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

Abstract

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.

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

“…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.…”

“…This step is very important as it allows the outcome of the forthcoming fusion measurement to be completely independent from the 2-qutrit input. Next, we do a single braid in order to retrieve the initial shape of the input using the fact that a full twist on (2222) 2 is a multiplication by a phase by [10] or [9]. It remains to dispose of the ancilla and separate the left and right qutrits.…”

“…If so, it remains to fuse anyons 4 and 5 on one hand and 6 and 7 on the other hand to be back to a 2-qutrit. If the interferometric measurement outcome is rather 4, it will suffice to fuse a pair of 4's into anyons 5 and 6, an operation called FFO after Michael Freedman in [9] and finish with the two fusions like previously. Finally, the outcome cannot be 2, because specific to this SU (2) Chern-Simons theory at level 4, the 6j-symbol with only 2's is zero.…”

“…In the latter case, we braid anyons 5 and 6 before remeasuring anyons 6,7,8,9 and iterate the process until we eventually measure 0. Obtain the gate written above instead of the one from the Theorem.…”

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

“…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"…”