Constructions of $\ell$-Adic $t$-Deletion-Correcting Quantum Codes

TL;DR

This paper introduces two systematic methods for constructing quantum deletion-correcting codes, one for single deletion with asymptotic limitations and another for multiple deletions with better asymptotic properties, including conversion from stabilizer codes.

Contribution

It presents novel constructions for quantum deletion-correcting codes, including a method to convert stabilizer codes and incorporate entanglement assistance.

Findings

First construction corrects one deletion but is asymptotically bad.

Second construction corrects multiple deletions and is asymptotically good.

Conversion of stabilizer codes to deletion-correcting codes is possible.

Abstract

We propose two systematic constructions of deletion-correcting codes for protecting quantum information. The first one works with qudits of any dimension, but only one deletion is corrected and the constructed codes are asymptotically bad. The second one corrects multiple deletions and can construct asymptotically good codes. The second one also allows conversion of stabilizer-based quantum codes to deletion-correcting codes, and entanglement assistance.