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

J. Phys. A: Math. Theor.

2014

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Abstract. We study a subgroup F r(162 × 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 (Z/18Z×Z/6Z) 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 (Z/18Z × Z/6Z) S 3 , also with respect to conjugation. We show that the two groups F r(162×4) and D(18, 1, 1; 2, 1, 1) are isomorphic and we provide an isomorphism between both groups. We prove that F r(162 × 4) is not isomorphic to the exceptional SU (3) subgroup Σ(216×3) 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 F r(162 × 4) may be isomorphic to belong to the (D)-series. Finally, we show that F r(162 × 4) and D(18, 1, 1; 2, 1, 1) are both conjugate under an orthogonal matrix which we provide.

Section: Structure Of the Groupmentioning

confidence: 93%

“…Our generators are the following: the two braid matrices from [1] which we recall below and the fusion matrix, which we will denote by F U M . Our result is the following.…”

Section: The Freedman Fusion Operationmentioning

confidence: 99%

Section: The Freedman Fusion Operationmentioning

confidence: 99%

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The Freedman group: a physical interpretation for theSU(3)-subgroupD(18, 1, 1; 2, 1, 1) of order 648

J. Phys. A: Math. Theor.

2014

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“…The action by the braid group on this qutrit as well as a pair fusion action due to Michael Freedman on the same qutrit are extensively studied in [2] and [9] respectively. The core ideas to produce an entangling gate on the 2-qutrit are to introduce an ancilla which plays the role of a mediator between the left and right qutrits and use it to entangle both qutrits before separating the two qutrits again.…”

Section: Introductionmentioning

confidence: 99%

“…The reader should keep in mind that the same actions are being performed simultaneously on the right input side. A σ 2 -braid on (0222) 2 or (4222) 2 is simply a multiplication by a phase while by [2] a σ 2 -braid on (2222) 2 results in a superposition of 0 and 4. We then transmit this information from the qutrit to the center by doing another full twist like on the drawing.…”

Section: Introductionmentioning

confidence: 99%

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

Bulletin des Sciences Mathématiques

2019

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AcknowledgementThis work was done with Michael Freedman. The author thanks him for his quite nice and fruitful mentoring. She thanks Stephen Bigelow for his comments and help with a better understanding of interferometry. AbstractThe 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 creations.

Topological quantum computation within the anyonic system the Kauffman–Jones version of SU(2) Chern–Simons theory at level 4

2016

Quantum Inf Process

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The third and last chapter deals with realizing all the 2-qubit permutation gates, some of which are entangling gates (like the CNOT gate). The chapter also relies on other published protocols which are not part of this paper. Chapter 1From ancilla to quantum gate Note This chapter grew out of many inspiring discussions with Michael Freedman and is the first stone in a series of two chapters leading to the production of irrational phase gates in the Kauffman-Jones version of SU (2) Chern-Simons theory at level 4. AbstractWe provide a way to turn an ancilla into a gate for specific ancillas in the Kauffman-Jones version of SU (2) Chern-Simons theory at level 4. We deal with both the qubit 1221 and the qutrit 2222. Together with ancilla preparations, our protocols are later used to make irrational phase gates, leading to universal single qubit and qutrit gates. MOTIVATION OF THE WORK5 interferometry, see [4], [5], [6]. For basic facts about recoupling theory, we refer the reader to [13], except the theory which we use is unitary. In particular we deal with unitary theta symbols and unitary 6j-symbols (see [21] and Appendix of [14]). The value of the Kauffman constant is, using the same notations as in [13],The main four moves which we use throughout the paper are summarized below.• The "F -move"The brackets are called unitary 6j-symbols.• The "R-move"• The "theta move" NoteThe author thanks Michael Freedman for providing these thrilling problems, for guiding her with endless generosity in time and encouragements and for communicating his own excitement during the pursuit of this research. AbstractIn Chapter 1, it is shown with respect to two different anyonic systems, the same as those considered in this paper, how to fuse an ancilla into the input in order to form a gate. In each case, the ancilla must satisfy to some adequate amplitude ratios properties. In the present chapter, we show how to prepare such ancillas, leading to infinite order phase gates. We obtain universal 1-qubit and 1-qutrit gates. Together with [16] and [17] in which are respectively shown how to make a 2-qubit and a 2-qutrit entangling gate, we get qubit and qutrit gate sets that are universal for quantum computation.