameters are specialized to some nonzero complex numbers, the representation is reducible. To do so, we construct a representation of the BMW algebra inside the Lawrence-Krammer space. As a representation of the braid group, this representation is equivalent to the Lawrence-Krammer representation, where the two parameters of the algebra are related to the parameters of the Lawrence-Krammer representation. We give all the complex values of the parameters for which the representation is reducible and describe the invariant subspaces in some cases. We show that for these values of the parameters and other values, the BMW algebra is not semisimple.

math.RT↗