Fig. 1: The construction steps for the defect-adaptive surface code.

a An example of a defective surface code lattice, where defective qubits and couplers are marked in red, and boundary qubits are marked with star symbols. b–d Display the surface code lattice after undergoing boundary deformation, internal defect disabling, and stabilizer patch, respectively. e Illustrates safe boundary data qubits and their frontiers, including X, Z, and corner boundary data qubits from top to bottom, along with their frontiers, encompassing couplers and syndrome qubits around data qubits. f Depicts the rules for internal defect disabling, showcasing the disabled qubits rules for defect syndrome qubits, data qubits, and couplers from top to bottom. The rules for defect data qubits and couplers are the same. g, h demonstrate the rules for bandage-like super-stabilizers. In scenarios where internal defect qubits are not clustered, they behave similarly to traditional super-stabilizers, as shown in (g). However, in clustered situations, these super-stabilizers can stretch across weight-1 and bridge syndrome qubits, as illustrated in (h). Additionally, (h) highlights a bridge syndrome qubit for illustration purposes.