# Can quantum deletion error correcting codes be constructed?

I'm wondering whether or not we can construct quantum deletion error correcting codes. The quantum deletion error is defined by the partial trace. If we can, could anyone give an example?

Such a code has not been discovered until a few months ago.

Ayumu Nakayama and Manabu Hagiwara discovered the first example. It encodes one qubit to eight qubits. The details are written in the following paper. The First Quantum Error-Correcting Code for Single Deletion Errors, Ayumu Nakayama, Manabu Hagiwara, IEICE Communications Express. Its DOI is https://doi.org/10.1587/comex.2019XBL0154

An improvement is found in the following paper. A Four-Qubits Code that is a Quantum Deletion Error-Correcting Code with the Optimal Length, Manabu Hagiwara, Ayumu Nakayama. The paper is available from https://arxiv.org/abs/2001.08405 This code encodes one qubit to four qubits. Its encoder and decoder are given. The paper proved there are neither two qubits deletion codes nor three qubits deletion codes. It means the four qubits code is optimal for code length.

These papers define deletion errors as partial trace operations.

• You should edit your answer to improve it, instead of deleting it and posting a new answer. (Note that deleted posts count against you.) – Norbert Schuch Feb 11 at 7:30
• Thanks for your advise. It is the first time for me to write comments at this website. Your advise is valuable for me. – quantum codes Feb 11 at 7:34

It is well known that quantum error correction can also correct erasure errors. In fact, a code which can correct $$k$$ general errors (in arbitrary locations) can correct erasure errors in $$2k$$ locations. Thus, any quantum error correction code serves as an example.

To learn more about that, you could e.g. consult Preskill's lecture notes or the book by Nielsen and Chuang.

• Thanks, but I think I led you to misunderstanding. In my sentence, "deletion error" is not "erasure error". A deletion error is an error that some quantum particles disappear. In classical coding theory, an erasure error is like 010→0e0, where e is the erasure symbol, and a deletion error is like 010→00. – user225954 Mar 26 '19 at 6:19
• I think this is not commonly considered, as usually one knows when a particle is lost. (Say, when you send photons you'd send them at a fixed rate, or - if the runtime is varying - you could accompany each by a classical "carrier" pulse e.g. on a different frequency.) – Norbert Schuch Mar 26 '19 at 11:14