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arXiv:2602.20635

Insertion Correcting Capability for Quantum Deletion-Correcting Codes

Ken-Ichi Nakamura, Kaito Sato, Takayuki Nozaki

Published Feb 24, 2026• 0 citations
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Abstract

It is conjectured that a quantum code $C$ corrects $t$ deletion errors if and only if $C$ corrects a total of $t$ 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.

DOI
10.48550/arxiv.2602.20635

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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.
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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.
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.
We introduce a model of quantum insertion-deletion (insdel) channels. Insdel channels are meant to represent, for example, synchronization errors arising in data transmission. In the classical setting, they represent a strict generalization of the better-understood corruption error channels, and until recently, had mostly resisted effort toward a similar understanding as their corruption counterparts. They have received considerable attention in recent years. Very recently, Haeupler and Shahrasbi developed a framework, using what they call synchronisation strings, that allows one to turn insdel-type errors into corruption-type errors. These can then be handled by the use of standard error-correcting codes. We show that their framework can be extended to the quantum setting, providing a way to turn quantum insdel errors into quantum corruption errors, which can be handled with standard quantum error-correcting codes.
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 any permutation-invariant quantum code that has a distance oft+1t+1t+1can correcttttquantum deletions for any positive integertttin 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 ofNNN-qubit permutation-invariant quantum codes, which we call shifted gnu codes. The main result of this work is that their encoding and decoding algorithms can be performed inO(N)O(N)O(N)andO(N2)O(N^{2})O(N2).