Analytical percolation theory for topological color codes under qubit loss

Quantum information theory has shown strong connections with classical statistical physics. For example, quantum error correcting codes like the surface and the color code present a tolerance to qubit loss that is related to the classical percolation threshold of the lattices where the codes are defined. Here we explore such connection to study analytically the tolerance of the color code when the protocol introduced in Vodola et al. [Phys. Rev. Lett. 121, 060501 (2018)] to correct qubit losses is applied. This protocol is based on the removal of the lost qubit from the code, a neighboring qubit, and the lattice edges where these two qubits reside. We first obtain analytically the average fraction of edges $r\left(p\right)$ that the protocol erases from the lattice to correct a fraction $p$ of qubit losses. Then, the threshold $p_c$ below which the logical information is protected corresponds to the value of $p$ at which $r\left(p\right)$ equals the bond-percolation threshold of the lattice. Moreover, we prove that the logical information is protected if and only if the set of lost qubits does not include the entire support of any logical operator. The results presented here open a route to an analytical understanding of the effects of qubit losses in topological quantum error codes.

Figure 1

Regular geometries of the color code. Regular trivalent and three-colorable lattices. (a) Lattice 4.8.8 where every node belongs to one square and two octagons. (b) Lattice 6.6.6 (or honeycomb) where every node belongs to three hexagons. (c) Lattice 4.6.12 where every node belongs to one square, one hexagon, and one dodecagon. The red shrunk lattice of the 4.8.8 geometry is (a.1) a square lattice, while the blue (a.2) and green (a.3) shrunk lattices are square lattices with double-bonds. The three shrunk lattices of the 6.6.6 geometry (b.1), (b.2), (b.3) are hexagonal lattices. The red, blue, and green shrunk lattices of the 4.6.12 geometry are (c.1) a kagome lattice, (c.2) a triangular lattice with double-bonds, and (c.3) a hexagonal lattice with double bonds, respectively.

Figure 2

Protocol to correct qubit losses on the color code. (i) Detect the lost qubit $i$ (orange circle). In this work we assume that the positions of the lost qubits are already known. We also show two string operators $l_q^{\sigma }$ (continuous line), and ${\tilde{l}}_q^{\sigma }=l_q^{\sigma }{g}_{\mathit{c}}^{\sigma }$ (dashed line) that differ by multiplication with the generator $g_{\mathit{c}}^{\sigma }$ defined on the face $\mathit{c}$. (ii) Choose a neighboring qubit $i_s$ as the sacrificed qubit (yellow circle), (iii) remove both $i$ and $i_s$ and modify the lattice: the faces $\mathit{a},\mathit{b}$ that contain both qubits are shrunk into ${\mathit{a}}^{\prime },{\mathit{b}}^{\prime }$, and the two faces $\mathit{c},\mathit{d}$ that contain only one of the removed qubits (lost and sacrificed) are merged into one face ${\mathit{c}}^{\prime }$. This correction erases the five edges adjacent to both qubits (dotted lines) and adds two new edges (dashed lines) such that all remaining qubits have an edge of each color. (iv) Check the existence of the logical information by searching for a well-defined logical operator (like ${\tilde{l}}_q^{\sigma }$) that does not have support on the removed qubits. (v) If the logical information exists, measure the redefined generators ${\mathit{a}}^{\prime },{\mathit{b}}^{\prime },{\mathit{c}}^{\prime }$. The well defined operators, like ${\tilde{l}}_q^{\sigma }$, remain valid logical operators in the redefined code.

Figure 3

Strings and string nets where logical operators have support. (a) 6.6.6 color code lattice with a blue string operator $l_q^{\sigma }$ on the continuous and discontinuous blue lines, and string-net operator ${\tilde{l}}_q^{\sigma }$. The string-net operator is composed by four paths represented by four continuous lines: (a.1) a red path in the red shrunk lattice, (a.2) two blue paths (the two continuous lines) in the blue shrunk lattice, (a.3) a green path in the green shrunk lattice. Here the blue string operator $l_q^{\sigma }$, which is not well defined because it has support on a lost qubit (the orange circle), is multiplied by the generator $g_{\mathit{f}}^{\sigma }$ on the blue face $\mathit{f}$ and transformed into the string-net operator ${\tilde{l}}_q^{\sigma }$ that does not have support on the lost qubit.

Figure 4

One loss corrections. There are three possible corrections $\mathit{\kappa }$ for an instance of one qubit loss $\mathit{i}=\left\{i\right\}$ (orange dot) depending on the selection of a neighboring qubit to sacrifice: (a) the qubit $i_s$ on the red edge, (b) the qubit $i_s^{\prime }$ on the blue edge, (c) the qubit $i_s^{″}$ on the green edge. We choose each correction with a probability $w_{\mathit{\kappa }}=1/3$. From left to right the number of red edges erased (red dotted lines) is ${\mathsf{R}}_{\mathit{\kappa }}=1$, ${\mathsf{R}}_{{\mathit{\kappa }}^{\prime }}=2$, and ${\mathsf{R}}_{{\mathit{\kappa }}^{″}}=2$. Therefore, the average number of edges erased from the red shrunk lattice by a one-loss event is $R_1=5/3$. This value is the same for every loss instance of one qubit loss and for every shrunk lattice.

