Reconstructing non-Abelian braiding and fusion without anyon transport

AI summary

Overview

Research area: Quantum physics — topological quantum computation, non-Abelian anyons, and quantum-double models implemented on trapped-ion hardware.

Technical level: Advanced. The paper assumes familiarity with anyon models, R- and F-symbols, ribbon operators, quantum doubles, and circuit-level quantum computing.

Scope (one sentence): The paper introduces and experimentally demonstrates a measurement-only, two-qutrit protocol on Quantinuum's H2 trapped-ion processors that reconstructs the braiding and fusion data of the non-Abelian quantum double model D(S₃) without physically transporting anyons.

What This Paper Is About

Non-Abelian anyons are promising for fault-tolerant quantum computing, but their defining braiding (R) and fusion (F) data are hard to access experimentally because conventional protocols demand controlled creation, physical motion and coherent fusion of anyonic excitations. The authors replace physical anyon transport with temporally ordered ribbon operations and local charge projections, compressing the relevant four-qudit D(S₃) plaquette physics into a two-qutrit protocol run on qubit hardware. The goal is to reconstruct the squared braiding phases and squared fusion amplitudes directly, and to show that combining these reconstructed primitives produces a genuinely non-Clifford resource.

Key Contributions

1. A measurement-only reconstruction protocol. The authors formulate a protocol based on temporally ordered ribbon operations and anyonic charge projections that reproduces braiding and fusion histories without moving anyons, giving access to squared braiding phases (Rⁱ)² and squared fusion amplitudes |F_ij|².

2. A dense reduction from four qudits to two qutrits. Starting from the minimal D(S₃) plaquette containing four d = 6 qudits, they show that the relevant ribbon and projection operators act non-trivially only on qudits 3 and 4, then reduce further to two qutrits using S₃ ≃ ℤ₃ ⋊ ℤ₂, and finally show the required overlaps equal (1/4)⟨ξ|𝒪|ξ⟩ on a simple product state |ξ⟩ rather than on the full plaquette ground state.

3. A qubit implementation with ancilla-assisted dilations. Each logical qutrit is encoded in the unit-Hamming-weight subspace of three physical qubits (|e⟩ ≡ |1,0,0⟩, |c⟩ ≡ |0,1,0⟩, |c²⟩ ≡ |0,0,1⟩), single-qutrit unitaries are compiled into two-qubit Givens rotations, and the non-unitary ribbon and charge-projection operators are realised probabilistically via ancillas and post-selection.

4. Experimental reconstruction on Quantinuum H2 and generation of a non-stabilizer resource. An adapted Hadamard test extracts the braiding phases, post-selected overlap and normalisation probabilities give the fusion amplitudes, and combining the two yields a non-Clifford braid acting on the vacuum fusion state to produce a state with positive second-order stabilizer Rényi entropy.

Main Findings

• Braiding phases reconstructed. For the composite Abelian sector, (R⁺_exp)² = −0.480 ± 0.095 − i(0.853 ± 0.095), consistent with the theoretical ω̄ ≈ −0.5 − i0.866. Using (R^G)² = (R^1)² − (R^+)², the non-Abelian phase was inferred as (R^G_exp)² = −0.418 ± 0.135 + i(0.846 ± 0.134), consistent with ω ≈ −0.5 + i0.866. Since (R^A)² = (R^B)² = (R^+)², the + measurement gives the common squared phase of the A and B channels.

• Significance above noise. Both reconstructed phases lie more than five standard deviations from the zero-amplitude value expected under complete depolarisation of the Hadamard ancilla.

• Fusion matrix reconstructed up to gauge. The measured F_exp has entries 0.497 ± 0.022 (the A/B block) and 0.768 ± 0.062 (the G off-diagonal entries), with F_GG = 0, to be compared with ideal values F_AA = F_AB = F_BA = F_BB = 1/2, F_AG = F_GA = 1/√2, F_BG = F_GB = −1/√2 and F_GG = 0.

• Post-selection cost quantified. The dominant uncertainty comes from the small normalisation probability p(n_G) = 4.6 × 10⁻⁴, which retains only 0.046% of trials; for 10⁴ shots about 5 accepted events are expected. No accepted p(m_GG) events were observed, consistent with F_GG = 0.

• High output-state fidelities. Averaging over 10⁵ Haar-random qutrit input states, the reconstructed transformations reproduce the ideal actions with average normalised output-state fidelities F̄_R = 0.9988 (braiding) and F̄_F = 0.9987 (fusion).

• Non-Clifford resource demonstrated. The composite braid B_s = F_s† R² F_s applied to the vacuum fusion state yields a state with measured M₂ = 0.258 ± 0.0886, against an ideal value of log(16/13) ≈ 0.208. Because the vacuum is a stabilizer state and Clifford operations preserve stabilizer states, M₂ > 0 demonstrates non-stabilizer resource generation.

• Gauge ambiguity is benign. The four admissible sign assignments of the reconstructed F-symbol produce only two distinct output states, differing solely in the relative sign of the |B⟩ component, and both have identical stabilizer Rényi entropy — so the braid is non-Clifford for every allowed sign choice.

• Resource footprints. The braiding Hadamard-test circuit for the + sector used 22 qubits with 229 single-qubit and 134 two-qubit gates, sampled with 7,000 shots per quadrature. Fusion-overlap circuits used 36 qubits (six computational plus 30 ancillas) with 322 single-qubit and 213 two-qubit gates, and normalisation circuits used 33 qubits with 275 single-qubit and 184 two-qubit gates, with 11,000 and 13,000 shots respectively.

