Universal gates from braiding and fusing anyons on quantum hardware

Key Points:

• The \({{\mathbb{Z}}}_{3}\) toric-code ground state is prepared via plaquette stabilizers, verified by expectation values of logical operators near theoretical values, with per-qudit fidelity quantified and low standard errors reported for local and nonlocal projectors.
• Braiding experiments involving control and target qutrits demonstrate high-fidelity state preparation, as confirmed by measured local projectors and W-flux correlators close to ideal noise-free values, indicating robust \({\mathcal{Z}}\) correlations.
• Measurement qutrit braiding around input qutrits in different logical states yields local and nonlocal projector values consistent with theoretical predictions, with nonlocal W-flux projectors distinguishing logical states and maintaining high fidelity despite superposition.
• Fusion of charge pairs around data and reference qutrits shows expected changes in vertex and plaquette stabilizer values, confirming logical qutrit endpoint presence and internal state distinctions via nonlocal projectors, with controlled error margins.
• A protocol defining logical qutrits in the \({\mathcal{Z}}\)-basis through flux comparisons and braiding measurements achieves prepared states closely approximating ideal noiseless states, with strong intra- and inter-pair correlations and acceptance rates matching theoretical expectations.
• Magic state creation via measurement-induced projection is experimentally realized, with local and nonlocal projector measurements showing changes consistent with theoretical values and high fidelities supported by near-unity two-body W-flux correlators and low measurement errors.