# Analytical percolation theory for topological color codes under qubit loss

## TL;DR

This paper analytically investigates the tolerance of topological color codes to qubit loss by connecting quantum error correction thresholds with classical percolation theory, providing a deeper understanding of their robustness.

## Contribution

It introduces an analytical method to determine the qubit loss threshold in color codes using percolation theory, advancing the understanding of their error correction capabilities.

## Key findings

- The threshold $p_c$ equals the bond-percolation threshold of the lattice.
- The protocol erases a fraction $r(p)$ of edges to correct loss.
- Logical information is protected if lost qubits do not fully support any logical operator.

## Abstract

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 [Phys. Rev. Lett. $\textbf{121}$, 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(p) $ 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(p) $ 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.