Methodology in Plain English

The authors wanted the numbers that define how non-Abelian anyons behave — how their worldlines twist around each other (braiding) and how they combine (fusion) — without having to build and move anyons.

Their strategy has three layers. First, they work with the smallest group that supports non-Abelian anyons, S₃, whose quantum double D(S₃) has a six-dimensional local Hilbert space. The full lattice description needs four six-state qudits per plaquette, but the operators that matter only touch two of them. Using the structure S₃ ≃ ℤ₃ ⋊ ℤ₂, they shrink the problem further to two three-state systems (qutrits), and show that the needed expectation values can be taken on an unentangled product state instead of an entangled ground state — a major simplification.

Second, since the hardware uses qubits, each qutrit is stored as a single particle spread across three qubits (one excitation in one of three positions). Ordinary qutrit gates become pairs of two-qubit rotations. The tricky part is that ribbon operators and charge projectors are not unitary (they do not conserve probability). The team implements them with extra ancilla qubits and keeps only the runs where the ancillas report success — a heralded, probabilistic approach.

Third, they measure. For braiding, they embed the whole sequence inside a Hadamard test so that the real and imaginary parts of the squared phase are read out from the ancilla's |0⟩ outcome probability, using an S† gate to switch between quadratures. They avoid needing a separate controlled G-projector by using the identity (R^G)² = (R^1)² − (R^+)², where R^1 is the same ribbon sequence with the projector replaced by the identity. For fusion, there is no Hadamard test: they simply run circuits and keep only bit strings meeting all the ribbon-ancilla, projection-ancilla and computational-output conditions, then take ratios of overlap to normalisation probabilities and square roots.

Finally, they check quality by feeding 10⁵ random qutrit states through the reconstructed matrices and comparing to the ideal outputs.

Why This Matters

Impact on research. The work shows that the defining data of a non-Abelian anyon model can be measured directly and compactly, without the overhead of preparing an extended topologically ordered state. It complements large-scale approaches — notably the 54-qubit realisation of D(S₃) topological order on a trapped-ion processor cited in the paper — by offering a modular, low-overhead way to characterise the elementary primitives before assembling larger structures. It also gives an explicit, experimentally validated recipe for converting non-unitary anyonic operators into heralded qubit circuits.

Real-world applications (prospective, as framed by the paper):

• Benchmarking and validation of topological quantum processors, by verifying braiding and fusion data against exact D(S₃) predictions.
• Building blocks for measurement-only topological quantum computation, where braids are generated by measurement sequences rather than physical transport.
• Magic-state preparation, since combining reconstructed braiding and fusion produces a non-stabilizer resource state.
• Characterisation of hardware platforms: the protocol stresses circuit depth, ancilla count and post-selection rates, making it a useful stress test for trapped-ion and other qubit architectures.

Industry relevance. The experiment ran on Quantinuum's H2 trapped-ion processors, and the protocol is tailored to the constraints of real hardware: shallow compiled circuits, economical ancilla use, and error detection through a distance-2 encoding in which a single physical bit flip leaves the logical subspace and can be rejected.

Future Directions

• Coupling modules. The present experiment addresses the topological data of a single encoded plaquette; the natural next step is to couple several such modules and apply ribbon and charge-projection operations between them, enlarging the fusion space and enabling genuine multi-anyon processes.

• Scaling the encoded processor. The authors frame the compact logical representation, high-fidelity primitives and modular measurement-only architecture as a route towards scaling braiding, fusion, magic-state preparation and logical gates within a larger encoded anyonic processor.

• Pushing past the post-selection bottleneck. The G-channel normalisation probability of 4.6 × 10⁻⁴ dominates the fusion uncertainties and the reported fusion uncertainties are correspondingly larger; improving acceptance rates or error mitigation would tighten the reconstruction.

• Reducing the sign ambiguity. Because only |F_ij|² is measured, the fusion matrix is fixed only up to four gauge-equivalent sign choices. Directly accessing the sign, rather than imposing symmetry and unitarity, would give a fully determined F-symbol.

Target Audience

Researchers in quantum information and topological quantum computation, particularly those working on non-Abelian anyons, quantum double models, and measurement-based or measurement-only schemes. It is also valuable to experimentalists on trapped-ion and other qubit platforms who need concrete circuit constructions for non-unitary operators, and to quantum algorithm and architecture developers interested in resource footprints (qubit counts, gate counts, shot counts and post-selection rates) for topological primitives. Readers without a background in anyon models and R/F-symbols will find the paper demanding.

Authors’ abstract

Non-Abelian anyons offer a route to fault-tolerant and universal quantum computing, but experimental access to their defining braiding and fusion data remains limited by the resource overhead of implementing extended anyonic processes on quantum hardware. Here we introduce and experimentally realise a measurement-only protocol based on temporally ordered ribbon operations that reconstructs the non-Abelian braiding and fusion primitives of the quantum double model $D(S_3)$ without physical anyon transport. We implement a reduced two-qutrit version of the protocol on Quantinuum's H2 trapped-ion processors, realising ancilla-assisted ribbon operations and anyonic charge projections in a qubit encoding. We reconstruct the squared braiding phases and fusion amplitudes using an adapted Hadamard test and post-selected measurements, respectively. The associated braiding and fusion transformations reproduce their ideal actions with average normalised output-state fidelities of $\overline{\mathcal{F}}_{R}=0.9988$ and $\overline{\mathcal{F}}_{F}=0.9987$. Combining these primitives produces a non-Clifford braid and a non-stabilizer resource state, supporting measurement-only anyonic encodings as building blocks for larger topologically encoded quantum processors.