Twins Percolation for Qubit Losses in Topological Color Codes

TL;DR

This paper links quantum error correction in topological color codes with classical percolation theory, introducing a protocol for qubit loss recovery and determining loss thresholds close to fundamental limits.

Contribution

It presents a novel protocol for qubit loss recovery in topological color codes and establishes a new connection between quantum error correction and classical percolation processes.

Findings

Qubit loss thresholds are close to the fundamental limit of 0.461.

The proposed protocol guarantees valid reconstructed color code lattices.

Color codes show high robustness against qubit loss.

Abstract

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. Our findings reveal a new connection between topological color codes and percolation theory, show high robustness of color codes against qubit loss, and are relevant for implementations of quantum error correction in various physical platforms.