Non-Abelian braiding of graph vertices in a superconducting processor

The authors state that 'the 2D braiding processes are physically taking place on the device, leading to actual non-Abelian exchange effects of local anyons in the many-body wavefunction, rather than matrix operations that simply follow the same algebra.' This is a crucial distinction, and the authors do a commendable job of justifying it through their experimental protocol. However, the manuscript would benefit from a more explicit discussion of how the lack of a spectral gap in this unitary-driven approach affects the robustness of the braiding compared to traditional Hamiltonian-based anyon systems. While the authors mention that 'the anyons are not quasi-particles in the sense of eigenstates that persist throughout a Hamiltonian evolution,' the implications of this for the 'topological' nature of the gates could be more thoroughly addressed.

Regarding the claim that 'the state fidelity is similar to the purity suggests that the infidelity is well described by a depolarizing error channel,' the authors should clarify if this holds across all steps of the braiding protocol. In Figure 3, the stabilizer values drop significantly during the sequence, and it is not immediately clear if the noise model (which assumes uniform errors) accounts for the spatial and temporal accumulation of errors during the 40 layers of CZ-gates. The authors should comment on whether the observed discrepancies in Figure 7 are purely due to 'inhomogeneity of the errors' or if there are other systematic effects (e.g., crosstalk or leakage) that are not fully captured by their depolarizing model.

The authors state: 'We therefore emphasize that the 2D braiding processes are physically taking place on the device, leading to actual non-Abelian exchange effects of local anyons in the many-body wavefunction, rather than matrix operations that simply follow the same algebra.' This is a crucial distinction. The authors should further clarify how the 'physical' nature of these anyons compares to the 'quasi-particle' excitations in fractional quantum Hall systems. While the unitary implementation is a valid way to explore the algebra, the lack of an energy gap protecting the anyons means that the 'topological' nature is currently a property of the circuit construction rather than an emergent property of a physical phase. A more explicit discussion of the difference between 'topological order' and 'topological simulation' would strengthen the manuscript's framing.

Regarding the claim that 'the double-braiding operation R^2 is a matrix that cannot be reduced to a scalar', the authors provide compelling evidence through the final fermion detection in Fig. 3. However, the manuscript could be improved by a more detailed analysis of the error budget. While the authors use a depolarizing channel model in Extended Data Fig. 7, it is not entirely clear how much of the observed signal is due to the specific braiding protocol versus the accumulation of gate errors over the 40 layers of CZ-gates. A more quantitative comparison between the measured fidelity and the expected fidelity from a circuit-level noise model (without relying on post-selection) would provide a more robust assessment of the protocol's scalability.

The authors state: 'Since the braiding of non-Abelian anyons in this platform is achieved through unitary gate control rather than adiabatic evolution of a Hamiltonian system, we note that the anyons are not quasi-particles in the sense of eigenstates that persist throughout a Hamiltonian evolution.' This is a critical distinction that the authors handle with commendable transparency. However, the manuscript could benefit from a more nuanced discussion on how this 'unitary braiding' relates to the long-term goal of topological protection. Specifically, how does the lack of a persistent spectral gap affect the robustness of the braiding operation against local noise compared to a true topological system?

Regarding the logical qubit encoding, the authors claim: 'Here we create two pairs of D3Vs, in addition to the four that are already present in the corners, to encode a total of three logical qubits.' While the math is sound, the manuscript would be strengthened by a clearer explanation of the physical interpretation of these logical qubits. For a reader less familiar with the specific graph gauge theory framework, it is not immediately intuitive why these specific defects encode the observed logical space. A slightly more pedagogical bridge between the 'microscopics' of the D3Vs and the resulting logical operators would improve accessibility.