Fig. 2: Measurement-based preparation of topological order.

a, A long-range entangled state of the toric-code type is prepared using a depth-3 circuit, independent of the system size, accompanied by mid-circuit readout of the ancilla qubits (I). The measurement results are random up to parity constraints along the periodic direction. A feedforward step realized through field-programmable gate array (FPGA)-triggered single-qubit Z rotations pairs the −1 outcomes (white hexagons). Finally, a parallel controlled-Y operation creates the weight-6 plaquettes (hexagons) and initializes the ZZ-link operators (ovals) to all be +1 (II). b, Expectation values of the weight-6 plaquettes and weight-2 ZZ links across the array of 72 data and 32 ancilla qubits with cylindrical boundary conditions. Inset: values averaged across the system. c, Parity expectation values of increasingly large loops, including loops that enclose one, two, three and four hexagons within a column and the loop around the cylinder (plotted for maximum decoding postselection; Extended Data Fig. 4c). The largest operator is equivalent to the product of two loops enclosing the cylinder. Error bars represent 68% confidence intervals. d, The product of plaquette operators in each column is equivalent to loops \({\bar{{\rm{Z}}}}_{{\rm{1}}}\) and \({\bar{{\rm{Z}}}}_{{\rm{2}}}\) enclosing the cylinder, which are +1 in the absence of errors. Postselection based on the parity of the measured ancilla values within each column improves the plaquette expectation values and the quality of the resulting fermion encoding.