Twins Percolation for Qubit Losses in Topological Color Codes

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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 lattice.
• Moreover, we show that determining the corresponding qubit loss error threshold is equivalent to a new generalized classical percolation process.
• We numerically compute the associated qubit loss thresholds for two families of 2D color code and find that these are close to satisfying the fundamental limit $p_\text{fund} = 0.461 \pm 0.005$ close to the 50% as imposed by the no-cloning theorem.