Universal Topological Gates from Braiding and Fusing Anyons on Quantum Hardware

TL;DR

The work demonstrates universal quantum computation using the $S_3$ quantum double on a scalable trapped-ion platform by encoding logical information in the nonlocal fusion space of non-Abelian anyons and using a three-pronged gate set: pull-through braiding, $\mathcal{X}$-basis measurement, and $\mathcal{Z}$-basis measurement. By preparing the $S_3$ ground state on a torus and explicitly braiding and fusing anyons, the authors realize a universal topological gate set and show magic-state preparation, all within a tractable, solvable non-Abelian TO. The combination of unitary ground-state preparation, coherent anyon transport, and measurement-based encoding (bureau of standards) enables efficient, scalable control of non-Abelian anyons, offering a practical path toward fault-tolerant universal TOP-QC. The results establish $S_3$ TO as a minimal yet powerful platform for exploring fusion-space encoding, topological gates, and real-time lattice gauge dynamics on hardware with existing quantum resources.

Abstract

Topological quantum computation encodes quantum information in the internal fusion space of non-Abelian anyonic quasiparticles, whose braiding implements logical gates. This goes beyond Abelian topological order (TO) such as the toric code, as its anyons lack internal structure. However, the simplest non-Abelian generalizations of the toric code do not support universality via braiding alone. Here we demonstrate that such minimally non-Abelian TOs can be made universal by treating anyon fusion as a computational primitive. We prepare a 54-qubit TO wavefunction associated with the smallest non-Abelian group, $S_3$, on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes $S_3$ TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.

Figures (25)

