We examine exact and approximate error correction for multi-mode Fock state codes protecting against the amplitude damping noise. Based on a new formalization of the truncated amplitude damping channel, we show the equivalence of exact and approximate error correction for Fock state codes against random photon losses. Leveraging the recently found construction method based on classical codes with large distance measured in the $\ell_1$ metric, we construct asymptotically good (exact and approximate) Fock state codes. These codes have an additional property of bounded per-mode occupancy, which increases the coherence lifetime of code states and reduces the photon loss probability, both of which have a positive impact on the stability of the system. Using the relation between Fock state code construction and permutation invariant (PI) codes, we also obtain families of asymptotically good qudit PI codes as well as codes in monolithic nuclear state spaces.

翻译：本文研究针对振幅阻尼噪声的多模福克态码的精确与近似纠错。基于截断振幅阻尼信道的新形式化描述，我们证明了福克态码在随机光子损失下的精确纠错与近似纠错的等价性。借助近期发现的基于$\ell_1$度量下大距离经典码的构造方法，我们构建了渐近优（精确与近似）的福克态码。这些码具有各模式占据数有界的附加特性，可延长码态的相干寿命并降低光子损失概率，二者均对系统稳定性产生积极影响。通过福克态码构造与置换不变（PI）码之间的关联，我们还得到了渐近优的qudit PI码族以及单体核态空间中的码族。