BB 码 · 参考文献BB Codes · References
QEC-Bank 条目导读 · BB 码 · 参考文献QEC-Bank entry guide · BB codes · References

三十二个键：
七个有名字，二十五个待登记 Thirty-two keys:
seven with metadata, twenty-five pending

「参考文献」侧面看起来是一张书单，其实是这个条目的地基：QEC-Bank 里凡不是机器重算出来的数字——Bravyi 等人的 0.8%（它是什么、怎么算出来的，见 4.3 节）、昆仑处理器上的 (8.91 ± 0.17)%、五条 claimed 的逻辑门、R5 上的四本代数词典、唯一那条关系——都必须挂在一个文献键上。BB 条目有 32 个键。打开 QEC-Bank 的参考文献页，你会看到一件不太体面的事：只有 7 条显示标题、作者、年份和 arXiv 链接，其余 25 条只有一行键名和一个灰色标签「metadata pending」。本篇先教你读懂一个键——它怎么造、登记在哪、在 card.yaml 的哪三种位置出现、被哪条记录引用、「metadata pending」精确地是什么意思；再把 7 条已登记的逐条讲清替谁作证；把 25 条待登记的按引用它的侧面分成五组，每条给出标题与 arXiv 号（本次修订逐条对照过 arXiv 摘要页），并一律标注「登记待补」；然后带你把一个键（sahay2026matching）从 card.yaml 一步一步追到导出页；最后诚实地说清：登记是这个条目目前最大的缺口——它意味着什么、不意味着什么、补起来要做什么。 The References facet looks like a reading list; it is really this entry's foundation. Every number in the QEC-Bank that the machine did not recompute, the 0.8% of Bravyi et al. (what it is and how it was obtained: section 4.3), the (8.91 ± 0.17)% on the Kunlun processor, the five claimed logical gates, the four algebraic dictionaries on R5, the single relation, must hang on a bibkey. The BB entry has 32 of them. Open the QEC-Bank references page and you meet something slightly embarrassing: only 7 show a title, authors, year and arXiv link; the other 25 show one line of key name and a grey chip, “metadata pending”. This guide first teaches you to read a key: how it is formed, where it is registered, in which three places of card.yaml it can appear, which record cites it, and what “metadata pending” means precisely. Then it explains, for each of the 7 registered keys, whose witness it is; groups the 25 pending keys by the facet that cites them, giving the title and arXiv id (each checked against its arXiv abstract page in this revision), each labelled “registration pending”; walks one key (sahay2026matching) step by step from card.yaml to the exported page; and finally says honestly that registration is this entry's largest open gap, what that means, what it does not, and what closing it takes.

前置：《BB 码 · 高中数学版》。这里没有新的物理，只有一套核对习惯：看到一个数字，先问它的键；看到一个键，先问它有没有名字。文献键的一般格式在《Surface Code · 参考文献》第 1 章里讲过，本页只讲 BB 条目特有的部分。 Prerequisite: “BB Codes · High-School Edition”. No new physics here, only a habit of verification: when you see a number, ask for its key; when you see a key, ask whether it has a name. The general format of a bibkey is covered in Chapter 1 of “Surface Code · References”; this page covers only what is specific to the BB entry.
开始 ↓Begin ↓ 全程约 35–45 分钟 · 在 QEC-Bank 中查看此条目 →About 35–45 minutes · Open in the QEC-Bank →
目录Contents
第 1 章Chapter 1

怎样读 BB 条目的文献表：键、登记表、三种住处How to read the BB entry's bibliography: keys, the registry, three homes

普通论文的 [12] 只告诉你「去文末找第 12 条」。QEC-Bank 的文献键是一个全局唯一、可被程序检查的名字：条目里每一句「文献说……」都挂在某个键上，每个键都应在一份受控登记表里有完整元数据。BB 条目把这套机制的两面都展示得很清楚——它有一半以上的键还没有完整元数据。这一章先解剖一个键，看登记表里两种不同的行，再数一数 32 个键在 card.yaml 里住在哪三种地方，最后列出本条目用到的全部来源词。 An ordinary paper's “[12]” only says “look up item 12 at the end”. A QEC-Bank bibkey is a globally unique, machine-checkable name: every “the literature says …” in the entry hangs on a key, and every key should have complete metadata in one controlled registry. The BB entry shows both sides of this mechanism, because more than half of its keys do not yet have complete metadata. This chapter dissects one key, looks at the two kinds of row in the registry, counts the three places where the 32 keys live in card.yaml, and lists every provenance word the entry uses.

1.1一个键的解剖，与登记表里的两种行Anatomy of a key, and the two kinds of registry row

看 bravyi2024bicycle：第一作者的姓（bravyi）+ 年份（2024）+ 一个选来帮助记忆的词（bicycle）。这个词常常取自标题，但不一定：这篇论文的标题是「High-Threshold and Low-Overhead Fault-Tolerant Quantum Memory」，里面没有 bicycle，这里取的是它研究的码族的名字（bivariate bicycle）。再看 wang2026lowoverhead：wang + 2026 + lowoverhead，这回取自标题（「Demonstration of low-overhead quantum error correction codes」里的 low-overhead 去掉连字符）。这个约定让键既好念又几乎不撞车——BB 条目里有三个姓 Wang 的键（wang2023abelian、wang2026coprime、wang2026lowoverhead），是三篇不同作者、不同年份或不同助记词的论文，靠后两段区分。所有键登记在 registry/refs.yaml。这份登记表里有两种行。完整的行有五个字段：id、title、authors、year、url（arXiv 链接），例如 bravyi2024bicycle 那一行。裸的行只有一个字段，写成一行 - {id: lin2023twoblock}：键已经被「占住」了名字，但没有任何元数据。站点构建脚本 tools/site/build.py 渲染参考文献页时只问一件事：title、authors、year、url 四个字段是否都在。都在，就显示带链接的标题、年份标签、作者和键；缺任何一个，就只显示 <code>键</code> 加一个灰色标签「metadata pending」。所以「metadata pending」的精确含义是：这个键在登记表里只有一行裸 id。它不是说论文不存在，也不是说引用它的那句话没有依据。第三种行是别名：- {id: bravyi2024high, alias_of: bravyi2024bicycle}，把一次文献调研用过的另一个键指向同一篇论文，而不算第二篇。BB 条目有两处用到别名：奠基扫描报告的备注写着「QEC-Bank 目前用 bravyi2024high，抓到的记录是 arXiv 2308.07915，另有任务把它改名为 bravyi2024bicycle」——现在改名已完成，旧键留作别名；译码扫描报告用 roffe2020bposd，卡片用 roffe2020decoding，登记表里 roffe2020bposd 是后者的别名。

Take bravyi2024bicycle: first author's surname (bravyi) + year (2024) + a word chosen as a mnemonic (bicycle). The word is often taken from the title, but not always: this paper is titled “High-Threshold and Low-Overhead Fault-Tolerant Quantum Memory”, which has no “bicycle” in it; here the mnemonic is the name of the code family it studies (bivariate bicycle). Or wang2026lowoverhead: wang + 2026 + lowoverhead, this time from the title (“low-overhead” in “Demonstration of low-overhead quantum error correction codes”, hyphen dropped). The convention keeps keys pronounceable and nearly collision-free; the BB entry has three keys surnamed Wang (wang2023abelian, wang2026coprime, wang2026lowoverhead), three papers by different authors, years or mnemonic words, told apart by the last two segments. All keys are registered in registry/refs.yaml, and that registry has two kinds of row. A complete row has five fields, id, title, authors, year, url (an arXiv link), like the row for bravyi2024bicycle. A bare row has one field, written on one line as - {id: lin2023twoblock}: the key's name is reserved, but no metadata is attached. When the site builder tools/site/build.py renders the references page it asks one question: are title, authors, year and url all present? If so it shows the linked title, a year chip, the authors and the key; if any is missing it shows only <code>key</code> plus a grey chip, “metadata pending”. So the precise meaning of “metadata pending” is: this key exists in the registry as a bare id only. It does not say the paper does not exist, nor that the sentence citing it lacks a source. A third kind of row is the alias: - {id: bravyi2024high, alias_of: bravyi2024bicycle} points a key used by some literature sweep at the same paper without counting a second paper. The BB entry uses aliases twice: the foundations scout report notes “QEC-Bank currently uses bravyi2024high; the fetched record is arXiv 2308.07915 and there is a separate task to rename it bravyi2024bicycle”, a rename since completed with the old key kept as an alias; and the decoders report uses roffe2020bposd where the card uses roffe2020decoding, the former being registered as an alias of the latter.

键key = 姓surname + 年份year + 助记词（常取自标题）mnemonic word (often from the title)
bravyi + 2024 + bicycle　·　wang + 2026 + lowoverhead
完整行complete row: { id, title, authors, year, url }
裸行bare row: { id }　·　别名行alias row: { id, alias_of }
title ∧ authors ∧ year ∧ url 都在 → 显示书目；
否则 → 「metadata pending」title ∧ authors ∧ year ∧ url all present → bibliographic display;
else → “metadata pending”

记号：id = 键本身；title、authors、year、url = 标题、作者、年份、链接四个元数据字段；alias_of = 「是……的别名」；∧ 读「且」，四个条件同时成立。BB 条目：32 个键中 7 个是完整行，25 个是裸行；2 个别名行指向其中的完整行。

Notation: id = the key itself; title, authors, year, url = the four metadata fields; alias_of = “alias of”; ∧ reads “and”, all four conditions holding at once. BB entry: of 32 keys, 7 are complete rows and 25 bare rows; 2 alias rows point at complete rows among them.

