论文标题
MDS纠缠辅助的量子代码是任意长度和任意距离的
MDS Entanglement-Assisted Quantum Codes of Arbitrary Lengths and Arbitrary Distances
论文作者
论文摘要
量子误差校正对于量子信息处理和计算至关重要。自Shor和Steane的开创性论文以来，已经研究和构建了量子误差校正代码。最佳（称为MDS）$ q $ q $ qubit量子代码达到量子单顿绑定的限制的长度非常有限，$ n \ leq q^2+1 $。提出了纠缠辅助量子误差校正（EAQEC）代码，以使用前共享的最大纠缠状态来增强误差校正能力。最近，达到了量子单例的MDS EAQEC代码的许多结构，这些构造的长度非常有限。在本文中，我们构建了此类MDS EAQEC $ [[N，K，D，C]] _ Q $代码，用于任意$ n $满足$ n \ leq q^2+1 $和任意距离$ d \ leq \ leq \ frac \ frac {n+2} {2} {2} $。事实证明，对于任何给定的长度$ n $满足$ o（q^2）= n \ leq q^2+1 $和任何给定的距离$ d $满足$ o（q^2）= d \ leq \ frac {n+2} {2} {2} {2} $，至少存在$ o（q^2）$ o（q^2）$ mdse $ cod $ cod $ code $ cod $ cod $ cod $ cod $ cod $ cod $ cod $ [ 参数。我们的结果表明，与MDS量子代码相比，MDS纠缠的量子代码要多得多，而无需消耗最大纠缠状态。从物理的角度来看，这是自然的。我们的方法还可以应用于从广义MDS扭曲的Reed-Solomon代码中构建MDS纠缠辅助量子代码。
Quantum error correction is fundamentally important for quantum information processing and computation. Quantum error correction codes have been studied and constructed since the pioneering papers of Shor and Steane. Optimal (called MDS) $q$-qubit quantum codes attaining the quantum Singleton bound were constructed for very restricted lengths $n \leq q^2+1$. Entanglement-assisted quantum error correction (EAQEC) code was proposed to use the pre-shared maximally entangled state for the enhancing of error correction capability. Recently there have been a lot of constructions of MDS EAQEC codes attaining the quantum Singleton bound for very restricted lengths. In this paper we construct such MDS EAQEC $[[n, k, d, c]]_q$ codes for arbitrary $n$ satisfying $n \leq q^2+1$ and arbitrary distance $d\leq \frac{n+2}{2}$. It is proved that for any given length $n$ satisfying $O(q^2)=n \leq q^2+1$ and any given distance $d$ satisfying $ O(q^2)=d \leq \frac{n+2}{2}$, there exist at least $O(q^2)$ MDS EAQEC $[[n, k, d, c]]_q$ codes with different $c$ parameters. Our results show that there are much more MDS entanglement-assisted quantum codes than MDS quantum codes without consumption of the maximally entangled state. This is natural from the physical point of view. Our method can also be applied to construct MDS entanglement-assisted quantum codes from the generalized MDS twisted Reed-Solomon codes.
