Unveiling the non-Abelian statistics of $D(S_3)$ anyons via photonic simulation

TL;DR

This work demonstrates a photonic simulation of $D(S_3)$ non-Abelian anyons using a minimal qudit encoding, showing that a single qutrit suffices to capture the core fusion and braiding of the $G$ anyon via ribbon operators. By implementing non-unitary ribbon operations with high fidelity in a photonic platform, the authors realize both $F^G_{\rho_0}$ and its products $F^G_{\rho_1}F^G_{\rho_2}$ and $F^G_{\rho_2}F^G_{\rho_1}$, and verify the resulting non-Abelian statistics through quantum process tomography. The experiments achieve process fidelities around 94–98% and purities above 94%, showcasing the viability of photonic non-unitary operations for simulating topological anyons. The work also outlines scalable avenues to larger lattices or alternative platforms and discusses potential integration with Hamiltonian-based fault tolerance for future quantum information processing.

Abstract

Simulators can realise novel phenomena by separating them from the complexities of a full physical implementation. Here we put forward a scheme that can simulate the exotic statistics of $D(S_3)$ non-Abelian anyons with minimal resources. The qudit lattice representation of this planar code supports local encoding of $D(S_3)$ anyons. As a proof-of-principle demonstration we employ a photonic simulator to encode a single qutrit and manipulate it to perform the fusion and braiding properties of non-Abelian $D(S_3)$ anyons. The photonic technology allows us to perform the required non-unitary operations with much higher fidelity than what can be achieved with current quantum computers. Our approach can be directly generalised to larger systems or to different anyonic models, thus enabling advances in the exploration of quantum error correction and fundamental physics alike.

Figures (3)

• Figure 1: Ribbon operators of $G$ anyons. (a) A dual triangle $\tau$ has support on link $e_\tau$ and creates excitations at $p_1$ and $p_2$ plaquettes. (b) A direct operator $\tau'$ has support on link $e_{\tau'}$ and create excitations at $v_1$ and $v_2$ vertices. (c,d) Compositions of dual and direct triangles give rise to ribbon operators $\rho$. Anyonic excitations of ribbon operators are dyons (dotted ovals) that are positioned at the endpoint plaquettes and vertices. (e) We define the elementary ribbon $\rho_0$ where both dual and direct triangles have support on the same link $e_0$.
• Figure 2: (a) Experimental Setup. A coherent light source (810 nm) is incident on a phase-only spatial light modulator ($\text{SLM}_1$) and then coupled into a 2 m-long graded-index (GRIN) multi-mode fiber (MMF) with core diameter $50~\mu m$. The output of the MMF is incident on $\text{SLM}_2$ followed by a CCD camera. The combination of a high-dimensional mode mixer (MMF) sandwiched between two phase planes ($\text{SLM}_1$ and $\text{SLM}_2$) serves as a programmable optical circuit that can encode any non-unitary operations as shown in goel2022inverse. Additionally, $\text{SLM}_2$ is used for performing projective measurements required for quantum process tomography (QPT) to check the fidelity of the implemented transformations $\mathbf{T} \in \{F^G_{\rho_0}, F^G_{\rho_1}F^G_{\rho_2}, F^G_{\rho_2}F^G_{\rho_1}\}$. (b) Qutrit Encoding. Images showing three-dimensional photonic transverse-spatial modes in the macro-pixel basis (M0) generated by $\text{SLM}_1$. Modes from all mutually unbiased bases (M1, M2, M3) of the three-dimensional macro-pixel basis are also shown, which are used for performing QPT.
• Figure 3: Experimentally Measured Operators. We obtain the leading Kraus operator from the tomographed process (top row) and the Choi state representation for each operator $F^G_{\rho_0}$ , $F^G_{\rho_1} F^G_{\rho_2}$ and $F_{\rho_2}^G F_{\rho_1}^G$ (bottom row) (see Supplementary Material). Insets show theoretically expected results in each plot. $|A|$ corresponds to the maximum amplitude for a given plot and respective inset.