1.2键在 card.yaml 里的三种住处，与两组数目Three homes for a key in card.yaml, and two counts

在条目源文件 codes/bivariate_bicycle/card.yaml 里，键以三种方式出现。其一，条目级的 refs 列表：32 个键按顺序排好，从 mackay2004sparse 到 chandra2026distributed——这就是 QEC-Bank 参考文献页显示的那张表，一条不多一条不少，顺序也一样。其二，侧面记录里的 refs 列表：8 条表示阶梯记录、3 条译码器记录、5 条实现记录、1 条关系各带自己的几个键。例如译码器记录 decoders[2]（mwpm，「BB 对称性约化到环面状圆柱」）的 refs 只有 [sahay2026matching]；关系 relations[0] 的 refs 是 [wang2023abelian, lin2023twoblock]。其三，origin.ref：一个记录的「出处」字段可以指向一个键。五条逻辑操作全部写成 origin: {kind: claimed, ref: KEY}——每个门只挂一个键，并诚实地写着 claimed；表示阶梯上五条 curated 记录也各挂一个 origin.ref（wang2023abelian、bravyi2024bicycle、postema2025existence、chen2025anyon、sahay2026matching）；分类标签 qldpc 的 origin 是 curated, ref: bravyi2024bicycle。把三种住处数一遍，得到两组数目。16 个键至少被一条侧面记录或一个 origin.ref 绑定；16 个键只在条目级列表里，不绑定任何记录——本页叫它们「仅条目级」（这是本页的叫法，不是 QEC-Bank 的官方词）。仅条目级的 16 个是：mackay2004sparse、voss2025multivariate、wang2026coprime、eberhardt2024pruning、liang2025planar、steffan2025tile、koutsioumpas2025autdec、blue2026machine、pakhunov2026convergence、pakhunov2026peeling、cohen2022longrange、xu2023constant、breuckmann2026cups、xu2025batched、chen2025fusion、chandra2026distributed。它们支撑的是六份文献扫描报告和 Level-2 页里的论证（例如「变体与边界地图」的七根轴），而不是某一条可执行的断言。被引最多的键是 bravyi2024bicycle：7 处——4 条表示阶梯记录的 refs、1 条的 origin.ref、1 条译码器记录、1 个分类标签。

In the entry source codes/bivariate_bicycle/card.yaml, keys appear in three ways. First, the entry-level refs list: 32 keys in order, from mackay2004sparse to chandra2026distributed; this is exactly the table the QEC-Bank references page shows, no more, no less, in the same order. Second, refs lists inside facet records: the 8 representation-ladder records, 3 decoder records, 5 realization records and 1 relation each carry their own few keys. The decoder record decoders[2] (mwpm, “BB symmetry reduction to toric-code-like cylinders”), for instance, has refs [sahay2026matching] only; the relation relations[0] has [wang2023abelian, lin2023twoblock]. Third, origin.ref: a record's provenance field can point at a key. All five logical operations are written origin: {kind: claimed, ref: KEY}, one key per gate, honestly labelled claimed; the five curated records on the representation ladder each carry an origin.ref too (wang2023abelian, bravyi2024bicycle, postema2025existence, chen2025anyon, sahay2026matching); and the category tag qldpc has origin curated, ref: bravyi2024bicycle. Counting the three homes gives two numbers. 16 keys are bound to at least one facet record or origin.ref; 16 keys sit only in the entry-level list, bound to no record, which this page calls “entry-level only” (this page's term, not an official QEC-Bank one). The 16 entry-level-only keys are mackay2004sparse, voss2025multivariate, wang2026coprime, eberhardt2024pruning, liang2025planar, steffan2025tile, koutsioumpas2025autdec, blue2026machine, pakhunov2026convergence, pakhunov2026peeling, cohen2022longrange, xu2023constant, breuckmann2026cups, xu2025batched, chen2025fusion, chandra2026distributed. They support arguments in the six scout reports and on the Level-2 pages (for instance the seven axes of the “variant and boundary map”), not any executable assertion. The most-cited key is bravyi2024bicycle, in 7 places: the refs of 4 representation-ladder records, 1 origin.ref, 1 decoder record and 1 category tag.

32 = 16 (绑定到记录bound to a record) + 16 (仅条目级entry-level only)
32 = 7 (完整登记fully registered) + 25 (metadata pending)
两种切分互不相关：已登记的 7 个里 3 个仅条目级、4 个绑定到记录；
待登记的 25 个里 12 个绑定到记录、13 个仅条目级the two splits are independent: of the 7 registered, 3 are entry-level only and 4 record-bound;
of the 25 pending, 12 are record-bound and 13 entry-level only

记号：「绑定到记录」= 出现在某条 views / decoders / realizations / relations 记录的 refs 里，或某条记录（操作、curated 视图、分类）的 origin.ref 里；数目由本页对 card.yaml 里 legacy_card 一节（卡片里按 decoders、views、realizations 等侧面分列记录的那一部分；4.1 节的路径 legacy_card.decoders[2] 就在其中）逐路径统计得到。核对：7 个已登记里，mackay2004sparse、steffan2025tile、koutsioumpas2025autdec 仅条目级（3 个），panteleev2021degenerate、bravyi2024bicycle、roffe2020decoding、sayginel2025automorphisms 绑定到记录（4 个）。于是 16 个绑定到记录 = 4 个已登记 + 12 个待登记，16 个仅条目级 = 3 个已登记 + 13 个待登记，两边各 16，合计 32。第 2、3 章逐条给出每个键的位置。

Notation: “bound to a record” = appears in the refs of some views / decoders / realizations / relations record, or in the origin.ref of some record (operation, curated view, category); counts come from this page's path-by-path tally of the legacy_card section of card.yaml (the part of the card that lists records facet by facet, decoders, views, realizations and so on; §4.1's path legacy_card.decoders[2] lies in it). Check: of the 7 registered, mackay2004sparse, steffan2025tile and koutsioumpas2025autdec are entry-level only (3) and panteleev2021degenerate, bravyi2024bicycle, roffe2020decoding and sayginel2025automorphisms are record-bound (4). Hence 16 record-bound = 4 registered + 12 pending, 16 entry-level only = 3 registered + 13 pending, 16 + 16 = 32. Chapters 2 and 3 give every key's location one by one.

1.3本条目用到的全部来源词Every provenance word the entry uses

为什么每条不能重算的论断都要指向一个键？因为 QEC-Bank 把「机器算出来的」和「文献说的」严格分开。BB 条目的分界写在直觉文稿「Why these parameters」一节：n = 144、k = 12、检查重量 6、比特度 6 是构造器对显示出来的矩阵重算的结果；gross 的 d = 12 分成两半——文献下界（bravyi2024bicycle）排除重量小于 12 的逻辑算符，QEC-Bank 用 rand-lowweight-v1（种子 5、2000 次）重跑出一个重量 12 的上界见证，「两个界相遇，所以记录的码距是 12，来源词是 bounded。QEC-Bank 没有复现文献的下界证书；这一处剩余的信任分工是明写的。」阈值、硬件结果、逻辑门则永远是文献论断。于是每个不能重算的数字都需要一个键，而每个键旁边都有一个来源词告诉你这句话有多硬。下表是 BB 条目里出现的全部来源词，每个都给出它在卡片上的确切位置：

Why must every claim that cannot be recomputed point at a key? Because the QEC-Bank keeps “what the machine computed” strictly apart from “what the literature says”. For the BB entry the split is written in the intuition essay's section “Why these parameters”: n = 144, k = 12, check weight 6 and qubit degree 6 are the constructor's recomputation of the displayed matrices; gross's d = 12 splits in two, a literature lower bound (bravyi2024bicycle) excluding logicals below weight 12 and a QEC-Bank rerun of rand-lowweight-v1 (seed 5, 2,000 iterations) finding a weight-12 upper witness: “The bounds meet, so the recorded distance is 12 with bounded provenance. The QEC-Bank has not reproduced the literature's lower-bound certificate; that remaining division of trust is explicit.” Thresholds, hardware results and logical gates are literature claims, always. Hence every number that cannot be recomputed needs a key, and beside every key a provenance word tells you how hard the sentence is. The table lists every provenance word in the BB entry with its exact location on the card:

词Word含义Meaning在 BB 卡片上的位置Where on the BB card
verified机器重算并断言相等machine-recomputed and asserted equalgross 的 n = 144、k = 12；toric_L3 的 n = 18、k = 2、d = 3；分类标签 css 的 origin.kindgross n = 144, k = 12; toric_L3 n = 18, k = 2, d = 3; origin.kind of the category tag css
bounded文献下界 + 机器见证上界相符，未穷举literature lower bound + machine witness upper bound agree, not exhaustivegross 的 d = 12（键 bravyi2024bicycle + rand-lowweight-v1 种子 5）gross d = 12 (key bravyi2024bicycle + rand-lowweight-v1 seed 5)
claimed来自所引论文，QEC-Bank 未重算from the cited paper, not recomputed全部 5 条逻辑操作的 origin.kind（键 yoder2025tour ×2、eberhardt2024fold、sayginel2025automorphisms、liang2025selfdual）origin.kind of all 5 logical operations (keys yoder2025tour ×2, eberhardt2024fold, sayginel2025automorphisms, liang2025selfdual)
demonstrated / proposed有实验论文报告 / 设计具体但无物理系统执行过reported by an experimental paper / concrete design, never executed physically5 条实现记录的 status：1 条 demonstrated（wang2026lowoverhead），4 条 proposed（poole2025rydberg、berthusen2025local、shaw2025morphing、yoder2025tour）status of the 5 realization records: 1 demonstrated (wang2026lowoverhead), 4 proposed (poole2025rydberg, berthusen2025local, shaw2025morphing, yoder2025tour)
standard / research-grade文献公认的基线译码器 / 研究级的专门译码器an accepted baseline decoder / a research-grade specialised decoder3 条译码器记录的 status：bp-osd standard（roffe2020decoding、panteleev2021degenerate）、bp-osd research-grade（bravyi2024bicycle）、mwpm research-grade（sahay2026matching）status of the 3 decoder records: bp-osd standard (roffe2020decoding, panteleev2021degenerate), bp-osd research-grade (bravyi2024bicycle), mwpm research-grade (sahay2026matching)
derived / canonical / curated转换器算出 / 照公认定义写下 / 从一篇文献整理（附 origin.ref）computed by a converter / written from an agreed definition / organised from one paper (with origin.ref)8 条表示阶梯记录：2 derived、1 canonical、5 curated；分类标签 qldpc 为 curated（表示阶梯导读 1.2 节）the 8 representation-ladder records: 2 derived, 1 canonical, 5 curated; the category tag qldpc is curated (representation-ladder guide, §1.2)
pending关系的目标条目尚不存在（构建时算出）the relation's target entry does not exist yet (computed at build time)relations[0] 的目标 two_bga（关系导读 1.3 节）——和「metadata pending」是两个不同的 pendingthe target two_bga of relations[0] (relations guide, §1.3); a different pending from “metadata pending”

记号：n、k、d = 数据比特数、逻辑比特数、码距（《从高考数学到 Toric Code》4.4 节与5.8 节）；origin.kind = 记录「出处」字段里的种类词；status = 译码器与实现记录的状态字段；rand-lowweight-v1 = QEC-Bank 随机低重量逻辑算符搜索器的名字与版本，种子 5 = 它的随机数起点。

Notation: n, k, d = data qubits, logical qubits, distance (§4.4 and §5.8 of “From High-School Math to the Toric Code”); origin.kind = the kind word in a record's provenance field; status = the state field of decoder and realization records; rand-lowweight-v1 = the name and version of the QEC-Bank's random low-weight logical search, seed 5 = its random-number starting point.

