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 {\em any} permutation-invariant quantum code that has a distance of $t+1$ can correct $t$ quantum deletions for any positive integer $t$ in 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 of $N$-qubit permutation-invariant quantum codes, which we call shifted gnu codes, and show that their encoding and decoding algorithms can be performed in $O(N)$ and $O(N^2)$.

翻译：量子删除比消化错误更难纠正,在许多现实环境中发生。因此,为量子删除渠道制定量子编码方案是相关的。迄今为止,对于哪些明显量子误差校正代码可以对抗量子删除,还不清楚。我们注意到,在Qubit和Qudit设置中,任何正整数$t的调整-异差量代码都可以纠正以美元计算的量子删除。在消化错误中,我们利用对调和-异差量代码编码编码的编码属性,在量子删除中,我们得出相应的调和-异差量代码的编码界限。我们把注意力集中在一个特定的组合,即$-Qbit 倍变异量代码,我们称之为调换 gnu 代码,并显示其编码和解密算法可以用$(N)和$O(N)美元进行。