On some projective unitary qutrit gates

Abstract 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 than the usual basis. With respect to that basis, the Freedman group of J. Phys. A: Math. Theor. 47, 285203 (2014) is identical to the D -group D (18, 1, 1; 2, 1, 1) from the Blichfeldt classification. We provide a physical realization of each Blichfeldt generator of the SU (3) finite subgroup D (18, 1, 1; 2, 1, 1). The link of this group to elementary particle physics was already well-known by the physicists. This paper uncovers its realization in quantum physics. Inspired by these new techniques for the qutrit, we are able to make new ancillas, namely $$\frac{1}{\sqrt{2}}(|1&gt;\,+|3&gt;)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> </mml:mfrac> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>|</mml:mo> <mml:mn>1</mml:mn> <mml:mo>&gt;</mml:mo> <mml:mspace/> </mml:mrow> <mml:mo>+</mml:mo> <mml:mrow> <mml:mo>|</mml:mo> <mml:mn>3</mml:mn> <mml:mo>&gt;</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and $$\frac{1}{\sqrt{2}}(|1&gt;\,-|3&gt;)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> </mml:mfrac> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>|</mml:mo> <mml:mn>1</mml:mn> <mml:mo>&gt;</mml:mo> <mml:mspace/> </mml:mrow> <mml:mo>-</mml:mo> <mml:mrow> <mml:mo>|</mml:mo> <mml:mn>3</mml:mn> <mml:mo>&gt;</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , for the qubit 1221.