关键想法Key idea

一个文献键是一份契约：「这个数字不是我算的，是这篇论文说的，你可以去核对。」BB 条目的 32 份契约都签了名（键都在），但有 25 份还没填地址（没有标题、作者、年份、链接）。接下来两章分别读有地址的 7 份和没地址的 25 份，每一份都回答同一个问题：它在 QEC-Bank 里替谁作证。

A bibkey is a contract: “I did not compute this number; this paper states it, and you may check.” All 32 of the BB entry's contracts are signed (the keys exist), but 25 still lack an address (no title, authors, year, link). The next two chapters read the 7 with an address and the 25 without, each answering the same question: whose witness is it in the QEC-Bank?

下一章Next2 · 七条已登记的文献2 · Seven registered references
第 2 章Chapter 2

七条已登记的文献：有名字、有链接、替谁作证Seven registered references: named, linked, and whose witness each is

「已登记」= 登记表里五个字段齐全，QEC-Bank 页面上显示带链接的标题。先说一件容易想错的事：登记与否和「重要不重要」无关。bravyi2024bicycle 登记了，可 BB 的直接父类 wang2023abelian、lin2023twoblock 没有；登记了的七条里，roffe2020decoding、koutsioumpas2025autdec、sayginel2025automorphisms 都是好几个条目共用的键。下面七条的标题、作者、年份一律照 registry/refs.yaml 原文，每条说明它在卡片上替哪条记录作证。 “Registered” = all five fields present in the registry, a linked title shown on the QEC-Bank page. First, a tempting mistake: registration has nothing to do with importance. bravyi2024bicycle is registered while BB's direct parents wang2023abelian and lin2023twoblock are not; among the seven, roffe2020decoding, koutsioumpas2025autdec and sayginel2025automorphisms are keys shared by several entries. Titles, authors and years below follow registry/refs.yaml verbatim, and each item says which record on the card it testifies for.

2.1祖先与奠基：这个码从哪来、是什么Ancestry and foundation: where the code comes from and what it is

  • David J. C. MacKay, Graeme Mitchison, and Paul L. McFadden, "Sparse Graph Codes for Quantum Error-Correction" (2004)
    mackay2004sparse · arXiv quant-ph/0304161 · bicycle 码的出生地：一个变量的循环矩阵 C，HX = HZ = [C | Cᵀ]。仅条目级：它不绑定任何侧面记录，支撑的是 Level-2 页「变体与边界地图」里的家谱一段与关系导读 2.1 节。扫描报告的提醒：它是「历史起点，但不是今天两变量 BB 版图的来源」。登记表里的标题写作「Error-Correction」（带连字符），扫描报告写作「Sparse-Graph Codes for Quantum Error Correction」；arXiv 摘要页的标题是「Sparse Graph Codes for Quantum Error-Correction」，与登记表一致。Birthplace of bicycle codes: one-variable circulant C, HX = HZ = [C | Cᵀ]. Entry-level only: bound to no facet record, it supports the ancestry paragraph of the Level-2 “variant and boundary map” and §2.1 of the relations guide. The scout report's caution: “the historical starting point, but not the source of the two-variable BB layout used today”. The registry spells the title “Error-Correction” with a hyphen, the scout report “Sparse-Graph Codes for Quantum Error Correction”; the arXiv abstract page has “Sparse Graph Codes for Quantum Error-Correction”, matching the registry.
  • Pavel Panteleev and Gleb Kalachev, "Degenerate Quantum LDPC Codes With Good Finite Length Performance" (2021)
    panteleev2021degenerate · arXiv 1904.02703 · 广义 bicycle 码（两个不同的一变量多项式、「交换再转置」的两块格式）、有限长度搜索、BP-OSD 译码器。绑定到 decoders[0]（bp-osd，status standard，噪声「memoryless Pauli and phenomenological qLDPC benchmarks」）：它替「BP 给软可靠度、OSD 修复退化与短圈造成的失败」这句话作证。既是祖先，也是工具（关系导读 2.2 节）。Generalized bicycle codes (two different one-variable polynomials, the “swap then transpose” two-block format), finite-length search, the BP-OSD decoder. Bound to decoders[0] (bp-osd, status standard, noise “memoryless Pauli and phenomenological qLDPC benchmarks”): it testifies for “BP supplies soft reliabilities and OSD repairs failures caused by degeneracy and short cycles”. Ancestor and tool at once (relations guide, §2.2).
  • Sergey Bravyi, Andrew W. Cross, Jay M. Gambetta, Dmitri Maslov, Patrick Rall, and Theodore J. Yoder, "High-Threshold and Low-Overhead Fault-Tolerant Quantum Memory" (2024)
    bravyi2024bicycle （别名 bravyi2024high）(alias bravyi2024high) · arXiv 2308.07915 · 条目的原始论文，被引最多的键，7 处：表示阶梯 R1 symplectic-css-checks、R5 bivariate-polynomial、R2 homological 的 refs；R4 tanner-circuit 的 origin.ref 与 refs（「curated from bravyi2024bicycle」：两层平面度 3 的耦合层）；decoders[1]（bp-osd，research-grade：卡片把 0.8% 叫作电路级阈值；论文里它是逐个码解出的盈亏平衡点，v1 与 v2 版本给的数不同，4.3 节逐项说清；另见《BB 码 · 高中数学版》6.4 节）；分类标签 qldpc 的 origin.ref。此外它还是 gross 码 d = 12 那个 bounded 里的下界来源（参数导读 3.3 节）。The entry's source paper and its most-cited key, in 7 places: the refs of the R1 symplectic-css-checks, R5 bivariate-polynomial and R2 homological ladder records; the origin.ref and refs of R4 tanner-circuit (“curated from bravyi2024bicycle”: two planar degree-3 coupling layers); decoders[1] (bp-osd, research-grade: the card calls 0.8% a circuit-level threshold; in the paper it is a break-even point solved code by code, and versions v1 and v2 print different numbers; section 4.3 goes through it; see also §6.4 of “BB Codes · High-School Edition”); and the origin.ref of the category tag qldpc. It is also the source of the lower bound inside the gross code's bounded d = 12 (parameters guide, §3.3).

2.2译码器：一条基线，一条对称性技巧Decoders: one baseline, one symmetry trick

  • Joschka Roffe, David R. White, Simon Burton, and Earl Campbell, "Decoding Across the Quantum Low-Density Parity-Check Code Landscape" (2020)
    roffe2020decoding （别名 roffe2020bposd）(alias roffe2020bposd) · arXiv 2005.07016 · BP-OSD 作为有限 qLDPC 码的稳健基线（《qLDPC 码 · 高中数学版》6.3 节）。绑定到 decoders[0]。表面码条目的 bp-osd 记录也用这个键，所以它早已登记。登记表的标题写「Quantum Low-Density Parity-Check Code Landscape」，arXiv 摘要页写的是「Decoding Across the Quantum LDPC Code Landscape」（发表于 Physical Review Research 2, 043423, 2020）：意思相同，字面不同，页面照登记表显示。注意一个细节：导出页上 decoders[1]（0.8% 那条）的 References 行也显示它，但卡片里那条记录的 refs 只有 bravyi2024bicycle——多出来的键来自译码器目录 registry/decoders.yaml 里 bp-osd 的默认 refs，导出时合并进来（4.1 节会再遇到同样的事）。BP-OSD as the robust baseline for finite qLDPC codes (§6.3 of “qLDPC Codes · High-School Edition”). Bound to decoders[0]. The surface-code entry's bp-osd record uses this key too, which is why it was registered long ago. The registry title says “Quantum Low-Density Parity-Check Code Landscape”, while the arXiv abstract page says “Decoding Across the Quantum LDPC Code Landscape” (published in Physical Review Research 2, 043423, 2020): same meaning, different wording; the page shows the registry's. One detail: on the exported page the References line of decoders[1] (the 0.8% record) also shows it, yet that record's refs in the card hold only bravyi2024bicycle; the extra key comes from the default refs of bp-osd in the decoder catalog registry/decoders.yaml, merged in at export (§4.1 meets the same thing again).
  • Stergios Koutsioumpas, Hasan Sayginel, Mark Webster, and Dan E Browne, "Automorphism Ensemble Decoding of Quantum LDPC Codes" (2025)
    koutsioumpas2025autdec · arXiv 2503.01738 · 在码的自同构上并行跑多个译码器再合并候选，利用 BB 的对称性而不是把每次 BP 失败当作独立事件。仅条目级：来自译码扫描报告，卡片上没有一条译码器记录引用它——QEC-Bank 的三条译码器记录只登了两种 BP-OSD 和一种匹配约化。Runs decoders in parallel across code automorphisms and merges candidates, exploiting BB symmetry instead of treating every failed BP instance independently. Entry-level only: it comes from the decoders scout report, and no decoder record on the card cites it; the QEC-Bank's three decoder records list only two BP-OSD settings and one matching reduction.

2.3逻辑操作与变体：一条 claimed 的门，一种开边界的瓷砖Operations and variants: one claimed gate, one open-boundary tile

  • Hasan Sayginel, Stergios Koutsioumpas, Mark Webster, Abhishek Rajput, and Dan E. Browne, "Fault-Tolerant Logical Clifford Gates from Code Automorphisms" (2025)
    sayginel2025automorphisms · arXiv 2409.18175 · 搜索自同构诱导的物理置换并算出它们的逻辑 Clifford 作用，含 BB 例子。绑定到 operations[2]：「logical Clifford operations through automorphisms」，mechanism fold-transversal，origin claimed，cost 原文「a certified Tanner-graph permutation and its physical routing; the induced symplectic action must be computed for the chosen block」。人话：Tanner 图的一个对称只有在算清它对逻辑 Pauli 的作用之后才算门——这正是《BB 码 · 高中数学版》6.5 节「平移是自同构，不是门」的出处之一。Searches automorphism-induced physical permutations and computes their logical Clifford action, with BB examples. Bound to operations[2]: “logical Clifford operations through automorphisms”, mechanism fold-transversal, origin claimed, cost verbatim “a certified Tanner-graph permutation and its physical routing; the induced symplectic action must be computed for the chosen block”. In plain words: a symmetry of the Tanner graph counts as a gate only once its action on logical Paulis is worked out, one source of “translations are automorphisms, not gates” in §6.5 of “BB Codes · High-School Edition”.
  • Vincent Steffan, Shin Ho Choe, Nikolas P. Breuckmann, Francisco Revson Fernandes Pereira, and Jens Niklas Eberhardt, "Tile Codes: High-Efficiency Quantum Codes on a Lattice with Boundary" (2025)
    steffan2025tile · arXiv 2504.09171 · 把反复出现的 BB 式稳定子瓷砖截成开边界平面码；扫描报告的结论「共享一块内部瓷砖不意味着与环面 BB 码相等」。仅条目级：支撑 Level-2 页边界轴的论证（关系导读 4.2 节），不绑定记录。它是七根变体轴牵涉的十个键里唯一已登记的。登记表里第二作者写作「Shin Ho Choe」、链接带 v1 后缀；扫描报告写作「Seung-Hoon Choe」。arXiv 摘要页（2026-09-24 核对）写的是「Shin Ho Choe」，与登记表一致，所以这里是扫描报告拼错了。Terminates recurring BB-like stabilizer tiles into open-boundary planar codes; the scout report's conclusion: “sharing a bulk tile does not imply equality to a toric BB code”. Entry-level only: it supports the boundary-axis argument of the Level-2 page (relations guide, §4.2) and is bound to no record. It is the only registered key among the ten the seven variant axes involve. The registry spells the second author “Shin Ho Choe” and the link carries a v1 suffix; the scout report writes “Seung-Hoon Choe”. The arXiv abstract page (checked 2026-09-24) says “Shin Ho Choe”, matching the registry, so the scout report has the spelling wrong.
数目核对Count check

