We study the error-correction properties of multi-mode Fock-state codes under amplitude-damping (AD) noise, focusing on the asymptotic regime in which the total excitation of the code states grows without limit and the number of photon losses induced by the noise scales linearly with it. In this setting, existing code families, which correct only sublinearly many photon losses, do not protect against amplitude-damping (AD) noise with a constant loss parameter. We address this gap by constructing asymptotically good Fock-state codes relying on random classical codes in the discrete simplex. Our approach is based on a new equivalence between approximate correction for the AD channel and exact or approximate correction of sufficiently many photon losses under a truncated AD channel. Unlike many standard constructions of random quantum codes, our construction introduces randomness through the underlying classical indexing structure. Randomization also enables another desirable feature: bounded per-mode occupancy, which limits the number of photons in any individual mode and thereby increases the coherence lifetime of the code states. Finally, via a relation between Fock-state codes and permutation-invariant codes, our results also yield asymptotically good families of qudit permutation-invariant codes as well as codes in monolithic nuclear state spaces.

翻译：我们研究了多模福克态码在振幅阻尼（AD）噪声下的纠错特性，重点关注渐近区域，其中码态的总激发无界增长，且噪声诱导的光子损失数量与之线性缩放。在此设定下，现有仅能纠正次线性光子损失的码族无法保护具有恒定损失参数的振幅阻尼（AD）噪声。我们通过利用离散单纯形中的随机经典码构建渐近优福克态码来填补这一空白。我们的方法基于AD信道的近似校正与截断AD信道下足够多次光子损失的精确或近似校正之间的新等价关系。与许多标准随机量子码构造不同，我们的构造通过底层经典索引结构引入随机性。随机化还带来了另一个理想的特性：有界每模占用率，它限制了任何单个模式中的光子数量，从而增加了码态的相干寿命。最后，通过福克态码与置换不变码之间的关系，我们的结果还产生了渐近优的qudit置换不变码族以及单一核态空间中的码族。