Singer-差集 Qudit 稳定子码：来自 PG(d,q) 中非退化二次曲面的构造、结构定理与蒙特卡洛性能
Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q)PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance
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中文总结 AI 辅助
本文提出基于 Singer 差集和非退化二次曲面的 q 元 qudit 稳定子码族 Q(q,d)，通过五个结构定理和蒙特卡洛验证，在去极化噪声下实现比 Steane 码高 70 倍的每逻辑 qudit 性能改进。
中文摘要 AI 辅助
我们提出 Q(q,d)，一族 q 元非 CSS qudit 稳定子码，其中 q 为素数幂，维度 d ≥ 2，由 Singer 差集和 PG(d,q) 中的非退化二次曲面构造。校验矩阵 H = (A | M_Q A) 将 Singer-循环关联矩阵 A 与二次曲面循环矩阵 M_Q 耦合，一个消去引理利用恒等式 (k - lambda) = q^{d-1} = 0 mod p 使 H 在 F_q 上成为辛-各向同性的。五个结构定理构成了新颖性：一个互相关模常值恒等式 nu = 1 mod q（q 为奇数，d ≥ 3），具有两值模式，证明了 2-挠距离上限；一个 CRT 零点计数恒等式；一个显式 Clifford 桥，将 Q(q,2) 识别为平凡 Singer CSS 码的图态编码，其中 2021 年低密度扩展（LDS）符号翻转模式由对称循环矩阵实现；一个配对列刚性等式 d_min = R_P + 1；以及一个下界 d_min(Q(7,2)) ≥ 8，来自穷举 Singer-Frobenius 搜索。Hamada-Smith 特化给出了闭式逻辑维度 k(p,d) = theta_d(p) - C(p+d-1, d) - 1，旗舰码 Q(p,2) = [[p^2 + p + 1, p(p+1)/2, d_min]]_p 达到渐近速率二分之一。我们猜想对于奇素数 d_min(Q(p,2)) = p + 1，在 p = 5 时已证明，在 p = 7 时下界为 ≥ 8。实例包括 Q(3,2) = [[13,6,3]]_3，Q(5,2) = [[31,15,6]]_5，Q(7,2) = [[57,28, d≥8]]_7，以及 Q(3,4) = [[121,105,3]]_3。在 1.5 × 10^6 次试验上的蒙特卡洛模拟显示，在去极化率 10^-3 下，Q(5,2) 每个逻辑 qudit 相比 Steane [[7,1,3]]_2 实现了 70 倍的改进，且零静默逻辑错误。使用了教科书式的有界距离译码器；Berlekamp-Massey 风格的循环译码留作自然后续工作。
英文摘要
We propose Q(q,d), a family of q-ary non-CSS qudit stabilizer codes with prime power q and dimension d >= 2, constructed from Singer difference sets and non-degenerate quadrics in PG(d,q). The parity check H = (A | M_Q A) couples the Singer-circulant incidence matrix A with the quadric circulant M_Q, and a Cancellation Lemma uses the identity (k - lambda) = q^{d-1} = 0 mod p to make H symplectic-isotropic over F_q. Five structural theorems carry the novelty: a cross-correlation modular-constancy identity nu = 1 mod q at q odd, d >= 3, with a two-value pattern that proves a 2-torsion distance ceiling; a CRT zero-count identity; an explicit Clifford bridge identifying Q(q,2) as a graph-state encoding of the trivial Singer CSS code, with the 2021 low-density-spreading (LDS) sign-flip pattern realised by a symmetric circulant; a paired column-rigidity equality d_min = R_P + 1; and a lower bound d_min(Q(7,2)) >= 8 from an exhaustive Singer-Frobenius search. A Hamada-Smith specialisation yields the closed-form logical dimension k(p,d) = theta_d(p) - C(p+d-1, d) - 1, and the flagship Q(p,2) = [[p^2 + p + 1, p(p+1)/2, d_min]]_p attains asymptotic rate one-half. We conjecture d_min(Q(p,2)) = p + 1 for odd primes, proven at p = 5 and lower-bounded to >= 8 at p = 7. Instances include Q(3,2) = [[13,6,3]]_3, Q(5,2) = [[31,15,6]]_5, Q(7,2) = [[57,28, d>=8]]_7, and Q(3,4) = [[121,105,3]]_3. Monte-Carlo over 1.5 x 10^6 trials shows Q(5,2) achieves a 70-fold per-logical-qudit improvement over Steane [[7,1,3]]_2 at depolarising rate 10^-3 with zero silent logical errors. A textbook bounded-distance decoder is used; Berlekamp-Massey-style cyclic decoding is left as the natural sequel.
发表机构
- Rice University（莱斯大学）
- University of Southampton（南安普顿大学）
机构由 AI 辅助整理，请以论文原文为准。