3 + 2 + 2 = 7，与 QEC-Bank 参考文献页上显示完整书目的七条一一对应。其中绑定到记录的 4 个：panteleev2021degenerate（decoders[0]）、bravyi2024bicycle（7 处）、roffe2020decoding（decoders[0]）、sayginel2025automorphisms（operations[2]）；仅条目级 3 个：mackay2004sparse、koutsioumpas2025autdec、steffan2025tile。七条的元数据全部照抄登记表，本页没有替任何一条「修正」拼写。

3 + 2 + 2 = 7, matching one to one the seven items with full bibliographic display on the QEC-Bank references page. Four are record-bound: panteleev2021degenerate (decoders[0]), bravyi2024bicycle (7 places), roffe2020decoding (decoders[0]), sayginel2025automorphisms (operations[2]); three are entry-level only: mackay2004sparse, koutsioumpas2025autdec, steffan2025tile. All seven sets of metadata copy the registry; this page has “corrected” no spelling.

上一章Back1 · 怎样读 BB 条目的文献表1 · Reading the BB bibliography 下一章Next3 · 二十五条待登记3 · Twenty-five pending
第 3 章Chapter 3

二十五条待登记的键，按引用它的侧面分组Twenty-five pending keys, grouped by the facet that cites them

QEC-Bank 页面上这 25 条只有键名。下面按「哪个侧面引用它」分成五组：关系与表示阶梯、译码器、逻辑操作、用例与实现、变体。每条给出作者、标题、年份与 arXiv 号，并一律标注「登记待补」；然后说它在卡片上替哪条记录作证（若不绑定记录，就说它支撑哪份报告或哪一页的论证），并链接到讨论它的那一页导读。六份扫描报告的检索日期都是 2026-07-26，每条都标 arxiv_verified: true（扫描时核对过 arXiv 编号）；本次修订又把 25 条逐一对照了 arXiv 摘要页（2026-09-24）。 On the QEC-Bank page these 25 show only a key name. Below they are sorted by the facet that cites them: relations and representation ladder, decoders, logical operations, use cases and realizations, variants. Each item gives authors, title, year and arXiv id, always labelled “registration pending”; then it says which record on the card it testifies for (or, if bound to none, which report or page's argument it supports) and links to the guide page that discusses it. All six scout reports were fetched on 2026-07-26 and mark every item arxiv_verified: true (the arXiv id was checked at scouting time); this revision checked all 25 against their arXiv abstract pages again (2026-09-24).

阅读约定Reading convention

本章每一条的粗体标题不是登记表里的内容——登记表里这些键只有一行裸 id。标题与作者最初取自 docs/lit/bivariate_bicycle/ 的六份扫描报告（QEC-Bank 为这个条目整理的文献综述，每条是一段 YAML 文本：一种「字段名: 值」逐行书写的数据格式）；本次修订已逐条对照 arXiv 摘要页，标题的大小写与作者全名按 arXiv 写，并改正了三处标题用词（chen2025anyon、poole2025rydberg、berthusen2025local）。键名里的年份是 QEC-Bank 定的，有的对应正式发表年份；arXiv 初版年份不同时写在括号里。标签 待补 的意思是「QEC-Bank 登记表里仍是裸行」。

The bold title of every item in this chapter is not registry content; in the registry these keys are bare ids. Titles and authors originally came from the six scout reports under docs/lit/bivariate_bicycle/ (the QEC-Bank's literature survey for this entry, each item a block of YAML text: a data format written line by line as “field: value”); this revision checked every item against its arXiv abstract page, writes title capitalisation and full author names as arXiv does, and corrects the wording of three titles (chen2025anyon, poole2025rydberg, berthusen2025local). The year in a key name is the QEC-Bank's choice and sometimes the year of formal publication; where the first arXiv version is from a different year it is given in brackets. The tag pending means “still a bare row in the QEC-Bank registry”.

3.1关系与表示阶梯：五本代数词典的出处Relations and representation ladder: sources of five algebraic dictionaries

