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.
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.