Extended Data Fig. 2: Initial state preparation.

a, Quantum circuit for preparing the initial cluster state for measuring \(\langle \mathop{Z}\limits^{ \sim }(t)\rangle \) in Fig. 2b,d and Extended Data Fig. 3a,c. The first three layers, including one layer of Hadamard gates and two layers of CZ gates, prepare the zero-temperature state in \(\{| {\Psi }_{0}\rangle \}\) with all stabilizers and \({\widetilde{Z}}_{{\rm{L}}},{\widetilde{Z}}_{{\rm{R}}}\) taking values +1. The last layer applies single-qubit π rotations around the x-axis of the Bloch sphere [X(π) gates] on the bulk qubits, each flipping two stabilizers and hence inducing two bulk excitations. The initial states in the manifold \(\{| {\Psi }_{{\rm{e}}}\rangle \}\) are obtained from applying the X(π) gates on qubits {Q₄, Q₁₉, Q₃₄, Q₃₉, Q₆₃, Q₆₉, Q₈₂, Q₉₇}, which induces 16 excitations in the bulk. In the bottom panel of Fig. 2c and Extended Data Fig. 3b, the X(π) gates are applied to {Q₆, Q₁₉, Q₃₄, Q₃₉, Q₆₃, Q₆₉, Q₈₂, Q₉₅} to observe the effect of varying excitation positions on the edge modes. b, Quantum circuit for preparing the initial state for measuring \(\langle \mathop{X}\limits^{ \sim }(t)\rangle \) in Fig. 2, Fig. 3 and Extended Data Fig. 3, with \({\widetilde{X}}_{{\rm{L}}},{\widetilde{X}}_{{\rm{R}}}\) taking values + 1. c, Expectation values of the stabilizers K_i (\({\sigma }_{i-1}^{z}{\sigma }_{i}^{x}{\sigma }_{i+1}^{z}\) in the bulk and \({\mathop{X}\limits^{ \sim }}_{{\rm{L}}},{\mathop{X}\limits^{ \sim }}_{{\rm{R}}}\) at the edges) for initial states in \(\{| {\Psi }_{0}\rangle \}\) (top panel) and in \(\{| {\Psi }_{{\rm{e}}}\rangle \}\) (bottom panel). The data are extracted from Extended Data Fig. 5 at t = 0. Error bars represent the standard deviation over fifteen rounds of measurements (five for the homogeneous case, five for the dimerized but resonant case, and five for the dimerized and off-resonant case), with each taking 20,000 shots. Grey dashed lines indicate the values of ±1.