Fig. 2: Illustration of the stabilizer measurement circuit building and related simulation results.

a Space-time lattice with super-stabilizers forming shells. The columns align along the temporal direction to show the measurement of super-stabilizers in the space-time lattice. Various types of super-stabilizers are shown: I. Super-stabilizers formed by a single data qubit defect. II. Bandage-like super-stabilizers formed by the nearby defect. III. Super-stabilizers formed by a single-syndrome qubit defect. IV. X stabilizer is unaffected by defects. V. Z stabilizer unaffected by defects. The shell size indicates the consecutive measurements of the same type of super-stabilizers. For regular stabilizers, X and Z stabilizers are measured in the same cycle. However, X and Z super-stabilizers cannot be measured in the same cycle. b The logical error rate (LER) of the surface code under different code sizes L and defect rates (DR). The box plot displays the logical error rates for defect rates of 0.005, 0.01, 0.015, and 0.02 at a physical error rate of p = 0.002. The whiskers extend to data within 1.5 times the interquartile range (IQR) from the box. Points beyond are fliers. The outlier arises from two factors: (1) sampling defective devices at a rate of p involves independently assigning each component a defect probability of p, rather than exactly pN defects across N components, leading to variation in defect counts even at a consistent defect rate; (2) the spatial distribution of defects, such as when multiple defects align in a straight line or cluster in a specific area, can significantly impact the code distance. The green, blue, and red dashed lines represent references for a perfect surface code with physical error rates of p = 0.002, 0.003, and 0.004, respectively. In simulations, we generated 100 devices with randomly distributed defects for each L and defect rate.