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10.55 跨码基准:A0/A1 几何诊断的普适性与三扇区结构

guobin ouyang

Published Sep 9, 2026• 0 citations
Publisher

Abstract

10.54 在 surface code 上验证了 A0/A1 拓扑分类的配对层显现(total_dist 3.00×、A1−A0 = L 标度律)。单码验证不足以确立"几何诊断"的普适性——本文把同一套管道(`diagnose_circuit`,电路无关入口)运行在结构完全不同的色码(color code,d=3,clorco 生成)上,与 surface code(L=4)做四噪声点扫描对照。核心结果:(1) **穿越提升(crossing lift)随噪声单调衰减**——双码一致:噪声使错误模式随机化,A0/A1 几何分离被冲掉,拓扑签名在阈值以下(码工作区)最强;(2) **色码(d=3,chromobius 解码)在噪声全程保持可区分**(crossing lift ≥ 1.80×,四噪声点全过 1.5 阈值),而 surface code 在 noise ≥ 0.02 后塌缩至 ~1.0×——三色(三扇区)结构对 A1 链的拓扑约束在高噪声区更持久;(3) **色码 cluster lift 低噪声最强(3.0×),高噪声稳定于 1.5× 平台**——A1 侧最大激发簇系统性大于 A0,但簇区分度(≤3.0×)弱于穿越率(≥1.8×),两者随噪声同向衰减。几何解读:A0/A1 分类是电路无关的普适诊断;三扇区结构(S₃ triality)越丰富,A1 链的几何分离越持久——这是几何论对量子纠错解码的可检验预言。全部数值可复现(§6)。

