Permutation-invariant quantum coding for quantum deletion channels

Abstract

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 $O(N^2)$.

arXiv comment5 pages, 2 figures

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• A Divide-and-Conquer Approach to Dicke State Preparation
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• Preparing Dicke states in a spin ensemble using phase estimation
• Imaging stars with quantum error correction
• Constructions of $\ell$-Adic $t$-Deletion-Correcting Quantum Codes
• Linear programming bounds for quantum channels acting on quantum error-correcting codes
• The equivalence between correctability of deletions and insertions of separable states in quantum codes