  • Renyu Wang, Hsiang-Ku Lin, and Leonid P. Pryadko, "Abelian and non-abelian quantum two-block codes" (2023)
    wang2023abelian · 待补pending · arXiv 2305.06890 · 两块框架的定义：对易方阵与群代数group algebra：系数取 0、1 的多项式，其中 xl = 1、ym = 1；每个单项式是环面上的一个平移。见补课 3a。，维数公式与距离界；BB 是群取 Zl × Zm（l × m 环面上的全部平移）的可交换特例。替两条记录作证：relations[0]（special_case_of → two_bga）与表示阶梯 R5 group-algebra（origin curated from wang2023abelian）。关系导读 1.2 节、表示阶梯导读 6.2 节。Defines the two-block framework: commuting square matrices and group algebrasPolynomials with coefficients 0 and 1 in which xl = 1 and ym = 1; each monomial is a shift of the torus. See catch-up 3a., dimension formulas and distance bounds; BB is the Abelian special case whose group is Zl × Zm (all shifts of the l × m torus). Testifies for two records: relations[0] (special_case_of → two_bga) and the R5 ladder record group-algebra (origin curated from wang2023abelian). Relations guide §1.2, representation-ladder guide §6.2.
  • Hsiang-Ku Lin and Leonid P. Pryadko, "Quantum two-block group algebra codes" (2023)
    lin2023twoblock · 待补pending · arXiv 2306.16400 · 把 2BGA（两块群代数码）放进提升积lifted product：先写一个元素是多项式的小矩阵，再把每个多项式换成它的平移矩阵，拼成 HX、HZ。两个 1 × 1 矩阵 [a]、[b] 拼出来就是两块码。见补课 3a。框架（摘要称 2BGA 为「最小的提升积码」，意思见补课 3a），枚举小的最优例子；「每个 BB 都是可交换 2BGA，不是每个 2BGA 都是 BB」。替三条记录作证：relations[0]、R5 group-algebra 的 refs、R2 homological 的 refs（把「X 检查 → 数据比特 → Z 检查」排成三层、用箭头表示「碰到」的视角，写作 R → R² → R，见补课 3a）。关系导读 2.4 节、表示阶梯导读 3.1 节。Places 2BGA (two-block group-algebra codes) inside the lifted-productWrite a small matrix whose entries are polynomials, replace each polynomial by its shift matrix, and assemble HX, HZ. Two 1 × 1 matrices [a], [b] give exactly the two-block code. See catch-up 3a. framework (the abstract calls 2BGA codes the “smallest lifted-product codes”; catch-up 3a says what that means) and enumerates small optimal examples; “every BB code is Abelian 2BGA, not every 2BGA code is bivariate bicycle”. Testifies for three records: relations[0], the refs of R5 group-algebra, and the refs of R2 homological (the view that lays out “X checks → data qubits → Z checks” as three layers with arrows meaning “touches”, written R → R² → R; see catch-up 3a). Relations guide §2.4, representation-ladder guide §3.1.
  • Jasper Johannes Postema and Servaas J. J. M. F. Kokkelmans, "Existence and Characterisation of Bivariate Bicycle Codes" (2025)
    postema2025existence · 待补pending · arXiv 2502.17052 · 用商环 R/(A, B) 的结构（它的维数的 2 倍就是 k，见补课 3a）预测维数与存在条件，并证明所研究子类渐近是坏的码越做越大时，k/n 或 d/n 至少有一个趋于 0；「渐近好」要求两者都保持在某个正数以上。（码长趋于无穷时码率 k/n 或相对码距 d/n 趋于 0）。替 R5 quotient-module 作证（origin curated from postema2025existence：「共同的商 R/(A,B) 控制编码比特数……为什么所研究子类不是渐近好的家族」）。直觉文稿「Design problem」与「Failure modes」也引用它。表示阶梯导读 6.3 节、参数导读 5.4 节。Uses the structure of the quotient ring R/(A, B) (twice its dimension is k; see catch-up 3a) to predict dimension and existence conditions, and proves that the studied subclass is asymptotically badAs the codes grow, k/n or d/n tends to 0; “asymptotically good” requires both to stay above some positive number. (as the length goes to infinity, the rate k/n or the relative distance d/n goes to 0). Testifies for R5 quotient-module (origin curated from postema2025existence: “the common quotient R/(A,B) controls the number of encoded qubits … why the studied subclass is not an asymptotically good family”). The intuition essay's “Design problem” and “Failure modes” cite it too. Representation-ladder guide §6.3, parameters guide §5.4.
  • Keyang Chen, Yuanting Liu, Yiming Zhang, Zijian Liang, Yu-An Chen, Ke Liu, and Hao Song, "Anyon Theory and Topological Frustration of High-Efficiency Quantum Low-Density Parity-Check Codes" (2025)
    chen2025anyon · 待补pending · arXiv 2503.04699 · 用 Gröbner 基、Koszul 复形（两者各算什么，见补课 3a）与任意子anyon：二维码里一个或一小团亮起的检查，可以被局部算符移动但不能被局部算符单独消掉；环面码有 e、m 两种。（能被移动、却不能被局部消掉的一小团警报）分析 BB 的拓扑序，发现「拓扑受挫」：环面上的逻辑比特数与任意子种类数对不上（摘要：「ground-state degeneracy on a torus deviates from the total anyon number」）。替 R5 anyon-koszul 作证（origin curated from chen2025anyon：「堵住把每个有限 BB 码宣布为独立环面码堆叠的捷径」）。表示阶梯导读 6.4 节。Analyses BB topological order with Gröbner bases, Koszul complexes (what each computes: catch-up 3a) and anyonsIn a 2D code, one lit check or a small cluster of them that local operators can move but cannot remove on their own; the toric code has two kinds, e and m. (clusters of alarms that can be moved but not removed locally), finding “topological frustration”: the number of logical qubits on a torus does not match the number of anyon types (abstract: “ground-state degeneracy on a torus deviates from the total anyon number”). Testifies for R5 anyon-koszul (origin curated from chen2025anyon: “blocks the shortcut of declaring every finite BB code a trivial stack of independent toric codes”). Representation-ladder guide §6.4.
  • Kaavya Sahay, Dominic J. Williamson, and Benjamin J. Brown, "A matching decoder for bivariate bicycle codes" (2026)
    sahay2026matching · 待补pending · arXiv 2602.22770 · 挑出一组乘积为单位算符的检查（论文叫一个「对称性」）：任何错误都让这组检查里亮起偶数条，所以亮起的检查可以两两配对，用最小权完美匹配MWPM：把亮起的检查两两配成对，使所有配对线的总长度（总权重）最小的算法；表面码的标准译码器。（MWPM）找最省的配法。再用「圆柱技巧」把配对结果翻成对逻辑比特的判断：把环面沿两条线切成两个圆柱 U 和 V，U 里这组检查的乘积只剩两条切线附近的两条算符 P₁、P₂，其中 P₁ 是一个逻辑算符，记作 L̄；数配对线里有几条从 U 连到 V、并穿过 L̄ 那条切线，这个数的奇偶就是译码器对「错误是否翻转了 L̄」的估计（图 3.2）。估计可能出错：匹配配错了对，数出的奇偶就错。所以有「过度匹配」：对几组对称性以及它们的组合各匹配一次，在同一条切线上读出的 b₁、b₂、b₁₊₂ 必须满足 b₁ ⊕ b₂ = b₁₊₂（⊕ 是模 2 加），不满足就说明某次匹配错了，再据此改正；另外用 BP 给匹配重新定权。报告的数值：v1（2026-02）是 code-capacity 比特翻转噪声（只有数据比特出错，读数完美），v2（2026-06）加了现象学噪声（读数也会出错）。替两条记录作证：R2 toric-cylinder（origin curated from sahay2026matching：「一本可操作的译码词典，不是全局的相等价」）与 decoders[2]（mwpm，research-grade）。第 4 章用它做追踪示范。表示阶梯导读 3.2 节、译码器导读。Picks a set of checks whose product is the identity (the paper calls it a “symmetry”): any error lights an even number of checks in this set, so the lit checks can be paired up, and minimum-weight perfect matchingMWPM: pairs up the lit checks so that the total length (total weight) of the pairing lines is smallest; the standard surface-code decoder. (MWPM) finds the cheapest pairing. The “cylinder trick” then turns the pairing into a statement about a logical qubit: cut the torus along two lines into cylinders U and V; the product of the set’s checks inside U reduces to two operators P₁ and P₂ near the two cut lines, and P₁ is a logical operator, written L̄. Count the pairing lines that join U to V across the cut line of L̄; the parity of that count is the decoder’s estimate of whether the error flipped L̄ (Fig. 3.2). The estimate can be wrong: if matching pairs the wrong checks, the parity is wrong. Hence “over-matching”: match once on each of several symmetries and on their combinations; the readings b₁, b₂, b₁₊₂ taken on the same cut must satisfy b₁ ⊕ b₂ = b₁₊₂ (⊕ is addition mod 2), and a violation shows that some matching went wrong, which is then corrected; BP is also used to reweight the matching. Reported numbers: v1 (2026-02) uses code-capacity bit-flip noise (only data qubits err, readout is perfect); v2 (2026-06) adds phenomenological noise (readout can also err). Testifies for two records: R2 toric-cylinder (origin curated from sahay2026matching: “an operational decoder dictionary, not a global phase equivalence”) and decoders[2] (mwpm, research-grade). Chapter 4 uses it as the tracing example. Representation-ladder guide §3.2, decoders guide.
① 0 条穿过 ① 0 crossings 圆柱 U cylinder U 圆柱 V cylinder V L̄ = P₁ P₂ P₂ b = 0：估计没有翻转 L̄ b = 0: estimate L̄ not flipped ② 1 条穿过 ② 1 crossing 圆柱 U cylinder U 圆柱 V cylinder V L̄ = P₁ P₂ P₂ b = 1：估计翻转了 L̄ b = 1: estimate L̄ flipped ③ 2 条穿过 ③ 2 crossings 圆柱 U cylinder U 圆柱 V cylinder V L̄ = P₁ P₂ P₂ b = 0：两次翻转相消 b = 0: the two flips cancel
图 3.2Fig. 3.2圆柱技巧怎么读（示意图，不是某个真实 BB 码的格子）。环面沿两条竖线切开：浅蓝的左半是圆柱 U，灰色的右半是圆柱 V；左右两条虚线其实是同一条线 P₂（环面在那里接回），中间的蓝线是逻辑算符 L̄ = P₁。洋红点是这组对称性里亮起的检查，洋红线是匹配给出的配对。只数从 U 连到 V、并穿过蓝线 L̄ 的配对线，留在 U 或 V 内部的不算：① 0 条，b = 0；② 1 条，b = 1，译码器估计错误翻转了 L̄，它的修正里要多补一个 L̄；③ 2 条，两次翻转相消，b = 0，和 ① 一样。b 只是估计：匹配把点配错，数出的奇偶也会错，过度匹配就是用几组读数之间的 b₁ ⊕ b₂ = b₁₊₂ 来发现这种错。How to read the cylinder trick (a schematic, not the lattice of an actual BB code). The torus is cut along two vertical lines: the light-blue left half is cylinder U, the grey right half is cylinder V; the two dashed edges are really one line, P₂ (the torus closes there), and the blue line in the middle is the logical operator L̄ = P₁. Magenta dots are the lit checks of this symmetry; magenta lines are the pairs matching chose. Count only the pairing lines that join U to V across the blue line L̄; lines that stay inside U or inside V do not count: ① 0 lines, b = 0; ② 1 line, b = 1, the decoder estimates that the error flipped L̄, and its correction adds one L̄; ③ 2 lines, the two flips cancel, b = 0, the same as ①. b is only an estimate: if matching pairs the wrong dots the parity is wrong too, and over-matching detects such mistakes through b₁ ⊕ b₂ = b₁₊₂ between several readings.
补课 · 本章的七个代数名词：群代数、提升积、R → R² → R、商、Gröbner 基、Koszul 复形、杯积Catch-up · The seven algebra words of this chapter: group algebra, lifted product, R → R² → R, quotient, Gröbner basis, Koszul complex, cup product

这些论文的工具都在回答同一类问题：给定两条多项式 a、b，码有几个逻辑比特，逻辑算符长什么样。下面每个名词给出它算什么、一个能手算的例子、以及它换来了什么。例子用一个小码：l = 2、m = 3，a = 1 + xy + y²，b = 1 + x（补课 3b 从另一个方向算同一个码），以及图 3.1 的 l = m = 4、a = b = x + y。

The tools of these papers all answer the same kind of question: given two polynomials a and b, how many logical qubits does the code have, and what do its logical operators look like? For each word below: what it computes, one example you can do by hand, and what it buys. The examples use a small code, l = 2, m = 3, a = 1 + xy + y², b = 1 + x (catch-up 3b computes the same code from another direction), and the code of Fig. 3.1, l = m = 4, a = b = x + y.

① 群代数 R① The group algebra R

R = 变量 x、y 的多项式，系数只取 0 或 1（1 + 1 = 0），并规定 xl = 1、ym = 1（向右走 l 步、向上走 m 步都回到原处）。一个单项式 xayb 就是环面上的格子 (a, b)，一条多项式就是一组格子；乘一个单项式就是把整组格子平移。例（l = 2, m = 3）：(1 + x)·xy = xy + x²y = xy + y，因为 x² = 1：格子 (1,1) 与 (0,1)。它换来的是：检查的腿、平移、相乘抵消，都变成多项式乘法。

R = polynomials in x and y with coefficients 0 or 1 only (1 + 1 = 0), with the rules xl = 1 and ym = 1 (l steps right or m steps up bring you back). A monomial xayb is the cell (a, b) of the torus, a polynomial is a set of cells, and multiplying by a monomial shifts the whole set. Example (l = 2, m = 3): (1 + x)·xy = xy + x²y = xy + y, because x² = 1: the cells (1,1) and (0,1). What it buys: legs of checks, shifts and cancellations all become polynomial multiplication.

② 提升积② Lifted product

先写一个元素是 R 中多项式的小矩阵，再把每个多项式「提升」成它的 lm × lm 平移矩阵（x 换成「向右走一步」的置换矩阵，y 换成「向上走一步」的），拿两个这样的矩阵拼出 HX、HZ（一般大小的矩阵按「超图积」的格式拼，本页用不到）。最小的情形是两个 1 × 1 矩阵 [a] 与 [b]，拼出来正是 HX = [A | B]、HZ = [Bᵀ | Aᵀ]，也就是两块码；Lin 与 Pryadko 说 2BGA 是「最小的提升积码」，指的就是这一点。例：a = x + y、l = m = 4 提升成 16 × 16 的 A，每行两个 1；图 3.1 的码就是 [a] 与 [a] 的提升积。它换来的是：更大的矩阵能造出更多的码，而两块码是其中最小的一格。

Write a small matrix whose entries are polynomials in R, then “lift” each polynomial to its lm × lm shift matrix (x becomes the permutation matrix “one step right”, y “one step up”), and assemble HX, HZ from two such matrices (matrices of general size are assembled in the “hypergraph product” layout, which this page does not need). The smallest case is two 1 × 1 matrices [a] and [b], which give exactly HX = [A | B], HZ = [Bᵀ | Aᵀ], the two-block code; that is what Lin and Pryadko mean by 2BGA codes being the “smallest lifted-product codes”. Example: a = x + y with l = m = 4 lifts to the 16 × 16 matrix A with two 1s per row; the code of Fig. 3.1 is the lifted product of [a] and [a]. What it buys: larger matrices give many more codes, and the two-block codes are the smallest cell of that family.

