Demonstration of magic state power of $\mathbf{D}(\mathbf{S}_{3})$ anyons with two qudits

TL;DR

The paper addresses realizing universal quantum computation with non-Abelian anyons by explicitly demonstrating magic-state generation in the D(S$_3$) quantum double model. It develops a measurement-only, lattice-based protocol using ribbon and projection operators to simulate braiding and fusion, and shows that the fundamental R and F matrices for the G anyons can be reconstructed. A key result is that all non-Clifford actions necessary for magic-state generation persist even under sign ambiguities in the F-matrix, enabling robust non-Clifford gates. Moreover, the authors compress the entire R and F reconstruction onto a two-qudit system, providing a compact, scalable blueprint compatible with current quantum platforms and enabling practical demonstrations of non-Abelian anyon-based universality.

Abstract

We consider a lattice of $d=6$ qudits that supports $\mathbf{D}(\mathbf{S}_3)$ non-Abelian anyons. We present a method for implementing both braiding and fusion evolutions using only the operators that create and measure anyons, without requiring additional dynamical control. This provides a minimal protocol demonstrating that $\mathbf{D}(\mathbf{S}_3)$ anyons can generate magic states, thereby establishing their universality for quantum computation. Furthermore, we show that the entire scheme can be encoded in just two qudits, offering a compact blueprint that is inherently scalable and readily implementable in current quantum platforms.

Figures (12)

• Figure 1: The $R$ and $F$ operations that characterise anyonic models. (a) The $R$ braid describes the phase, $R^{ab}_c$, gained by exchanging two anyons, $a$ and $b$ when they have a fixed fusion outcome $c$. (b) The $(F^d_{abc})^i_j$ matrix elements give the relation between the states of three fusing anyons $a$, $b$, and $c$ with fixed fusion outcome $d$ when the order of fusion is changed.
• Figure 2: (a) The states (i) $\ket{i}$ and (ii) $\ket{j}$ belong to alternative bases for the fusion space of four $G$ anyons. The generators $b_{1,G}$ and $b_{2,G}$ for the three-strand braid group $\mathcal{B}_{3}$ act on the states $\ket{i}$ as shown in (iii) and (iv) respectively. (b) Repeated application of the generator $b_{2,G}$ generates a complete evolution of $G_{2}$ around $G_{3}$ as represented by the operation $B_{2,G}\equiv (b_{2,G})^{2}$.
• Figure 3: The action of the braiding operator $b_{2,G}$ on the basis state $\ket{i}$ can be understood through a series of $R$ and $F$ moves as shown.
• Figure 4: The lattice construction of the quantum double model $\mathbf{D}(\mathbf{S}_{3})$. The solid lines denote the direct lattice with underlying orientation as indicated by arrows. The dual lattice is shown with dotted lines. At the top of (a), an example of a direct triangle is shown as outlined in equation \ref{['eq:T']}. The direct triangle, $\tau$, creates excitations on vertices $v_{1}$ and $v_{2}$ as shown. Below, is a dual triangle as defined in equation \ref{['eq:L']}. The dual triangle, $\tau^{\prime}$, creates a pair of excitations on plaquettes $p_{1}$ and $p_{2}$. In (b) closed loops of these dual (direct) triangles form the vertex (plaquette) operators. The orientation of each of the component operators depends on the lattice enumeration and ensures that all $A^{g}(v)$ and $B^{h}(p)$ mutually commute. (c) The two $G$ anyonic ribbons $\rho_{1}$ and $\rho_{2}$ that will be used throughout the paper. The dyons created by the associated ribbon operators $F^{G}_{\rho_{1}}$ and $F^{G}_{\rho_{2}}$ are situated along the sites formed of the vertices and plaquettes at the endpoints of each ribbon as highlighted.
• Figure 5: The anyonic ribbon operator $F^G_{\rho_{1}}$ creates a pair of dyons $G$ and $G^{\prime}$ at the endpoints of the ribbon $\rho_{1}$ when acting on the ground state $\ket{\zeta}$. In the $\{A,B,G\}$ sub-model, anyons can be measured with the application of the vertex projection operators such as in \ref{['eq:fusions_of_r1']}. In this way, the state $F^{G}_{\rho_{1}}\ket{\zeta}$ is preserved under the action of the measurement operators $A^{G}(v_{1})$ and $A^{G}(v_{2})$.