Fig. 8

Boundary rotation process. a A boundary rotation starting from Fig. 1b. Ancilla qubits colored in dark gray are initialized in \(\left| 0\right\rangle \). b Expanding X boundary on the top to the right. The minimum weight of logical operators remains \(d=3\). After expansion, 1d rounds of surface code cycles are required. The qubit in the right corner is initialized in \(\left| +\right\rangle \). c Patch deformation by moving corners. The top X boundary is changed to a Z boundary by turning on Z stabilizers adjacent to previous X stabilizers. Additional 1d cycles are required. d Measuring physical qubits colored in green in X basis to shrink the patch. e Preparation to extend Z boundaries. Ancilla qubits colored in light gray are initialized in \(\left| +\right\rangle \). f Expanding Z boundaries on the top and bottom. Although the weight of logical X has increased, the code distance remains \(d=3\) because the patch cannot correct error patterns that would have been corrected using a code with \(d=5\). The parity measurement of the stabilizer is repeated 1d times. g Shrink the patch by measuring physical qubits colored in green in X basis. Total surface code cycles for boundary rotation are 3d cycles (Color figure online)