Non-Abelian Topological Order and Anyons on a Trapped-Ion Processor

TL;DR

This work reports the first unambiguous experimental realization of non-Abelian topological order using a 27-qubit trapped-ion platform to implement the $D_4$ twisted quantum double on a kagome lattice, achieving high fidelity and enabling controlled anyon braiding. By preparing the ground state with a measurement-assisted, adaptive circuit and performing non-Abelian anyon creation, movement, and fusion, the authors demonstrate Borromean-ring braiding and observe a 22-fold ground-state degeneracy consistent with non-Abelian statistics. They also show that non-Abelian anyon braiding affects fusion channels and that logic sectors on the torus can be accessed via non-Abelian operations, including the appearance of single Abelian anyons in certain sectors. The results establish a scalable path to studying non-Abelian topological phases on quantum devices and open avenues toward topologically protected quantum computation.

Abstract

Non-Abelian topological order (TO) is a coveted state of matter with remarkable properties, including quasiparticles that can remember the sequence in which they are exchanged. These anyonic excitations are promising building blocks of fault-tolerant quantum computers. However, despite extensive efforts, non-Abelian TO and its excitations have remained elusive, unlike the simpler quasiparticles or defects in Abelian TO. In this work, we present the first unambiguous realization of non-Abelian TO and demonstrate control of its anyons. Using an adaptive circuit on Quantinuum's H2 trapped-ion quantum processor, we create the ground state wavefunction of $D_4$ TO on a kagome lattice of 27 qubits, with fidelity per site exceeding $98.4\%$. By creating and moving anyons along Borromean rings in spacetime, anyon interferometry detects an intrinsically non-Abelian braiding process. Furthermore, tunneling non-Abelions around a torus creates all 22 ground states, as well as an excited state with a single anyon -- a peculiar feature of non-Abelian TO. This work illustrates the counterintuitive nature of non-Abelions and enables their study in quantum devices.

Figures (11)

• Figure 1: Creating and controlling non-Abelian wavefunctions. (a) We entangle 27 ions to create the ground and excited states of a Hamiltonian with $D_4$ topological order on a kagome lattice with periodic boundary conditions. (b) Its excitations go beyond Abelian anyons, whose spacetime braiding depends only on pairwise linking, as exemplified by the $e$- and $m$-anyons of the toric code. (c) We create and control non-Abelian anyons $m_{R,G,B}$ which can detect Borromean ring braiding via anyon interferometry; see Fig. \ref{['fig_fuse2red_borromean']}b of this work.
• Figure 2: The non-Abelian $D_4$ model and its logical operators. We consider the model (\ref{['eq_hamiltonian']}) on qubits that live on the vertices of a kagome lattice with periodic boundary conditions. Each kagome star is associated with three local operators: a 12-body star operator $A_s = \prod_{i=1}^6 CZ_{i,i+1} X^{\otimes 6}$ and two 3-body triangle operators $B_t = Z^{\otimes 3}$. It is convenient to choose a vertex coloring and assign a corresponding color to each of the qubits. For each of the three colors and directions along the torus, there are two logical string operators. The logical $\mathcal{Z}$-operators are products of local Pauli-$Z$ acting on all qubits of the respective colour in the chosen direction ($\mathcal{Z}_{GH}$ and $\mathcal{Z}_{BV}$ highlighted). For the logical $\mathcal{X}$-operator ($\mathcal{X}_{BV}$ shown) a product of $X$ is applied connecting stars of the the other two colors and decorated with a linear-depth circuit of $CZ$. More precisely, after choosing a starting point for the vertical string (here, bottom of figure) and direction (here, blue$\to$red$\to$green), we act with $CZ$ gates connecting every green vertex with preceding red vertices on the path. In the experiment, we implement the system in the black dashed lines containing $3\times3$ stars (27 qubits) and periodic boundary conditions.
• Figure 3: Ground state preparation protocol and experimental data. (a) Starting with a $|+\rangle^{\otimes N_P}$ product state on the plaquettes of the kagome lattice, we entangle all qubits via non-Clifford three-qubit $\exp(\pm i \pi/8 ZZZ)$ gates; the sign is $+1$ ($-1$) for up-pointing (down-pointing) triangles. (b) The plaquette qubits are subsequently entangled with qubits living on the vertices of the kagome lattice (see Eq. \ref{['eq_gs_prep']}) via $\textsc{CNOT}$ gates (colored lines). (c) Measuring the plaquette qubits in the $X$-basis prepares a state with non-Abelian topological order on the kagome lattice, albeit with a random pattern of Abelian anyons where the 12-body term $A_s=-1$. (d) A feed-forward layer of conditional $Z$ gates is applied to pair up the Abelions, giving a state with $A_s = B_t = 1$. Dashed lines indicate the periodic boundary conditions. (e) After preparing the state, we experimentally measure the expectation values of star ($A_s$), triangle ($B_t$) and logical $\mathcal{Z}$ operators (see Fig. \ref{['fig_overview']} for definitions). The measured energy density with respect to Eq. \ref{['eq_hamiltonian']} is $-0.946(4)$. The average (maximal) standard error on the mean of the star and triangle operators is 0.015 (0.022). The average (maximal) standard error on logical operators is 0.004 (0.005).
• Figure 4: Creating and fusing a non-Abelian anyon pair. (a) A non-Abelion pair is created from the vacuum by an $X$-operator that toggles the two adjacent blue triangles $B_t$ (shaded). Exciting such a pair locally leads to $A_s = 0$. Physically, this indeterminate value of $A_s$ corresponds to the internal space of the non-Abelion. (b,c) Moving a non-Abelion requires a linear depth quantum circuit. To move a blue non-Abelion two stars over, we apply Pauli-$X$ operators on blue qubits (blue dashed) and $CZ$ operators between each red qubit and all preceding green qubits (thick blue). (d) Fusion of the blue non-Abelion pair in the identity channel brings the state back to $\ket{\psi_0}$. The average (maximum) standard error on the mean of star and triangle operators is 0.023 (0.068). The average (maximum) standard error on the mean of logical operators in (d) is 0.006 (0.008).
• Figure 5: Braiding non-Abelian anyons. (a) A pair of blue non-Abelian fluxes $m_B$ is created, indicated by excited triangle operators. Subsequently, one partner of a pair of green $m_G$ is braided around one of the $m_B$ and annihilated, leaving behind an $e_R$ (shaded) due to the fusion rules (\ref{['eq_fusion_rules']}). The upper blue anyon is then brought back to its partner, creating another $e_R$. The average (maximum) standard error on the mean of star and triangle operators is 0.025 (0.069). (b) Borromean braiding in spacetime. Pairs of $m_B$, $m_G$ and $m_R$ are created and braided in such a way that each pair is unlinked. Thumbnails show the operators applied at different points in time. The creation and movement of the the $m_B$ is controlled on an ancilla, allowing to extract the phase $1.02(2)\pi$, with a modulus $r = 0.80(2)$. For Abelian anyons the phase of the corresponding diagram would be 0 (see Fig. \ref{['fig_conceptual']}).