Figure 5

Corrections of a loss instance with two qubit losses. To correct a loss instance $\mathit{i}=\{i_1,i_2\}$ like the one in (a) composed by two losses indicated with orange dots, the protocol first chooses the order in which the losses are going to be corrected. In this case, the order $i_1,i_2$ is chosen with a probability of $1/2$. To correct the first loss $i_1$, any of the three neighboring qubits can be chosen with a probability $1/3$ as the sacrificed qubit $i_{s_1}$. In (b.1) the loss $i_2$ has been chosen as the sacrificed qubit, so there is no need to correct the loss $i_2$. The correction is $\mathit{\kappa }=\left[i_{s_1}\right]$. The probability of this correction is $w_{\mathit{\kappa }}=(1/2)(1/3)=1/6$ and ${\mathsf{R}}_{\mathit{\kappa }}=1$ red edges are erased (red dotted lines). In (b.2) a qubit different from the loss $i_2$ has been chosen as the sacrificed qubit $i_{s_1}^{\prime }$ (yellow dot), and the lattice has been modified accordingly. Then, in (b.2.1) a sacrificed qubit $i_{s_2}^{\prime }$ has been chosen to correct the loss $i_2$ producing the final erasure of ${\mathsf{R}}_{{\mathit{\kappa }}^{\prime }}=3$ red edges with a probability $w_{{\mathit{\kappa }}^{\prime }}=(1/2){(1/3)}^2=1/18$, where the correction is ${\mathit{\kappa }}^{\prime }=[i_{s_1}^{\prime },i_{s_2}^{\prime }]$. Note that the new red edge added in (b.2) has not been counted as an erased edge in (b.2.1), because in ${\mathsf{R}}_{\mathit{\kappa }}$ we count only those edges erased from the original lattice.

Figure 6

Fraction of edges erased $r\left(p\right)$ as a function of the qubit loss rate $p$. The points correspond to the numerical estimation, while curves are the analytical estimation. The analytical results with the first three coefficients: $r\left(p\right)\simeq {\alpha }_{1}p+{\alpha }_2{p}^2+{\alpha }_3{p}^3$ for the red, blue, and green shrunk lattices are represented by the red, blue, and green points and curves respectively. The three curves are almost superposed. By comparing the analytical curves with the bond-percolation thresholds $r_c$ (horizontal lines that from the top correspond to the blue, green, and red shrunk lattices) taken from Table 1 we obtain the loss thresholds $p_c$ of the shrunk lattices. The numerical data are obtained by a Monte Carlo sampling of losses at various values of the qubit loss rate $p$. The points for the three colors are also almost superposed.

Figure 7

Energies $E_{\mathit{i}}$ of instances $\mathit{i}\in {\mathcal{I}}_{i_1}^{\text{(f-i)}}$ of two losses $i_1$, $i_2$ for the red shrunk lattice of the 6.6.6 geometry of the color code. In the horizontal axis we indicate the value of the interacting energies computed from the averaged number of edge erased [Eq. (10)]. These energies are rescaled by a factor of $2!\times 3^2=18$ that represents the number of all possible corrections for each of the two-loss instances. In the vertical axis we indicate the occurrence of each energy, i.e., the number of instances $\mathit{i}\in {\mathcal{I}}_{i_1}^{\text{(f-i)}}$ that have the same energy $E_{\mathit{i}}$. The unique instance that has the biggest energy (in absolute value) is depicted in (a), while one of the four instances with the smallest energy (in absolute value) is depicted in (b). The other three instances with the same energy as (b) can be found by lattice symmetries. The instance in (b) corresponds to an interacting instance since the red link between the two sacrificed qubits (yellow circles) is erased to correct both qubit losses.

Figure 8

Fraction of edges erased $r\left(p\right)$ as a function of the qubit loss rate $p$ for every shrunk lattice of the three regular geometries of the color code. The continuous lines correspond to the first three orders in the expansion of $r\left(p\right)$ in powers of $p$ [Eq. (7)]. The coefficients of this curve were obtained analytically without performing any approximation. The numerical data (dots) are obtained by a Monte Carlo sampling of losses at various values of the qubit loss rate $p$.

Figure 9

Convergence of the first three orders in the power expansion of the average fraction of edges erased $r\left(p\right)$ for the red shrunk lattice of the 4.8.8 geometry of the color code. We compute analytically the first three coefficients ${\alpha }_1,{\alpha }_2,{\alpha }_3$ in Eq. (7). The dotted line is the first order of the power expansion, the dashed line contains up to the second order, and the continuous line up to the third order. The lines approach the numerical data (red dots) as more orders are added. The numerical data are obtained by a Monte Carlo sampling of losses at various values of the qubit loss rate $p$ and a posterior scaling analysis.

Figure 10

Critical qubit loss rate $p_c$ and fundamental qubit loss rate $p_f$ obtained numerically. By sampling loss instances with a Monte Carlo method, we compute the values of $p_c$ (percolation), and $p_f$ (fundamental) for various code distances $L$ of the three regular geometries of the color code. The thresholds are plotted as a function of $1/L^{1/\nu }$ with a critical exponent $\nu =4/3$ as expected from the percolation theory. Red circles, blue squares, and green triangles represent the numerical data for the red, blue, and green shrunk lattices, respectively. The continuous lines fit the points and their intercepts (marked with the same symbols as the data) give the critical threshold in the limit $L\to \infty$. In the graphs (a) and (b) for the 4.8.8 lattice, the green shrunk lattice is not represented because it has the same geometry as the blue. In (c) and (d) the blue and the green shrunk lattices of the 6.6.6 lattice have the same geometry as the red, so only the red is represented. In (e) and (f) for the 4.6.12 lattice, the three shrunk lattices are represented.