Native topological readout on qubit hardware: a Fibonacci-chain benchmark of measurement-compilation trade-offs

Recent demonstrations of non-Abelian braiding of graph vertices on noisy intermediate-scale quantum (NISQ) superconducting processor, and the experimental realization of topological order in general on various quantum hardware platforms necessitate an important question: when does a native (topological) fusion readout genuinely help for topological anyonic Hamiltonians implemented on NISQ hardware? We use the Fibonacci anyons chain as a concrete model for understanding the trade-off between measurement cost and compilation cost in that setting. The comparison is made against a simple grouped-Pauli baseline, and is scored by a covariance-aware mean-squared-error (MSE) of the full energy estimator. We based our benchmark on two different important classes of quantum circuits, namely Floquet time-evolved and variational quantum eigensolver quantum circuits, with the underlying Hamiltonian consisting of both braiding and fusion interaction. Our analysis found that there is not a uniform best method across both problems: the fusion readout method performed better on Floquet-type circuits on both the MSE and covariance-aware sampling variance, while the grouped Pauli method performed better on VQE on the MSE but worse on sampling variance. We derive scaling laws, and compute shot-budget crossover points, where one method is operationally favored above the other. The relevance of this work extends beyond Fibonacci chains to two-dimensional topological models compiled on superconducting and other qubit-native platforms, and can be used as a guide in answering the question of when one should measure in the native operator basis of the target physics, or when it is better to fall back on Pauli-basis reconstruction.