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arXiv:2105.07214 · IEEE Information Theory Workshop (ITW)

The equivalence between correctability of deletions and insertions of separable states in quantum codes

Taro Shibayama, Yingkai Ouyang

Published Oct 17, 2021• 12 citations
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Abstract

In this paper, we prove the equivalence of inserting separable quantum states and deletions. Hence any quantum code that corrects deletions automatically corrects separable insertions. First, we describe the quantum insertion/deletion error using the Kraus operators. Next, we develop an algebra for commuting Kraus operators corresponding to insertions and deletions. Using this algebra, we prove the equivalence between quantum insertion codes and quantum deletion codes using the Knill-Laflamme conditions.

DOI
10.1109/itw48936.2021.9611450

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In this paper, we prove the equivalence of inserting separable quantum states and deletions. Hence any quantum code that corrects deletions automatically corrects separable insertions. First, we describe the quantum insertion/deletion error using the Kraus operators. Next, we develop an algebra for commuting Kraus operators corresponding to insertions and deletions. Using this algebra, we prove the equivalence between quantum insertion codes and quantum deletion codes using the Knill-Laflamme conditions.
Quantum synchronisation errors are a class of 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 ttt-insertion error-correctable. We extend these conditions to quantum insertion-deletion errors, formulating a more restrictive set of conditions under which permutation-invariant codes are (t,s)(t,s)(t,s)-insdel error-correctable. For completeness, we extend our approach to qudit errors on qudit permutation-invariant codes. Our work resolves many of the outstanding questions regarding the quantum error correction of synchronisation errors.
It is conjectured that a quantum code CCC corrects ttt deletion errors if and only if CCC corrects a total of ttt insertion and deletion errors. Although this conjecture has been proved only under certain conditions, it has not been proved in general. In this paper, we prove this conjecture for general quantum codes. Specifically, we show that a quantum state after an insertion error is represented as a product state of the original multi-qudit system and the inserted quantum state. Then, we prove this conjecture by combining the previous results and this property. In addition, we define generalized insertion errors by removing the quantum channel requirement from insertion errors and present their properties.
In this paper, we discuss a construction method of quantum deletion error-correcting codes. First of all, we define deletion errors for quantum states, an encoder, a decoder, and two conditions which is expressed by only the combinatorial language. Then, we prove that quantum deletion error-correcting codes can be constructed by two sets that satisfy the conditions. In other words, problems that correct the deletion errors for quantum states are reduced to problems that find the sets satisfying the condition by this paper. Also, we performed experiment of the codes over IBM Quantum Experience.