DOI
10.5281/zenodo.22682670

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10.54 在 surface code 上验证了 A0/A1 拓扑分类的配对层显现(total_dist 3.00×、A1−A0 = L 标度律)。单码验证不足以确立"几何诊断"的普适性——本文把同一套管道(diagnose_circuit,电路无关入口)运行在结构完全不同的色码(color code,d=3,clorco 生成)上,与 surface code(L=4)做四噪声点扫描对照。核心结果:(1) 穿越提升(crossing lift)随噪声单调衰减——双码一致:噪声使错误模式随机化,A0/A1 几何分离被冲掉,拓扑签名在阈值以下(码工作区)最强;(2) 色码(d=3,chromobius 解码)在噪声全程保持可区分(crossing lift ≥ 1.80×,四噪声点全过 1.5 阈值),而 surface code 在 noise ≥ 0.02 后塌缩至 ~1.0×——三色(三扇区)结构对 A1 链的拓扑约束在高噪声区更持久;(3) 色码 cluster lift 低噪声最强(3.0×),高噪声稳定于 1.5× 平台——A1 侧最大激发簇系统性大于 A0,但簇区分度(≤3.0×)弱于穿越率(≥1.8×),两者随噪声同向衰减。几何解读:A0/A1 分类是电路无关的普适诊断;三扇区结构(S₃ triality)越丰富,A1 链的几何分离越持久——这是几何论对量子纠错解码的可检验预言。全部数值可复现(§6)。
几何论的简并类方法(10.30/10.83)此前只在 AG 完备码(自家码族)上验证。本文把同一套方法应用到**主流码**——表面码(surface code)与 LDPC(hypergraph product 族),回答三个桥接问题:(1) 主流码的权重 1/2 层简并结构如何(唯一率、类大小、fail(2));(2) 与 AG 完备码(r≥2 零简并 fail=0;r=1 部分简并 fail≈0.3)对照,主流码处于什么位置;(3) A0/A1 拓扑诊断在 LDPC 上的行为(延续 10.55 的非格点退化发现)。核心结果:**主流码权重 1 错误全部 100% 恢复(与 AG 一致);权重 2 恢复率 HGP LDPC 58.7%→87.7%(随码长上升);AG r≥2 零简并(100%)。注(260827 勘误):初版 surface w2=33.3% 基于非合法构造(X/Z 同面不对易),已撤回待权威工具重算——'surface 结构性弱码'结论不成立。depolarizing 噪声验证:零简并码 AG(6,2)/AG(8,3) 的 p_L 理论值远低;sinter.Decoder 集成完成——AG 查表经 sinter.collect 的 p_L 低于 AG+MWPM(p=0.01 差 9.8×),解码器-码结构匹配清晰**。 ---
本文提出一个源自因果量度几何框架(CMG)的、可直接检验的定量预测:在有效维度可连续调节的人工量子系统(各向异性光晶格或超导量子处理器)中,多体关联的特征建立时间 τ 满足 τ(d_eff)/τ(3) = C(d_eff)/C(3),其中 C(d) = 2√π · Γ((d+1)/2) / Γ(d/2),C(3) = 4等价地,关联传播速度 v ∝ 1/C(d)。该预测为零参数拟合——曲线形状完全由 Γ 函数确定,无可调参数。关键数值:维度从 3 降到 2,τ 缩短 21.5%(τ(2)/τ(3)=0.785);升到 4,τ 延长 17.8%(1.178)。副预测:角关联零点 θ_d = arccos(1/√d),在 d=3 精确给出标准魔角 54.73561°,与已知物理对表,并非任意构造。初步对照显示现有 1D/2D 数据方向与预测相反,但受 U/J 未控制、速度定义未对齐等混杂因素限制,尚不能判定;本文据此强调需要"固定 U/J、只变维度"的控制变量实验。方案 A(各向异性光晶格连续维度 crossover)、方案 B(超导量子处理器,含非整数维 Sierpiński 构型 d≈1.89)、方案 C(角关联零点测量,纯几何量、无混杂因素、建议优先)均可实施。本文预先声明四条证伪判据 F1–F4,并诚实标注三处待厘清的口径。即便 CMG 框架最终被证伪,该维度标度曲线的系统测量本身亦构成新的实验事实,是对多体非平衡动力学理论的独立检验。
Color codes are promising alternatives to surface codes for universal fault-tolerant quantum computing due to their simpler lattice-surgery protocols and the transversal implementation of logical Clifford gates. However, their practical deployment has been limited by slower decoding algorithms and worse logical failure rates and thresholds compared to surface codes. Although AI-based logical-flip decoders have recently been proposed to address these challenges, no clear framework currently exists for implementing such decoders within the parallel block-wise decoding schemes in both space and time required for large-scale fault-tolerant computation. AI-based pre-decoders offer a scalable alternative due to their local nature. By performing spacelike corrections on physical qubits and timelike corrections on stabilizer measurements, pre-decoders are naturally compatible with parallel block-wise decoding schemes and lattice-surgery protocols. In this work, we introduce AI-based pre-decoders for triangular color codes. We present a novel neural-network architecture for their implementation and develop methods to simplify the complex training data generated by color-code syndrome-extraction circuits containing feedforward operations. Remarkably, we find that both logical failure rates (LERs) and runtimes improve relative to raw Chromobius decoding as the code distance increases. For example, at code distance d=31 and physical error rate p=0.3%p=0.3\%p=0.3%, our pre-decoder + Chromobius pipeline improves the logical failure rate by a factor of 347x while reducing runtime by 7.33x compared to raw Chromobius decoding alone. These results demonstrate that AI-based pre-decoding can substantially narrow the performance gap between color codes and surface codes, bringing color codes closer to practical large-scale fault-tolerant quantum computation.
We present two new color code circuits: one inspired by superdense coding and the other based on a middle-out strategy where the color code state appears halfway between measurements. We also present ``Chromobius'', an open source implementation of the möbius color code decoder. Using Chromobius, we show our new circuits reduce the performance gap between color codes and surface codes. Under uniform depolarizing noise with a noise strength of 0.1%0.1\%0.1%, the middle-out color code circuit achieves a teraquop footprint of 1250 qubits (vs 650 for surface codes decoded by correlated matching). Finally, we highlight that Chromobius decodes toric color codes better when given *less* information, suggesting there's substantial room for improvement in color code decoders.
10.86–10.92 建立了"扇区投影 → NO 基坍缩 → Bruhat 波包 WCI → 约束子流形 Newton"的求解器链,实测三项独立压缩(3.7×10⁶、289×、228×),乘积 ≈2.4×10¹¹,口号是"把精确化学精度的小分子 FCI 代价降 ~10¹⁰"。 本文把口号拆成**两条可证伪定理**并逐项裁决: - **定理 I(波包表示定理)**:类 C 分子的基态可用 poly(n)·log(1/ε) 个 Bruhat 波包表示到化学精度 ε。 - **定理 II(谱间隙下界)**:有隙共价类的扇区基态谱间隙 Δ_n ≥ c/n^α(不遭遇贫瘠高原)。 裁决结果(含一次 h1e 表象 bug 的修正,expB/C 旧数字作废): 1. **定理 II:实证成立**(expC 修正后,(H₂)_m 链 gap 收敛于 ~0.57 Ha,几乎与规模无关,α≈0)。 2. **定理 I 朴素版(裸行列式 poly):被决定性证伪**(expD/expE)。根因是**尺寸一致性**:m 个近独立单元的基态是乘积态,其行列式展开承载化学精度所需项数随 m **指数增长**(N_det ×~6.4/二聚体;加 EN-PT2 也只压到 ×~4.5)。这不是基选得不好(NO 基、PT2 均无效),而是**截断 CI 对广延多体态的内在缺陷**(= FCI 的 #P 困难)。 3. **重定向(定理 I†,开放纲领)**:10¹⁰ 不在"更多离散行列式",而在**紧表示**——有隙 + 局域 ⇒ 基态可用 **poly 键维张量网络 / CC 型指数 ansatz**(其行列式展开是指数个,参数却 poly 个)表示;"准确的尺子"正名 = **5.12 的离散→连续过渡**(Schubert/Bruhat 粗粒化 → 连续极限),恰是"一体两面"的连续面。 **方法论收获**:①"单粒子伪铺展"(基可消除)与"多体关联尾"(基不可消除)必须分开度量——NO 基只压前者(引理 2 闭合);②尺寸一致性要求任何"能量误差 ≤ ε 的行列式数"测量必须用加性测度;③expB/C/D/E 曾因 h1e 表象 bug 全数污染,修正后图景完全不同——已全程记录。 ---