③ R → R² → R：链复形与同调③ R → R² → R: chain complex and homology

三层对象：X 检查（每格一条，一份 R）、数据比特（左右两层，两份 R，写作 R²）、Z 检查（一份 R）。箭头表示「碰到」。第一支箭头把一组 X 检查 f 送到它们碰到的比特：(a·f, b·f)（左层、右层），矩阵是 HXᵀ；第二支把一组比特 (u, v) 送到碰它们奇数次的 Z 检查：b·u + a·v，矩阵是 HZ。连走两步：b·a·f + a·b·f = 2ab·f = 0，因为多项式乘法可以交换次序，而 1 + 1 = 0。这句话的意思是每条 X 检查与每条 Z 检查共用偶数个比特，也就是检查两两对易。逻辑比特是剩下的部分：第二支箭头送到 0 的比特组（没有 Z 检查发现它），去掉第一支箭头能得到的（X 检查的乘积），剩下的就叫这个复形的同调，k 是它的维数（X 与 Z 两种各一半）。例：图 3.1 的 X 串是 (u, v) = (1, 1)（多项式 1 = 格子 (0,0)，即 L(0,0) 与 R(0,0)），第二支箭头给出 b + a = 2(x + y) = 0，没有警报；它又不在第一支箭头的像里（左层奇偶性不对），所以它代表一个非零的同调类。它换来的是：「几个逻辑比特」变成拓扑里算同调的问题，现成的定理可以直接用。

Three layers: X checks (one per cell, one copy of R), data qubits (left and right layers, two copies, written R²), Z checks (one copy of R). Arrows mean “touches”. The first arrow sends a set of X checks f to the qubits they touch, (a·f, b·f) (left layer, right layer); its matrix is HXᵀ. The second sends a set of qubits (u, v) to the Z checks that touch it an odd number of times, b·u + a·v; its matrix is HZ. Two steps in a row: b·a·f + a·b·f = 2ab·f = 0, because polynomial multiplication can be done in either order and 1 + 1 = 0. In words: every X check shares an even number of qubits with every Z check, i.e. the checks commute. The logical qubits are what is left: the qubit sets the second arrow sends to 0 (no Z check detects them), minus those the first arrow produces (products of X checks); what remains is the homology of the complex, and k is its dimension (half from X, half from Z). Example: the X string of Fig. 3.1 is (u, v) = (1, 1) (the polynomial 1 = the cell (0,0), i.e. L(0,0) and R(0,0)); the second arrow gives b + a = 2(x + y) = 0, no alarm; and it is not produced by the first arrow (wrong left-layer parity), so it represents a nonzero homology class. What it buys: “how many logical qubits” becomes a homology computation from topology, where ready-made theorems apply.

④ 商 R/(a, b)④ The quotient R/(a, b)

(a, b) 是所有 f·a + g·b 组成的集合（f、g 取遍 R），叫 a、b 生成的理想；R/(a, b) 是把这个集合里的多项式都当作 0 之后剩下的东西。它的维数（还剩几个互相独立的多项式）直接给出 k：k = 2 · dim R/(a, b)。理由：③ 里第二支箭头 (u, v) ↦ b·u + a·v 的全部输出恰好就是理想 (a, b)，所以 (a, b) 的维数是 rank HZ，dim R/(a, b) = lm − rank HZ；BB 码的 rank HX 与 rank HZ 相等，所以 k = 2lm − rank HX − rank HZ = 2(lm − rank HZ) = 2 · dim R/(a, b)。例：gross 的 rank HX = rank HZ = 66，dim R/(a, b) = 72 − 66 = 6，k = 12（本页脚本）。它换来的是：k 变成一个只含 a、b 的代数问题，可以在不写出 144 列矩阵的情况下讨论。

(a, b) is the set of all f·a + g·b (f, g ranging over R), called the ideal generated by a and b; R/(a, b) is what is left once every polynomial in that set is treated as 0. Its dimension (how many independent polynomials remain) gives k directly: k = 2 · dim R/(a, b). Reason: the outputs of the second arrow of ③, (u, v) ↦ b·u + a·v, are exactly the ideal (a, b), so the dimension of (a, b) is rank HZ and dim R/(a, b) = lm − rank HZ; for BB codes rank HX equals rank HZ, so k = 2lm − rank HX − rank HZ = 2(lm − rank HZ) = 2 · dim R/(a, b). Example: gross has rank HX = rank HZ = 66, so dim R/(a, b) = 72 − 66 = 6 and k = 12 (this page’s script). What it buys: k becomes an algebra question about a and b alone, which can be discussed without writing out a 144-column matrix.

⑤ Gröbner 基⑤ Gröbner basis

理想的一组特别挑过的生成元，使「商有几维」可以机械地数出来：每个生成元有一个最高项，不能被任何一个最高项整除的单项式的个数就是 dim R/(a, b)。例（l = 2, m = 3，a = 1 + xy + y²，b = 1 + x）：b = 0 意味着 x = 1（模 2 里 −1 = 1）；代进 a 得 1 + y + y² = 0；而 y³ = 1 自动成立，因为 1 + y³ = (1 + y)(1 + y + y²)（展开：1 + y + y² + y + y² + y³，中间两对抵消）。所以这个理想由 x + 1 与 y² + y + 1 生成，这就是一组 Gröbner 基。最高项是 x 与 y²，不被它们整除的单项式只剩 1 和 y，所以 dim R/(a, b) = 2，k = 2 × 2 = 4。本页脚本按矩阵秩直接算，也是 k = 4（n = 12）。它换来的是：大的 l、m 上也能不列矩阵就数出 k。

A specially chosen set of generators of the ideal that makes “how many dimensions does the quotient have” a mechanical count: each generator has a leading term, and the number of monomials divisible by no leading term is dim R/(a, b). Example (l = 2, m = 3, a = 1 + xy + y², b = 1 + x): b = 0 means x = 1 (mod 2, −1 = 1); substituting into a gives 1 + y + y² = 0; and y³ = 1 then holds automatically, since 1 + y³ = (1 + y)(1 + y + y²) (expand: 1 + y + y² + y + y² + y³, the middle pairs cancel). So the ideal is generated by x + 1 and y² + y + 1, and that is a Gröbner basis. The leading terms are x and y²; the only monomials divisible by neither are 1 and y, so dim R/(a, b) = 2 and k = 2 × 2 = 4. This page’s script computes the matrix ranks directly and also gets k = 4 (n = 12). What it buys: k can be counted for large l, m without writing a matrix.

⑥ Koszul 复形⑥ Koszul complex

③ 里的三层链有一个代数里现成的名字：给两条多项式，第一支箭头把 f 同时乘上这两条（f ↦ (a·f, b·f)），第二支再交叉乘回去相加（(u, v) ↦ b·u + a·v），连走两步必为 0。这种链叫这两条多项式的 Koszul 复形，所以 BB 码的检查结构就是 (a, b) 的 Koszul 复形。例：图 3.1 的码 a = b = x + y，f = 1（(0,0) 格的一条 X 检查）经第一支箭头到 (x + y, x + y)，即 L(1,0)、L(0,1)、R(1,0)、R(0,1)，和图下正文列的腿一致；再经第二支箭头到 (x + y)² + (x + y)² = 0。它换来的是：代数里关于 Koszul 复形的定理可以直接拿来用；chen2025anyon 的摘要说他们用它把任意子与逻辑算符对应起来（「extend the connection between anyons and logical operators using Koszul complex theory」）。

The three-layer chain of ③ has a ready-made name in algebra: given two polynomials, the first arrow multiplies f by both (f ↦ (a·f, b·f)), the second multiplies crosswise and adds ((u, v) ↦ b·u + a·v), and two steps always give 0. Such a chain is the Koszul complex of the two polynomials, so the check structure of a BB code is the Koszul complex of (a, b). Example: for the code of Fig. 3.1, a = b = x + y, f = 1 (the X check of cell (0,0)) goes by the first arrow to (x + y, x + y), i.e. L(1,0), L(0,1), R(1,0), R(0,1), the legs listed in the text under the figure; the second arrow then gives (x + y)² + (x + y)² = 0. What it buys: theorems about Koszul complexes apply directly; the chen2025anyon abstract says they use it to relate anyons to logical operators (“extend the connection between anyons and logical operators using Koszul complex theory”).

⑦ 上同调不变量与杯积⑦ Cohomology invariants and cup products

把 ③ 的箭头反过来读，得到同一套计数的对偶版本，叫上同调；它数的是另一种字母的逻辑算符。一个不变量是一条规则：输入逻辑算符，输出 0 或 1，而且把输入乘上任何检查，输出不变，所以它只看逻辑信息。最简单的例子是「一个 X 型逻辑与一个 Z 型逻辑共用比特数的奇偶」。图 3.1 的码里，X 串 v = {L(0,0), R(0,0)} 与 Z 型逻辑 w = {L(0,0), L(1,3), L(2,2), L(3,1)} 共用 1 个比特，是奇数；w 与每条 X 检查共用偶数个比特，所以把 v 乘上任何一条 X 检查，奇偶不变（脚本：乘上一条 X 检查后 v 变成重量 6，共用数仍是 1）。杯积是把两个或更多上同调量相乘、得到新的上同调量的规则，由它能造出更高次的不变量。Breuckmann 等人给一般 CSS 码定义了杯积；按这种不变量的值给态乘上 ±1 的门是对角的逻辑门，可以用常数深度的电路实现；Λ 个量的杯积作用在同一个码的 Λ 份拷贝上，给出 Clifford 层级第 Λ 层的门（第 1 层是 Pauli 门，第 2 层是 Clifford 门，第 3 层含 CCZ 这类门），论文叫它 copy-cup 门。杯积在 CSS 码上具体怎样定义，超出本页，见原论文（arXiv 2410.16250）。

Reading the arrows of ③ backwards gives the dual version of the same count, called cohomology; it counts the logical operators of the other letter. An invariant is a rule that takes logical operators and returns 0 or 1, and whose output does not change when an input is multiplied by any check, so it sees only logical information. The simplest example is “the parity of the number of qubits shared by an X-type and a Z-type logical”. In the code of Fig. 3.1 the X string v = {L(0,0), R(0,0)} and the Z-type logical w = {L(0,0), L(1,3), L(2,2), L(3,1)} share 1 qubit, an odd number; w shares an even number of qubits with every X check, so multiplying v by any X check keeps the parity (script: after one X check v has weight 6 and still shares 1 qubit with w). A cup product is a rule that multiplies two or more cohomology quantities into a new one, and it yields higher-degree invariants. Breuckmann et al. define cup products for general CSS codes; the gate that multiplies a state by ±1 according to such an invariant is a diagonal logical gate that a constant-depth circuit can implement; a cup product of Λ quantities acting on Λ copies of the same code gives a gate in level Λ of the Clifford hierarchy (level 1 = Pauli gates, level 2 = Clifford gates, level 3 includes gates such as CCZ), which the paper calls the copy-cup gate. How the cup product is defined on a CSS code is beyond this page; see the paper (arXiv 2410.16250).

