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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4768_Article_IEq1.gif" Format="GIF" Height="26" Rendition="HTML" Resolution="72" Type="Linedraw" Width="115" /> </InlineMediaObject> <EquationSource Format="TEX">$\frac{1}{\sqrt{2}}(|1>\,+|3>)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> <mrow> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mn>1</mn> <mo>></mo> <mspace width="0.166667em" /> </mrow> <mo>+</mo> <mrow> <mo stretchy="false">|</mo> <mn>3</mn> <mo>></mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4768_Article_IEq2.gif" Format="GIF" Height="26" Rendition="HTML" Resolution="72" Type="Linedraw" Width="116" /> </InlineMediaObject> <EquationSource Format="TEX">$\frac{1}{\sqrt{2}}(|1>\,-|3>)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> <mrow> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mn>1</mn> <mo>></mo> <mspace width="0.166667em" /> </mrow> <mo>-</mo> <mrow> <mo stretchy="false">|</mo> <mn>3</mn> <mo>></mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, for the qubit 1221.

On some projective unitary qutrit gates

期刊论文
出版模式：

开放获取
发表日期：

2025年5月28日

查看全文

Claire Levaillant

摘要

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 $\frac{1}{\sqrt{2}}(|1>\,+|3>)$ $\frac{1}{\sqrt{2}} (| 1 > + | 3 >)$ and $\frac{1}{\sqrt{2}}(|1>\,-|3>)$ $\frac{1}{\sqrt{2}} (| 1 > - | 3 >)$ , for the qubit 1221.