Figure 1

(Color online) (a) Measurement pattern on the first and last slice of $\mathcal{C}$, for $l=5$. The resulting state is in the code space of the planar code. (The unmeasured qubits are displayed as shaded circles.) (b) Lattice for the planar code. (c) and (d) Measurement pattern for even and odd inner slices.Reuse & Permissions

Figure 2

(Color online) The measurement pattern on the cluster $\mathcal{C}$. The sublattices $T_e$ and $T_o$ are displayed (thick lines). For reference, the cluster lattice is also shown (thin lines) and the axis labeling shows the cluster coordinates. Cluster qubits measured in the $Z$ basis (on the sites of $T_o$ and $T_e$) are displayed in black, and qubits measured in the $X$ basis (on the edges of $T_o$ and $T_e$) are displayed in gray (red). The large circles to left and to the right denote the unmeasured qubits which form the encoded Bell pair. The measurement pattern has a bcc symmetry.Reuse & Permissions

Figure 3

(Color online) A homologically nontrivial and a homologically trivial error cycle on the lattice $T_o$. The nontrivial error cycle stretches from one rough face to the opposite one while the trivial error has both ends on the same face. Only the qubits belonging to $T_o$ are shown and the qubits important for establishing the ${\overline{X}}_A{\overline{X}}_B$-correlation are displayed enlarged.Reuse & Permissions

Figure 4

This figure plots data for simulations of error correction on an $L\times L\times d_{\text{toric}}$ lattice, with periodic boundary conditions in the first two directions, for various $L$ and $d_{\text{toric}}$ $(d=2d_{\text{toric}}+1,l=2L)$. The error rate is $p=0.01$. The logarithms have are base $e$. Two standard deviations above and below the computed values (as given by statistical noise due to the sample sizes) are shown by the error bars. The solid lines each have slope 1, and they are spaced equally apart. This lends good support to the model of fidelity $F\sim \exp [-d{k}_1\exp (-l{k}_2)]$ for error rates below threshold.Reuse & Permissions

Figure 5

(Color online) (a) Correspondence between physical and virtual qubits. Domains are shown by dashed lines. (b) A bipartite cut of a cubic lattice. The regions $A$ and $B$ are highlighted.Reuse & Permissions