The equivalence of quantum deletion and insertion errors on permutation-invariant codes

TL;DR

This paper establishes the conditions under which permutation-invariant quantum codes can correct insertion and deletion errors, advancing the understanding of quantum synchronization error correction.

Contribution

It provides the first detailed conditions for quantum insertion-deletion error correction on permutation-invariant codes, addressing a longstanding open problem.

Findings

Conditions for $t$-insertion error correction are derived.

Extended conditions for quantum insdel errors are formulated.

The work resolves key questions in quantum error correction of synchronization errors.

Abstract

Quantum synchronisation errors are a class of quantum errors that change the number of qubits in a quantum system. The classical error correction of synchronisation errors has been well-studied, including an insertion-deletion equivalence more than a half-century ago, but little progress has been made towards the quantum counterpart since the birth of quantum error correction. We address the longstanding problem of a quantum insertion-deletion equivalence on permutation-invariant codes, detailing the conditions under which such codes are $t$-insertion error-correctable. We extend these conditions to quantum insdel errors, formulating a more restrictive set of conditions under which permutation-invariant codes are $(t,s)$-insdel error-correctable. Our work resolves many of the outstanding questions regarding the quantum error correction of synchronisation errors.