3.2译码器：扫描报告带进来、卡片没有引用的三个键Decoders: three keys the scout report brought in and the card does not cite

  • John Blue, Harshil Avlani, Zhiyang He, Liu Ziyin, and Isaac L. Chuang, "Machine Learning Decoding of Circuit-Level Noise for Bivariate Bicycle Codes" (arXiv v1 2025; Quantum 10, 2149, 2026)
    blue2026machine · 待补pending · arXiv 2504.13043 · 基于 Transformer 的电路级 BB 译码器。仅条目级。扫描报告的警句：「训练分布与延迟是译码器规格的一部分，必须随性能声明一起给出」。A transformer-based circuit-level decoder for BB syndrome histories. Entry-level only. The scout report's caveat: “training distribution and latency are part of the decoder specification and must accompany performance claims”.
  • Anton Pakhunov, "Belief Propagation Convergence Prediction for Bivariate Bicycle Quantum Error Correction Codes" (2026)
    pakhunov2026convergence · 待补pending · arXiv 2604.07995 · 用「症状缺陷数模列重量」预测 BP 是否收敛，在五个 BB 码上验证；是一层调度 / 分流，可以省掉不必要的 OSD 调用，「不是完整的替代译码器」。仅条目级。Predicts BP convergence from the syndrome defect count modulo column weight, validated on five BB codes; a scheduling/triage layer that can avoid unnecessary OSD calls, “not a complete replacement decoder”. Entry-level only.
  • Anton Pakhunov, "Analytical Theory of Greedy Peeling for Bivariate Bicycle Codes and Two-Shot Streaming Decoding" (2026)
    pakhunov2026peeling · 待补pending · arXiv 2604.11352 · 分析电路级噪声下的贪心剥皮greedy peeling：先找出那些「亮起的检查只能由它解释」的单个错误，一个个认定、从症状里剥掉；剥不动时再交给别的译码器。译码：先把能单独认出的错误一个个剥掉，剩下的再交给 BP-OSD。摘要的说法：p = 10−3 时延迟比 BP 低 330 倍，逻辑错误率相同。正文表 5 实际测到的：gross 码 [[144,12,12]]，Stim 模拟的电路级噪声，12 轮症状提取，每个 p 采样 10,000 次；p = 10−3 时，完整的三段式译码器（剥皮 → 残余很小时枚举单个与成对错误 → 回退到串行 min-sum BP 加 OSD-CS-2）每次译码的中位延迟 1.4 μs，作对照的 BP+OSD 为 164.9 μs，比值 118 倍；两者的逻辑错误率都只报告为「< 0.04%」，10,000 次采样的精度分不出两者，所以「相同」只是「都低于这个上限」。330 倍对应另一个计时：优化后「只剥皮」那一段的中位 500 ns（论文 §VII 注明是 peeling-only；164.9 μs ÷ 500 ns ≈ 330，这个除法论文没有写出，是本页的推算）。计时都在 Apple Silicon 的通用处理器上、用 Rust 实现，不是专用译码硬件。另有一个论文标作 [[32,8,6]] 的码上的两轮流式方案。仅条目级；这些数字属于该论文自己的基准，QEC-Bank 未重算。注意那个标签：论文 §VI 给的构造是 l = m = 4、A = B = x + y，这时存在重量 2 的逻辑算符，码距至多 2，与「6」矛盾；下面的图 3.1 一步一步画出来（本页脚本核对，译码器导读 3.5 节也记录了这处不一致）。Analyses greedy peelingFind single errors that are the only possible explanation of the checks they light, accept them one at a time and strip them from the syndrome; hand whatever cannot be stripped to another decoder. decoding under circuit-level noise: strip off the errors that can be identified one at a time, and hand the rest to BP-OSD. The abstract says: at p = 10−3, 330× lower latency than BP with identical logical error rate. What Table 5 actually measures: the gross code [[144,12,12]], circuit-level noise simulated with Stim, 12 rounds of syndrome extraction, 10,000 shots per p. At p = 10−3 the full three-stage decoder (peeling → enumerate single and paired faults when the residue is small → fall back to serial min-sum BP with OSD-CS-2) has a median latency of 1.4 μs per decode, the BP+OSD reference 164.9 μs, a ratio of 118×. Both logical error rates are reported only as “< 0.04%”; 10,000 shots cannot tell the two apart, so “identical” means “both below that bound”. The 330× belongs to a different timing: the optimized peeling-only stage, median 500 ns (the paper’s §VII says peeling-only; 164.9 μs ÷ 500 ns ≈ 330 is this page’s arithmetic, the paper does not write the division out). All timings are on a general-purpose Apple Silicon processor with a Rust implementation, not dedicated decoding hardware. There is also a two-shot streaming scheme on a code the paper labels [[32,8,6]]. Entry-level only; the numbers belong to that paper's own benchmark and are not recomputed by the QEC-Bank. Mind that label: the construction in the paper's §VI is l = m = 4, A = B = x + y, and that code has a weight-2 logical operator, so its distance is at most 2, contradicting the “6”; Fig. 3.1 below draws it step by step (checked by this page's script; decoders guide §3.5 records the same inconsistency).

记号：p = 每个电路位置的物理错误率；[[32,8,6]] = 论文给一块 BB 码的标签，读作「32 个数据比特、8 个逻辑比特、码距 6」；n = 32 与 k = 8 本页脚本核对无误，码距那一项与论文的构造矛盾（见图 3.1）；它不是 QEC-Bank 的实例；330 倍与 118 倍 = 该论文报告的延迟比，各自对应哪两个计时见上一条；μs = 微秒，ns = 纳秒（千分之一微秒）。译码扫描报告的另外两个键 roffe2020bposd（别名）与 shaw2025morphing（3.4 节）不在这一组。

Notation: p = physical error rate per circuit location; [[32,8,6]] = the paper's label for a BB block, read “32 data qubits, 8 logical qubits, distance 6”; n = 32 and k = 8 check out in this page's script, the distance entry contradicts the paper's construction (see Fig. 3.1); it is not a QEC-Bank instance; 330× and 118× = latency ratios the paper reports, and the item above says which two timings each compares; μs = microsecond, ns = nanosecond (a thousandth of a microsecond). The decoders report's other two keys, roffe2020bposd (an alias) and shaw2025morphing (§3.4), are not in this group.

左层 L（数据比特） left layer L (data qubits) X Z₁ Z₂ Z₁ Z₂ 0 1 2 3 0 1 2 3 a（x 的次数）→ a (power of x) → b（y 的次数）→b (power of y) → Z 检查（每格一条） Z checks (one per cell) Z₂ Z₁ 0 1 2 3 0 1 2 3 a（x 的次数）→ a (power of x) → b（y 的次数）→b (power of y) → 右层 R（数据比特） right layer R (data qubits) X Z₁ Z₂ Z₁ Z₂ 0 1 2 3 0 1 2 3 a（x 的次数）→ a (power of x) → b（y 的次数）→b (power of y) → Z₁ = (1,0) 格的 Z 检查，腿：L(0,0) L(1,3) R(0,0) R(1,3) Z₁ = the Z check of cell (1,0); legs L(0,0) L(1,3) R(0,0) R(1,3) Z₂ = (0,1) 格的 Z 检查，腿：L(0,0) L(3,1) R(0,0) R(3,1) Z₂ = the Z check of cell (0,1); legs L(0,0) L(3,1) R(0,0) R(3,1)
图 3.1Fig. 3.1论文 §VI 给「[[32,8,6]]」的构造：l = m = 4，A = B = x + y。环面上每个格子 (a, b) 放一个左层比特 L(a, b)、一个右层比特 R(a, b)、一条 X 检查和一条 Z 检查（中间那张只画 Z 检查）。Z 检查从自己的格子往回走一步取腿：左层 (a−1, b)、(a, b−1)，右层同样两格（HZ = [Bᵀ | Aᵀ]，转置就是反方向；坐标按 4 取余，所以 −1 就是 3）。橙色：一条 X 串，X 作用在 L(0,0) 与 R(0,0)。蓝色：碰到这两个比特的只有 Z₁、Z₂ 两条 Z 检查，格子里的标签写出每个比特被哪条碰。Z₁ 与 X 串共用 L(0,0)、R(0,0) 两个比特；每个共用比特让这条检查的读数变一次号，变两次回到原样，2 ≡ 0 (mod 2)，所以对易；Z₂ 同理；其余 14 条 Z 检查与它不共用比特。结论：没有一条 Z 检查能发现这条 X 串。The construction the paper’s §VI gives for “[[32,8,6]]”: l = m = 4, A = B = x + y. Each cell (a, b) of the torus holds a left-layer qubit L(a, b), a right-layer qubit R(a, b), one X check and one Z check (the middle grid shows only the Z checks). A Z check takes its legs one step backwards from its cell: left-layer (a−1, b) and (a, b−1), and the same two cells in the right layer (HZ = [Bᵀ | Aᵀ]; transposing reverses the direction; coordinates are taken mod 4, so −1 means 3). Orange: an X string, X on L(0,0) and R(0,0). Blue: only two Z checks, Z₁ and Z₂, touch those qubits; the label in each cell says which of them touches it. Z₁ shares two qubits with the X string, L(0,0) and R(0,0); each shared qubit flips the sign of that check’s reading once, and two flips restore it, 2 ≡ 0 (mod 2), so they commute; the same holds for Z₂; the other 14 Z checks share no qubit with it. Conclusion: no Z check can detect this X string.

它又为什么不是 X 检查的乘积（乘积是检查本身，不改变逻辑信息）？X 检查从自己的格子往前走一步取腿：左层 (a+1, b)、(a, b+1)，右层同样两格，所以每条 X 检查在左层恰好碰 2 个比特。把几条 X 检查乘起来，同一个比特上的两个 X 相乘等于什么都不做（X·X = 1），也就是按模 2 相加。例：(0,0) 格的 X 检查碰左层 L(1,0)、L(0,1)；(1,3) 格的碰 L(2,3)、L(1,0)。两者相乘，L(1,0) 被碰两次消掉，左层剩 L(0,1)、L(2,3)，仍是 2 个。一般地，一个比特留下与否只看它被碰的次数是奇是偶；所有左层比特被碰次数的总和 = 2 × 检查条数，是偶数，所以被碰奇数次的比特也有偶数个。任何一组 X 检查的乘积在左层都占偶数个比特。图中的 X 串在左层只占 L(0,0) 一个，是奇数，所以它不是 X 检查的乘积。它既不被任何 Z 检查发现，又不是检查本身，那它就是一个会改变逻辑信息的逻辑算符，而重量只有 2：这个码的码距至多 2。n = 32、k = 8 与上面每一步都由本页脚本（notes/references/verify2.py）逐项算过。

And why is it not a product of X checks (such a product is itself a check and leaves the logical information alone)? An X check takes its legs one step forwards: left-layer (a+1, b) and (a, b+1), and the same two cells in the right layer, so every X check touches exactly 2 left-layer qubits. When X checks are multiplied, two X’s on the same qubit multiply to nothing (X·X = 1), which is addition mod 2. Example: the X check of cell (0,0) touches L(1,0) and L(0,1); that of cell (1,3) touches L(2,3) and L(1,0). In their product L(1,0) is hit twice and drops out, leaving L(0,1) and L(2,3) on the left, still 2. In general a qubit survives exactly when it is hit an odd number of times; the hits on left-layer qubits add up to 2 × (number of checks), an even number, so the qubits hit an odd number of times are also even in number. Every product of X checks occupies an even number of left-layer qubits. The X string in the figure occupies one, L(0,0), an odd number, so it is not a product of X checks. It escapes every Z check and is not itself a check, so it is a logical operator that changes the logical information, of weight only 2: the code’s distance is at most 2. n = 32, k = 8 and every step above were computed by this page’s script (notes/references/verify2.py).

