A generalization of the four qubits single insertion error-correcting code

Abstract

The first quantum insertion code is the four qubits code given by Hagiwara in 2021. Since then, several quantum insertion codes have been proposed, but they are all unrelated to Hagiwara’s four qubits code. This article provides a new class of quantum single insertion codes that include the Hagiwara code. Furthermore, this construction gives a new decoder for insertion errors in the Hagiwara codes.