Fig. 3: Kitaev model on a honeycomb lattice.

a, The honeycomb model, consisting of anisotropic spin interactions along the three directions of the honeycomb lattice (J_X + J_Y + J_Z = 1), exhibits topological order with three Abelian (A) phases and a single non-Abelian phase B. The toric-code state in Fig. 2 is the fixed-point state \({{\rm{A}}}_{{\rm{Z}}}^{{\rm{I}}}\) of the Abelian A_Z phase (J_Z = 1). b, Starting from this initial state, we prepare different states on the phase diagram. Top: numerically optimized sequence of two-qubit gates used to prepare the non-Abelian phase B. Each circuit layer includes CPHASE (CP) gates and global single-qubit rotations. Bottom: plaquette parity during the evolution, with postselection based on atom loss and decoding (dec.). The plaquette parity is lower than in Fig. 2 owing to a longer sequence of gates and atom movements (Methods). c, Pauli strings of different lengths measured on the three studied states and averaged over the bulk of the system (Extended Data Fig. 6a,b). Error bars represent 68% confidence intervals. d, In the fermion representation, the link operators are proportional to nearest-neighbour Majorana hoppings, \({K}_{ij}^{{\rm{X/Y/Z}}}\propto {\rm{i}}{c}_{i}{c}_{j}\). Longer Pauli strings constructed from their products result in longer-range hopping operators. e, The free-fermion parent Hamiltonian \({H}_{{\bf{k}}}\) can be reconstructed from measured two-point Majorana correlations. f, The Chern number C is evaluated using the learned parent Hamiltonians of the string distributions in c, resulting in C = 0 in phase A and C = 1 in phase B; the robustness of this procedure is explored in Methods (Extended Data Fig. 6f–h). The non-Abelian phase B is characterized by the underlying topological order and an odd Chern number³.