Extended Data Fig. 6: Operator transformations on a single plaquette and plaquette-by-plaquette transformation of the Hamiltonian.

a, The table shows the different transformations of the operators on a plaquette when transforming them with the C-NOT gates of the variational circuit acting on that plaquette. The left side shows the transformations of the different vertex operators. Note that, when a vertex operator has some remaining Z gates not supported on this specific plaquette, they are left unchanged by the C-NOT gates. The right side shows the transformations of the different onsite Pauli-Z operators. The last diagram at the bottom shows the transformation of the plaquette operators. b, We can use the results of the table in a to transform the original Hamiltonian plaquette by plaquette. At each step, the plaquette transformed next by the C-NOT gates in the variational circuit is highlighted in yellow. Note that the plaquettes of the Hamiltonian are transformed in the opposite order of how they are applied in the quantum circuit in Extended Data Fig. 5b. Because the C-NOT gates in the circuit applied to plaquettes in the same row commute, we can transform a whole row of the Hamiltonian at the same time.