This article contributes to an unsolved quantum information problem: the equivalence between insertion error-correctability and deletion error-correctability. The solution in Shibayama and Ouyang’s sense is provided in the single-error under the assumption of the purity of quantum codeword states On the other hand, a class of codes that can correct both single deletion and single insertion is provided.

This paper presents conditions for constructing permutation-invariant quantum codes for deletion errors and provides a method for constructing them. Our codes include examples of quantum codes that can correct two or more deletion errors, which have not been studied before. We also give examples of quantum codes that can correct both multiple-qubit errors and multiple-deletion errors. We discuss a generalization of the construction of our codes at the end.

In this paper, we prove the equivalence of inserting separable quantum states and deletions. Hence any quantum code that corrects deletions automatically corrects separable insertions. First, we describe the quantum insertion/deletion error using the Kraus operators. Next, we develop an algebra for commuting Kraus operators corresponding to insertions and deletions. Using this algebra, we prove the equivalence between quantum insertion codes and quantum deletion codes using the Knill-Laflamme conditions.