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

Lewis Bulled, Yingkai Ouyang

TL;DR

This work addresses quantum synchronization errors—insertions and deletions—in quantum error correction by focusing on permutation-invariant (PI) codes. The authors prove a quantum insertion–deletion equivalence on PI codes: a $t$-insertion error-correctable PI code is also $t$-deletion error-correctable, by showing the $t$-insertion conditions (C1),(C2) are equivalent to the $2t$-deletion conditions; they connect these to the Knill–Laflamme framework via Vandermonde decomposition. They then extend the framework to insdel errors by analyzing semi-insdel and full-insdel channels, deriving conditions (C3)-(C6) and establishing a channel-swap decomposition that reduces full-insdel correction to shifted semi-insdel/insertion and deletion criteria. These results imply that deletion PI codes with distance at least $t+1$ automatically serve as insertion codes, and they open questions about universality of insertion–deletion equivalence beyond PI codes and for more general error models. The findings provide a rigorous route to quantum synchronization-error correction in PI codes, with potential impact on quantum communication and DNA-like quantum storage paradigms.

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.

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

TL;DR

This work addresses quantum synchronization errors—insertions and deletions—in quantum error correction by focusing on permutation-invariant (PI) codes. The authors prove a quantum insertion–deletion equivalence on PI codes: a -insertion error-correctable PI code is also -deletion error-correctable, by showing the -insertion conditions (C1),(C2) are equivalent to the -deletion conditions; they connect these to the Knill–Laflamme framework via Vandermonde decomposition. They then extend the framework to insdel errors by analyzing semi-insdel and full-insdel channels, deriving conditions (C3)-(C6) and establishing a channel-swap decomposition that reduces full-insdel correction to shifted semi-insdel/insertion and deletion criteria. These results imply that deletion PI codes with distance at least automatically serve as insertion codes, and they open questions about universality of insertion–deletion equivalence beyond PI codes and for more general error models. The findings provide a rigorous route to quantum synchronization-error correction in PI codes, with potential impact on quantum communication and DNA-like quantum storage paradigms.

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 -insertion error-correctable. We extend these conditions to quantum insdel errors, formulating a more restrictive set of conditions under which permutation-invariant codes are -insdel error-correctable. Our work resolves many of the outstanding questions regarding the quantum error correction of synchronisation errors.
Paper Structure (6 sections, 4 theorems, 46 equations, 1 figure)

This paper contains 6 sections, 4 theorems, 46 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Permutation-invariant quantum codes
  3. Quantum insertion errors
  4. Quantum insdel errors
  5. Discussions & conclusions
  6. Acknowledgements

Key Result

Theorem 1

Let $\mathcal{C}$ be a PI code with logical coefficients $\vec{\alpha}, \vec{\beta}$. Then $\mathcal{C}$ is $t$-insertion error-correctable if and only if for all $0 \leq a', b' \leq t-j$ and $0 \leq j \leq t$, $\vec{\alpha}, \vec{\beta}$ satisfy

Figures (1)

  • Figure 1: Respective action of the permuted Kraus operators $\sigma K_{\vec{a}, \vec{v}}, \sigma K_{\vec{b}, \vec{u}}$ on the logical codewords $\ket{0_L}, \ket{1_L}$. The permutation $\sigma \coloneqq \sigma_3 \circ \sigma_2 \circ \sigma_1$ is such that the qubits overlap by $j$, where $0 \leq j \leq t$, with the qubit count underbraced. Here, $\sigma_1$ acts non-trivially on the full $(N+t)$-qubit register and arranges the $t$ inserted qubits on the top row leftwards; $\sigma_2$ then acts non-trivially on the first $t$ qubits and arranges $j$ inserted qubits on the bottom row rightwards; finally, $\sigma_3$ acts non-trivially on the last $N$ qubits and arranges the remaining $t-j$ inserted qubits on the bottom row leftwards. The top row depicts $\sigma \, K_{\vec{a}, \vec{v}} \ket{0_L}$, and likewise the bottom row $\sigma \, K_{\vec{b}, \vec{u}} \ket{1_L}$.

Theorems & Definitions (7)

  • Theorem 1: $t$-insertion conditions
  • proof
  • Lemma 1: Semi-insdel conditions
  • Lemma 2
  • proof
  • Theorem 2: Full-insdel conditions
  • proof