发表机构
Chicago Quantum Institute and Pritzker School of Molecular Engineering, University of Chicago（芝加哥量子研究所与普林茨克分子工程学院，芝加哥大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过有限群对称性研究双模玻色子损耗码，推导Knill-Laflamme条件并构造最优码，证明其达到最大损耗距离，提出无限码族猜想。
AI 中文摘要
光子损耗是玻色子量子硬件（包括超导腔）中的主要噪声过程。在两个玻色子模式中，固定总光子数的量子比特编码保留了双轨量子比特的损耗检测优势，同时支持光子损耗校正。在此设置中，对任意编码器和解码器优化纠缠保真度（4≤n≤25）揭示了每个最佳码中的有限群结构：11个对应于二维不可约表示，11个对应于可约表示。受此启发，我们推导了任意有限群不变码的充分必要条件（Knill-Laflamme条件），将其构造简化为表示多重性空间上的方程。对于SU(2)的每个有限子群和每个张量秩，我们显式构造了施加所有相应损耗校正约束所需的最小算子数量。这些结果产生了大多数数值码的解析对应物，并预测了初始搜索遗漏的构造。精确的MacWilliams-Farkas证书表明，我们的码在22个扇区中的21个达到最大可能的损耗距离，并且我们证明了所得距离界随总光子数单调递增。在扫描之外，我们识别了一个二元多面体序列，其光子数n_d=⌈(3d^2+1)/4⌉，并构造了每个相应码直至距离d=10。据我们所知，构造的(n,d)=(28,6),(49,8),(76,10)码在其各自距离上给出了最小的报告n；通过d=9的所有成员都达到了固定n的LP距离界。总之，这些构造、证书和对称性缩减的约束计数为无限码族提供了证据，该码族被推测在最小光子数下达到每个距离。
英文摘要
Photon loss is a dominant noise process in bosonic quantum hardware, including superconducting cavities. Fixed-total-photon-number qubit encodings in two bosonic modes retain the loss-detection advantage of dual-rail qubits while supporting photon loss correction. Optimizing entanglement fidelity in this setting for $4\leq n\leq25$ over arbitrary encoders and decoders reveals finite-group structure in every best-found code: 11 correspond to two-dimensional irreducible representations and 11 to reducible ones. Motivated by this emergence, we derive the necessary-and-sufficient Knill-Laflamme conditions for arbitrary finite-group-invariant codes, reducing their construction to equations on representation multiplicity spaces. For every finite subgroup of $SU(2)$ and every tensor rank, we explicitly construct the minimum number of operators required to impose all corresponding loss-correction constraints. These results produce analytic counterparts to most numerical codes and predict constructions missed by the initial search. Exact MacWilliams-Farkas certificates show that our codes achieve the maximum possible loss distance in 21 of the 22 sectors, and we prove that the resulting distance bound is monotone in total photon number. Beyond the scan, we identify a binary-polyhedral sequence with photon number $n_d=\lceil(3d^2+1)/4\rceil$ and construct each corresponding code through distance $d=10$. To our knowledge, the constructed $(n,d)=(28,6),(49,8),(76,10)$ codes give the smallest reported $n$ for their respective distances; all members through $d=9$ attain the fixed-$n$ LP distance bound. Together, these constructions, certificates, and a symmetry-reduced constraint count provide evidence for an infinite code family conjectured to attain every distance at the minimum photon number.
Comments7-page Letter and 65-page Supplemental Material; 3 main figures and 28 supplementary figures
