Protocol for making a 22-qutrit entangling gate in the Kauffman-Jones version of SU(2)4SU(2)_4

The following paper provides a protocol to physically generate a $2$-qutrit entangling gate in the Kauffman-Jones version of $SU(2)$ Chern-Simons theory at level $4$. The protocol uses elementary operations on anyons consisting of braids, interferometric measurements, fusions and unfusions and ancilla pair creation.Comment: 8 pages, 4 figure

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 $1221$.Comment: 17 pages, 18 figure

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.Comment: 37 page

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 $\Sigma(216\times 3)$ 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.Comment: 42 pages, 8 figure

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.Comment: 17 page

Irreducibility of the Lawrence-Krammer representation of the BMW algebra of type An−1A_{n-1}

It is known that the Lawrence-Krammer representation of the Artin group of type $A_{n-1}$ based on the two parameters $t$ and $q$ that was used by Krammer and independently by Bigelow to show the linearity of the braid group on $n$ strands is generically irreducible. Here, we recover this result and show further that for some complex specializations of the parameters the representation is reducible. We give all the values of the parameters for which the representation is reducible as well as the dimensions of the invariant subspaces. We deduce some results of semisimplicity of the Birman-Murakami-Wenzl algebra of type $A_{n-1}$.Comment: 8 page

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.Comment: 21 pages, 33 figure

A quantum combinatorial approach for computing a tetrahedral network of Jones-Wenzl projectors

Trivalent plane graphs are used in various areas of mathematics which relate for instance to the colored Jones polynomial, invariants of 3-manifolds and quantum computation. Their evaluation is based on computations in the Temperley-Lieb algebra and more specifically the Jones-Wenzl projectors. We use the work by Kauffman-Lins to present a quantum combinatorial approach for evaluating a tetrahedral net. On the way we recover two equivalent definitions for the unsigned Stirling numbers of the first kind and we provide an equality for the quantized factorial using these numbers.Comment: 25 pages, 17 figure

Tangles of type EnE_n and a reducibility criterion for the Cohen-Wales representation of the Artin group of type E6E_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.Comment: 34 pages, 17 figure

A new set of generators and a physical interpretation for the SU(3) finite subgroup D(9,1,1;2,1,1)

After 100 years of effort, the classification of all the finite subgroups of SU(3) is yet incomplete. The most recently updated list can be found in P.O. Ludl, J. Phys. A: Math. Theor. 44 255204 (2011), where the structure of the series (C) and (D) of SU(3)-subgroups is studied. We provide a minimal set of generators for one of these groups which has order 162. These generators appear up to phase as the image of an irreducible unitary braid group representation issued from the Jones-Kauffman version of SU(2) Chern-Simons theory at level 4. In light of these new generators, we study the structure of the group in detail and recover the fact that it is isomorphic to the semidirect product Z_9 \times Z_3 \rtimes S_3 with respect to conjugation.Comment: 14 page