Error correction and degeneracy in surface codes suffering loss

Many proposals for quantum information processing are subject to detectable loss errors. In this paper, we give a detailed account of recent results in which we showed that topological quantum memories can simultaneously tolerate both loss errors and computational errors, with a graceful tradeoff between the threshold for each. We further discuss a number of subtleties that arise when implementing error correction on topological memories. We particularly focus on the role played by degeneracy in the matching algorithms and present a systematic study of its effects on thresholds. We also discuss some of the implications of degeneracy for estimating phase transition temperatures in the random bond Ising model.

Figure 1

(a) Physical qubits (arrows) reside on the edges of a square lattice (dashed lines). A plaquette operator is a product of four $Z$ operators acting on the qubits within the plaquette (blue), and a star operator is a product of four $X$ operators acting on the qubits within the star (red). The logical $\mathrm{Z̄}$ operator is a product of $Z$ operators acting on qubits along the solid vertical (blue) line, and the logical $\mathrm{X̄}$ operator is a product of $X$ operators acting on qubits along the solid horizontal (red) line. (b) In the event of a qubit loss, an equivalent logical operator $\mathrm{Z̃}$ can be routed around the loss.

Figure 2

(a) Lattice with two lost physical qubits. The physical qubits are represented by the green lines between plaquettes. Representative physical qubits are labeled $1,\dots ,8$, and representative plaquettes are labeled $\mathsf{A},\mathsf{B},\mathsf{C},\mathsf{D},\mathsf{E}$. (b) Connectivity of (super)plaquettes, where each edge represents a physical qubit. (c) Degraded lattice showing superedges (thick lines). (d) Restored lattice with zero-weight edges (dashed) and edges between elements of superplaquettes with irregular weights (thick lines). Also shown are the test lines for calculating the homology class of $C=E+E^{\text{'}}$ on the restored lattice.

Figure 3

(Top) Statistical sampling of $p_{\mathrm{fail}}$ vs $p_{\mathrm{comp}}$. (Bottom) Same data scaled according to $x=(p_{\mathrm{comp}}-p_{c0})L^{1/{\nu }_0}$. Then $p_{\mathrm{fail}}(x)=a+bx$ is fitted to the scaled data. Sample standard-error bars are shown. Red circles are for $L=16$, green squares are for $L=24,$ and blue diamonds are for $L=32$.

Figure 4

Correctability phase diagram. The shaded region is correctable in the limit $L\to \infty$. Large, red points are the crossing point of $p_{\mathrm{fail}}$ calculated on $L=16$ and $L=24$ lattices; medium, green points are the crossing of $p_{\mathrm{fail}}$ (vs $p_{\mathrm{comp}}$) on $L=16$ and $L=32$ lattices; and small, blue points are the crossing point of $p_{\mathrm{fail}}$ on $L=24$ and $L=32$ lattices. The curve is a quadratic fit to the points in the interval $0⩽p_{\mathrm{loss}}⩽0.4$ for which finite size effects are negligible. It extrapolates through the point $(p_{\mathrm{loss}},p_{\mathrm{comp}})=(0.5,0)$. The dashed line is the linearized phase boundary for $p_{\mathrm{loss}}$ much less than the percolation threshold, given in Eq. (9).

Figure 5

Fit parameters $a,$ $b$, and ${\nu }_0$ for data collapse in Fig. 3 (bottom).

Figure 6

(Left) Scaling law for the size of the largest superplaquette on an $L\times L$ lattice. ${\mu }_L$ is the number of sites in the largest superplaquette, $s_{p_{\mathrm{loss}}}$ is the number of sites within a “percolated-correlation length,” $s_p=|p_{\mathrm{loss}}-p_{\mathrm{perc}}|{}^{\nu D}$, where $p_{\mathrm{perc}}=1/2$ is the percolation threshold for the lattice, $\nu =4/3$ is the correlation length scaling exponent, $D=91/48$ is the fractal dimension of the largest percolated cluster, and $d=2$ is the dimensions of the lattice. Points are statistical samples on different lattice sizes ranging from $L=2$ to 64, and the solid curve is a fitted scaling law, given by $\Phi (x)=0.116\log (7.74\hspace{0.5em}{x}^{D/d}+1)$ (adapted from Ref. [24]). (Right) Solving ${\mu }_L(p_{\mathrm{scale}})=L^2/2$ for $p_{\mathrm{scale}}$ as a function of $L$. When $p_{\mathrm{loss}}=p_{\mathrm{scale}}$, the largest superplaquette occupies half of the lattice.

Figure 7

The two minimum-weight matchings of the numbered nodes: $\{\{1,2\},\{3,4\}\}$ or $\{\{1,3\},\{2,4\}\}$. The former has a single minimum-weight matching path (green, dashed lines); the latter has nine (including the thick, blue line).

Figure 8

The critical error probability as a function of the temperature parameter $\tau$. The blue squares were calculated on even lattices with $L=16,$ 20, 24, 28, and 32. The red circles were calculated on odd lattices with $L=15,$ 19, 23, 27, and $31$. The same error instances were used to calculate $p_c$ for all $0.1⩽\tau ⩽1.5$, so the error bars are correlated. The values at $\tau =0$ are taken from [15]. The value at $\tau =2$ is shown to demonstrate the drop in the threshold as $\tau$ increases further.

Figure 9

The fraction of matchings that are minimum distance, for different values of $\tau$. As $\tau$ increases, an increasing proportion of the selected matchings are longer than the minimum distance matchings (i.e., ground-state matchings), but have sufficiently high degeneracy that their free energy is lower. All simulations shown were done at $p_{\mathrm{comp}}=0.1035$, on different sized lattices, as indicated. Note the disparity between the minimum-distance matching fraction on even (blue) and odd (red) lattices.