Necessary and sufficient condition for constructing a single qudit insertion/deletion code and its decoding algorithm

TL;DR

This paper extends quantum error correction theory to include insertion and deletion errors, providing a necessary and sufficient condition for constructing single qudit insertion/deletion codes and their decoding algorithms.

Contribution

It generalizes correction conditions for insertion and deletion errors to a necessary and sufficient level and constructs a new code with its decoding algorithm.

Findings

Knill-Laflamme condition applies to particle number-changing errors.

Correctabilities of single insertion and deletion errors are equivalent.

A new single qudit insertion/deletion code with decoding algorithm is constructed.

Abstract

This paper shows that Knill-Laflamme condition, known as a necessary and sufficient condition for quantum error-correction, can be applied to quantum errors where the number of particles changes before and after the error. This fact shows that correctabilities of single deletion errors and single insertion errors are equivalent. By applying Knill-Laflamme condition, we generalize the previously known correction conditions for single insertion and deletion errors to necessary and sufficient level. By giving an example that satisfies this condition, we construct a new single qudit insertion/deletion code and explain its decoding algorithm.