• Figure 1: Topological order based on $S_3$. (a) Symmetry group $S_3 = \mathbb Z_3 \rtimes \mathbb Z_2$ of an equilateral triangle, generated by rotation $\mu$ and reflection $\sigma$. (b) Logical state encoded in the flux-neutral fusion space of a pair of $C_2$ fluxes. (c) Anyon braiding together with fusion realizes a universal gate set for quantum computation. (d) The code is defined on a square lattice on a torus with oriented edges hosting $d\!=\!6$ qudits, composed of a qubit (solid circle) and a qutrit (solid triangle). The triangle direction specifies the edge orientation. Physically, each qubit is encoded in the hyperfine states of $^{171}\mathrm{Yb}^+$ ions. (e--k) The stabilizers decompose into Pauli operators $X,Z$ on qubits and generalized Pauli operators $\mathcal{X},\mathcal{Z}$ together with charge conjugation $\mathcal{C}$ on qutrits (see Section \ref{['app:qubit-qutrit-encoding']}). There are (e) qubit vertex stabilizers $S^{\mathbb Z_2}_A$, (f) qutrit vertex stabilizers $S^{\mathbb Z_3}_A$ (these act on qutrits, conditioned on qubits), (g) qubit plaquette stabilizers ($S_B^{\mathbb Z_2}$) and (h) qutrit plaquette stabilizers ($S_B^{\mathbb Z_3}$). There are also non-contractible versions of the (i) $S^{\mathbb Z_3}_Z$ and (j) $S^{\mathbb Z_2}_Z$ stabilizers. The stabilizer $S_{W,p}$ associated with the global flux $W$ is defined as the product of $S_3$ group elements $x_i$ along the directed loop in (k), beginning at the origin $o$ and encircling exactly one plaquette $p$ (see Section \ref{['app:z2z3-decomposition']}).
• Figure 2: Ground and single anyon states on 54 qubits. (a) $\mathbb Z_2$ (bottom) and $\mathbb Z_3$ (top) vertex and plaquette projector expectation values on a $3\times3$ periodic square lattice (see Eqs. \ref{['eq:qutrit-qubit-vertex-projectors']}, \ref{['eq:qutrit-qubit-plaquette-projectors']}). Lighter (darker) squares (rhombi) denote plaquette (vertex) projectors. The mean $\mathbb Z_2$ and $\mathbb Z_3$ projector values are 0.987(2) and 0.962(4), respectively and the average (maximum) standard error is 0.014 (0.029). For the non-contractible projectors, the average (maximum) standard error is 0.017 (0.022). (b) Single $C_3$ anyon creation by applying a horizontal $\hat{X}_{C_3}$ to a ground state with vertical $C_2$ flux loop. (c) The anyon is identified by a single excited $B^{\mathbb Z_3}$ plaquette projector; average (maximum) standard error is 0.020 (0.034).
• Figure 3: Pull-through gate entangles topological qutrits. (a) Braiding between the logical pairs (Eq. \ref{['eq:absolute-logical']}) gives an entangling gate (Eq. \ref{['eq:pullthrough']}) via the conjugation $(\mu^a \sigma) \mu^b \sigma (\mu^a \sigma)^{-1}$. (b) Spacetime diagram of the protocol on the $3\times3$ torus. Two pairs of $C_2$ flux anyons, control (blue worldlines) and target (red worldlines), are initialized in the logical state $\ket{\tilde{0}}_L\ket{0}_L$. One endpoint of the control qutrit is braided (pulled through) around both endpoints of the target and returned to its initial position at $t=6$. Oriented gray triangles indicate the ribbon operators implementing anyon motion. (c) Expectation values of the local projectors $B^{\mathbb{Z}_3}$ remain close to $1$ except on plaquettes hosting logical qutrit endpoints, where they are approximately $1/3$; the average (maximum) standard error is $0.028$ ($0.044$). The nonlocal correlator $\langle \Pi^{\mathbb{Z}_3}_{W_2 W_3}\rangle$ certifies $\mathcal{Z}$ correlations between the control and target qutrits. Its measured value is consistent with the predicted value of $1/3$ at $t=0$ and increases to $0.86(3)$ at $t=6$, in good agreement with the theoretical value of $1$ for the prepared Bell state.
• Figure 4: $\mathcal{X}$-basis measurement. (a) A symmetric "measurement" flux pair (blue worldlines), created from the vacuum in the state $\ket{\tilde{0}}_{\! L}$, is braided around an endpoint of the input qutrit's flux pair (yellow worldlines) and then fused. The fusion outcome diagnoses the input qutrit state: annihilation to the vacuum corresponds to $\ket{\tilde{0}}_{\! L}$, while a remnant charge anyon indicates $\ket{\tilde{1}}_{\! L}$. (b) Experimental verification at $t=0,2,4$. Left (right) column: input state $\ket{\tilde{0}}_{\! L}$ ($\ket{\tilde{1}}_{\! L}$). Local stabilizers remain insensitive to the logical content of the qutrits, with $B^{\mathbb Z_3}$ values near $1/3$ at all flux endpoints (average/maximum standard error $0.034/0.059$). At fusion $(t=5)$, the vertex projector $A^{\mathbb Z_3}$ yields $0.84(4)$ for $\ket{\tilde{0}}_{\! L}$, consistent with vacuum annihilation, and $0.06(3)$ for $\ket{\tilde{1}}_{\! L}$, indicating a remnant charge anyon.
• Figure 5: $\mathcal{Z}$-basis measurement. (a) Worldlines on the $3\times2$ lattice for a data qutrit (blue) and two reference qutrits: reference 1 (red) in $\ket{0}_L$ and reference 2 (green) in $\ket{1}_L$. A $[2]$ charge pair (yellow) encircles the data and reference-2 qutrits between $t=2$ and $t=3$; because the logical states differ, the pair annihilates to the vacuum with probability $1/4$ or leaves a remnant charge with probability $3/4$. Between $t=3$ and $t=4$, a second $[2]$ charge pair encircles the data and reference-1 qutrits, both in $\ket{0}_L$, resulting in deterministic vacuum annihilation. (b) Experimental demonstration. Local plaquette projectors $B^{\mathbb Z_2}$ remain near zero at $t=2,3,4$, confirming persistent qutrit endpoints (avg/max error $0.014/0.021$). At $t=3$, the vertex projector $A^{\mathbb Z_2}$ at the first fusion site yields $0.32(4)$, consistent with the expected value $1/4$; at $t=4$, $A^{\mathbb Z_2}$ at the second fusion site is $0.92(2)$, indicating deterministic annihilation to the vacuum.