Permutation-invariant quantum coding for quantum deletion channels

TL;DR

This paper explores permutation-invariant quantum codes, especially shifted gnu codes, demonstrating their ability to correct quantum deletions with efficient encoding and decoding algorithms, thus advancing quantum error correction methods.

Contribution

It establishes that any permutation-invariant quantum code with a certain distance can correct quantum deletions and introduces shifted gnu codes with efficient algorithms.

Findings

Permutation-invariant codes can correct quantum deletions.

Shifted gnu codes have encoding complexity O(N).

Decoding algorithms for these codes run in O(N^2).

Abstract

Quantum deletions, which are harder to correct than erasure errors, occur in many realistic settings. It is therefore pertinent to develop quantum coding schemes for quantum deletion channels. To date, not much is known about which explicit quantum error correction codes can combat quantum deletions. We note that {\em any} permutation-invariant quantum code that has a distance of $t+1$ can correct $t$ quantum deletions for any positive integer $t$ in both the qubit and the qudit setting. Leveraging on coding properties of permutation-invariant quantum codes under erasure errors, we derive corresponding coding bounds for permutation-invariant quantum codes under quantum deletions. We focus our attention on a specific family of $N$-qubit permutation-invariant quantum codes, which we call shifted gnu codes, and show that their encoding and decoding algorithms can be performed in $O(N)$ and $O(N^2)$.