Figure 1

The stabilizers of the error correction substrate. Dots indicate the location of qubits. The dashed box is a unit cell of the lattice containing eight qubits and associated with eight independent stabilizers.Reuse & Permissions

Figure 2

Examples of errors in a color code. White dots indicate the locations of errors. White circles indicate stabilizers of changed sign. As drawn, each chain of errors is of a single type $X$ or $Z$ and the neighboring stabilizers of changed sign are of type $Z$ or $X$, respectively.Reuse & Permissions

Figure 3

Circuit showing how an additional syndrome qubit (top line of each figure) is used to measure red (dark) face (a) $Z$ stabilizers and (b) $X$ stabilizers.Reuse & Permissions

Figure 4

Underlying lattice of physical qubits. Dots represent qubits, lines represent tunable interactions between qubits. The numbered qubits are used in Figs. 6 and 7.Reuse & Permissions

Figure 5

Preparation of a four-qubit cat state using the five qubits indicated in Fig. 4. The states initially stored in qubits 1–4 are manipulated to form a cat state stored on qubits 2–5. The state stored in qubit 5 is used to check the cat state and is read out using qubit 1.Reuse & Permissions

Figure 6

Circuit giving the eigenvalue of the eight-qubit $X$ stabilizer associated with a green (light) or blue (medium) face.Reuse & Permissions

Figure 7

Circuit giving the eigenvalue of the eight-qubit $Z$ stabilizer associated with a green (light) or blue (medium) face.Reuse & Permissions

Figure 8

Examples of the three colors of boundaries and error chains of each color starting at each boundary and meeting at a single qubit.Reuse & Permissions

Figure 9

(a) Red (dark) defect created by measuring $\mathit{XX}$ and $\mathit{ZZ}$ on qubits indicated by a white line. The signs of these measurements determine the signs of the stabilizers of the neighboring faces of reduced size. (b) Red (dark) defect created by measuring a complete face. Equivalent blue (dashed square) and green (dashed diamond) boundary stabilizers with positive sign independent of the measurement results assuming no errors. These boundary stabilizers are always positive as they are equal to the product of the five complete face stabilizers contained within the defect. (c) Blue (medium) defect. (d) Green (light) defect.Reuse & Permissions

Figure 10

A defect of each color forming (a) a primal qubit and (b) a dual qubit. Note that the six primal $Z_L$ and dual $X_L$ operators are equivalent.Reuse & Permissions

Figure 11

Examples of error chains likely to lead to logical errors. (a) Half encircling a defect. (b) Half connecting a defect to a boundary of the same color. (c) Half connecting the three defects. (d) Half connecting a defect to another of the same color.Reuse & Permissions

Figure 12

(a) Procedure for expanding a red (dark) defect. It is conceptually simpler to imagine that the operators $\mathit{XX}$ and $\mathit{ZZ}$ are measured directly on the qubits on the white lines, but in practice these qubits can be ignored and only the stabilizers of the indicated partial faces measured. (b) Procedure for contracting a red (dark) defect. The full stabilizers of the indicated faces are measured once more. The signs of these face stabilizers are then corrected using the regular error correction procedure.Reuse & Permissions

Figure 13

Step-by-step deformation of a ring operator into a pair of tree operators via deformation of the associated defects to create an isolated region of lattice.Reuse & Permissions

Figure 14

Logical CNOT via braiding with a primal qubit (top) as control and dual qubit (bottom) as target. The appropriate mappings of logical stabilizers $\mathit{XI}\mapsto \mathit{XX}$, $\mathit{IX}\mapsto \mathit{IX}$, $\mathit{ZI}\mapsto \mathit{ZI}$, and $\mathit{IZ}\mapsto \mathit{ZZ}$ can be seen by tracing the deformation of one logical stabilizer at a time.Reuse & Permissions

Figure 15

Circuit constructed from available components implementing primal-primal logical qubit CNOT.Reuse & Permissions