Twins Percolation for Qubit Losses in Topological Color Codes

In this Letter, we establish and explore a new connection between quantum information theory and classical statistical mechanics by studying the problem of qubit losses in 2D topological color codes. We introduce a protocol to cope with qubit losses, which is based on the identification and removal of a twin qubit from the code, and which guarantees the recovery of a valid three-colorable and trivalent reconstructed color code. Moreover, we show that determining the corresponding qubit loss error threshold is equivalent to a new generalized classical percolation problem. We numerically compute the associated qubit loss thresholds for two families of 2D color code and find that with $p=0.461\pm 0.005$ these are close to satisfying the fundamental limit of 50% as imposed by the no-cloning theorem. Our findings reveal a new connection between topological color codes and percolation theory, show high robustness of color codes against qubit loss, and are directly relevant for implementations of topological quantum error correction in various physical platforms.

Figure 1

(a) Excerpt of a hexagonal 2D color code. Physical qubits reside at the vertices of the lattice, and each plaquette $P$ hosts an $X$ and $Z$-type generator $S_P^X$, $S_P^Z$ of the stabilizer group of the code. Logical (string) operators ${\mathcal{O}}_G$ and ${\mathcal{O}}_R$ can be deformed by stabilizers, to evade qubit loss locations (white circles): for example, ${\mathcal{O}}_G$ is deformed by the generator $A$ into the lighter green path, whereas the red string ${\mathcal{O}}_R$ branches into an equivalent green and blue string by the action of generators $B$ and $C$. For clarity, only qubits on which ${\mathcal{O}}_G$ and ${\mathcal{O}}_R$ have support are shown. Note that the branched red operator belongs to all three shrunk lattices of plaquettes as shown in panels (b),(c),(d).

Figure 2

(a) Representation of a distance $d=6$ color code, on a 4.8.8 lattice, affected by three qubit losses (white circles). This code stores two logical qubits, with four logical operators defined along green and blue strings that connect opposite borders of the corresponding colors. Panels (b)–(d): Loss recovery protocol (see also main text). (b) Twin identification. (c) Link removal and lattice reconstruction. (d) Plaquette redefinition. (e) Reconstructed lattice after loss correction. A logical operator (e.g., the blue one) potentially affected by a loss can be transformed into an equivalent one by multiplication with a stabilizer. (f) Loss thresholds for a 4.8.8 lattice computed by checking percolation for the three logical operators (filled circles), branching (filled triangle) and the existence of solutions of Eq. (1) (filled square) only for the red logical operator. Thresholds are plotted as a function of $1/d^{1/\nu }$, with $d$ the logical distance and $\nu =4/3$ for the methods (I,II) while $\nu =1$ for the method (III). The intercepts of the black lines (marked with the same symbols as the data) represent the critical threshold for $d\to \infty$. (g) Probability to find a logical red operator using the methods (I,II,III) as a function of the loss rate $p$ for a 4.8.8 lattice of distance $d=36$. (h) Fraction of qubits left in the lattice for each of the three methods as a function of $1/d$. For methods (II) and (III), the fractions approach the fundamental 50% limit.