Construction of Quantum Error Correcting Codes for Single Deletion Errors

Taro Shibayama

Construction conditions of quantum error-correcting codes for various types of insertion/deletion errors

Markus Grassl

Taro Shibayama

This paper provides new instances of quantum error-correcting codes. According to recent results by Nakayama and Hagiwara, by using two sets of binary sequences that satisfy three conditions, we can construct quantum error-correcting codes for single deletion errors. However, only two instances of two sets that satisfy the three conditions have been found to date. The present study constructs two infinite series of sets that satisfy the three conditions. In other words, this paper constructs new infinite series of quantum error-correcting codes for single deletion errors.

Manabu Hagiwara

This manuscript is an extended abstract version of the paper entitled ``Quantum Deletion Codes derived from Classical Deletion Codes.'' The paper contributes to the fundamental theory for quantum deletion error-correcting codes. The paper proposes a code construction condition for a partition of classical deletion error-correcting codes to derive quantum deletion error-correcting codes. The construction methods in this paper give examples of quantum codes that can correct single-quantum deletion errors and have a code rate arbitrarily close to 1, while the previously known quantum deletion code rates are close to 0 for long length. This manuscript omits the proofs of the statements in the paper.

Ayumu Nakayama, Manabu Hagiwara

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.

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