3.3逻辑操作：三条 claimed 的门的键，与四个基础设施键Logical operations: keys of three claimed gates, and four infrastructure keys

  • Jens Niklas Eberhardt and Vincent Steffan, "Logical Operators and Fold-Transversal Gates of Bivariate Bicycle Codes" (2024)
    eberhardt2024fold · 待补pending · arXiv 2407.03973 · 为对称的 BB 例子（含 [[98,6,12]]、[[162,8,12]]）构造显式逻辑基与折叠横向 Clifford 门。替 operations[1] 作证：「selected logical Clifford generators」，fold-transversal，claimed，cost「只有具有已算清逻辑作用的对称 BB 实例才符合」。Constructs explicit logical bases and fold-transversal Clifford gates for symmetric BB examples, including [[98,6,12]] and [[162,8,12]]. Testifies for operations[1]: “selected logical Clifford generators”, fold-transversal, claimed, cost “only symmetric BB instances with a resolved logical action qualify”.
  • Theodore J. Yoder et al., "Tour de gross: A modular quantum computer based on bivariate bicycle codes" (2025)
    yoder2025tour · 待补pending · arXiv 2506.03094 · d = 12 与 d = 18 的 BB 模块的指令集、编译与端到端资源估计。替三条记录作证：operations[0]（相同 BB 块之间的横向 CNOT，claimed）、operations[3]（块间 Pauli 测量与 Clifford 手术，claimed）、realizations[4]（modular gross-code architecture，proposed）。实现导读 4.2 节。Instruction set, compilation and end-to-end resource estimates for distance-12 and distance-18 BB modules. Testifies for three records: operations[0] (transversal CNOT between identical BB blocks, claimed), operations[3] (inter-block Pauli measurements and Clifford surgery, claimed), realizations[4] (modular gross-code architecture, proposed). Realizations guide §4.2.
  • Zijian Liang and Yu-An Chen, "Self-dual bivariate bicycle codes with transversal Clifford gates" (2025)
    liang2025selfdual · 待补pending · arXiv 2510.05211 · 扭环面上重量 8 的自对偶 BB 子族，带横向 CNOT、H、S。替 operations[4] 作证：「H / S / CNOT in the self-dual BB subfamily」，claimed，cost 明写「not a gross-code claim」。也是变体轴「对偶性」的键（关系导读 4.3 节）。A weight-8 self-dual BB subfamily on twisted tori with transversal CNOT, H and S. Testifies for operations[4]: “H / S / CNOT in the self-dual BB subfamily”, claimed, cost stating “not a gross-code claim”. Also the key of the “duality” variant axis (relations guide §4.3).
  • Lawrence Z. Cohen, Isaac H. Kim, Stephen D. Bartlett, and Benjamin J. Brown, "Low-overhead fault-tolerant quantum computing using long-range connectivity" (arXiv v1 2021; Science Advances 8, eabn1717, 2022)
    cohen2022longrange · 待补pending · arXiv 2110.10794 · 长程格点手术原语，用于初始化与耦合高码率块；「是一般的基础设施，不是任意 BB 逻辑门横向的证明」。仅条目级。Long-range lattice-surgery primitives for initialising and coupling high-rate blocks; “general infrastructure, not a proof that arbitrary BB logical gates are transversal”. Entry-level only.
  • Qian Xu et al., "Constant-Overhead Fault-Tolerant Quantum Computation with Reconfigurable Atom Arrays" (2023)
    xu2023constant · 待补pending · arXiv 2308.08648 · 用原子重排做 qLDPC 态制备、搬运与逻辑操作的架构；BB 是其中一个目标家族。仅条目级。Architecture for qLDPC state preparation, movement and logical operations using atom rearrangement; BB codes are one target family. Entry-level only.
  • Nikolas P. Breuckmann, Margarita Davydova, Jens N. Eberhardt, and Nathanan Tantivasadakarn, "Cups and Gates I: Cohomology invariants and logical quantum operations" (arXiv v1 2024, v2 2025)
    breuckmann2026cups · 待补pending · arXiv 2410.16250 · 用杯积 / 上同调不变量（两者是什么，见补课 3a 最后一条）为具备所需结构的码构造常深度逻辑操作（摘要讲的是一般 CSS 码与上同调不变量，没有提 BB 码；「含选定的 BB 码」是扫描报告的说法，本页未在正文里核对）；「代数不变量就是门的证书」。仅条目级。Uses cup-product/cohomology invariants (what they are: the last item of catch-up 3a) to construct constant-depth logical operations for codes with the required structure (the abstract speaks of general CSS codes and cohomology invariants and does not mention BB codes; “including selected BB codes” is the scout report's wording, not checked in the paper body here); “the algebraic invariant is the gate certificate”. Entry-level only.
  • Qian Xu et al., "Batched high-rate logical operations for quantum LDPC codes" (2025)
    xu2025batched · 待补pending · arXiv 2510.06159 · 为 CSS qLDPC 码构造常数时空开销的批量纠错、手术、切换与 Clifford 组件，并把近期方案特化到自对偶 BB 块加并行魔法态培育。仅条目级。Constant-space-time-overhead batched error correction, surgeries, switching and Clifford gadgets for CSS qLDPC codes, with a near-term proposal specialised to self-dual BB blocks plus parallel magic-state cultivation. Entry-level only.

操作扫描报告的综合判断值得整段照抄，因为它解释了为什么这一组有七个键却只对应三条 claimed 记录：「一般的 BB 块作为存储器最强。逻辑计算需要四种额外结构之一：一块表面码辅助与手术；一个物理自同构 / 折叠；非平凡的杯积数据；或一个自对偶子族。操作侧面必须指明哪份证书适用于哪个具体的块。」负结果：「没有任何被抓取的文献为原始的 [[144,12,12]] gross 码提供普适的横向门集。」逻辑操作导读逐条展开五条 claimed 记录。

The operations report's synthesis deserves quoting in full, because it explains why this group has seven keys but only three claimed records: “Generic BB blocks are strongest as memories. Logical computation requires one of four extra structures: a surface-code ancilla and surgery; a physical automorphism/fold; nontrivial cup-product data; or a self-dual subfamily. The operations facet must name which certificate applies to which concrete block.” Negative result: “No fetched source provides a universal transversal gate set for the original [[144,12,12]] gross code.” The logical-operations guide unfolds the five claimed records one by one.

3.4用例与实现：一次真机、三份提案、两个扫描键Use cases and realizations: one experiment, three proposals, two scout keys

  • Ke Wang et al., "Demonstration of low-overhead quantum error correction codes" (arXiv v1 2025; Nature Physics, 2026)
    wang2026lowoverhead · 待补pending · arXiv 2505.09684 · 昆仑长程耦合超导处理器（32 个量子比特）上的码距 4、4 个逻辑比特的 BB 码 [[18,4,4]]，18 个数据比特加 14 个检查比特，连续做最多 6 轮症状提取。报告的每个逻辑比特每轮逻辑错误率 (8.91 ± 0.17)% 是在这些条件下得到的：丢掉检测到泄漏leakage：超导比特跑出了用来存 0 和 1 的两个能级，落到第三个能级上。论文用能分辨三个能级的读出把这些轮次挑出来丢掉。（比特跑出存 0、1 的两个能级）的实验轮次，每个轮数 t = 1 到 6 各留 40,000 次以上；用扩展到电路级去极化噪声模型的 BP-OSD 译码；把失败概率 PL(t) 对 t = 1…6 拟合出整块每轮的失败率 pL，再折成每个逻辑比特 ε = 1 − (1 − pL)1/4；逻辑 Z 基与 X 基两组存储实验分别得 (9.15 ± 0.27)% 与 (8.67 ± 0.21)%，8.91% 是两者的平均（(9.15 + 8.67) / 2 = 8.91）；论文写明 ± 是一个标准差，含统计与拟合两部分不确定度。（同一篇还有一个码距 3、6 个逻辑比特的 qLDPC 码，(7.77 ± 0.12)%。）替 realizations[0] 作证——本条目唯一的 demonstrated，记录原文强调「establishes feasibility, not below-threshold scaling and not a gross-code run」。实现导读第 3 章。A distance-4 BB code with 4 logical qubits, [[18,4,4]], on the Kunlun long-range-coupled superconducting processor (32 qubits), 18 data qubits plus 14 check qubits, running up to 6 consecutive syndrome-extraction cycles. The reported logical error rate of (8.91 ± 0.17)% per logical qubit per cycle holds under these conditions: runs in which leakageA superconducting qubit leaves the two levels that store 0 and 1 and lands on a third level. The paper flags such runs with a readout that tells three levels apart and discards them. was detected (a qubit leaving the two levels that store 0 and 1) are discarded, keeping more than 40,000 runs for each number of cycles t = 1 to 6; decoding uses BP-OSD extended to a circuit-level depolarizing noise model; the failure probability PL(t) is fitted over t = 1…6 to give the whole block’s failure rate per cycle pL, then converted per logical qubit as ε = 1 − (1 − pL)1/4; the logical-Z-basis and X-basis memory experiments give (9.15 ± 0.27)% and (8.67 ± 0.21)%, and 8.91% is their average ((9.15 + 8.67) / 2 = 8.91); the paper states that ± is one standard deviation covering statistical and fit uncertainty. (The same paper also runs a distance-3 qLDPC code with 6 logical qubits, (7.77 ± 0.12)%.) Testifies for realizations[0], the entry's only demonstrated record, whose text stresses “establishes feasibility, not below-threshold scaling and not a gross-code run”. Realizations guide, Chapter 3.
  • C. Poole, T. M. Graham, M. A. Perlin, M. Otten, and M. Saffman, "Architecture for fast implementation of qLDPC codes with optimized Rydberg gates" (arXiv v1 2024; Phys. Rev. A 111, 022433, 2025)
    poole2025rydberg · 待补pending · arXiv 2404.18809 · 为 BB 连接性优化的 Rydberg 版图与门路由。替 realizations[1] 作证（optimized Rydberg neutral-atom layout，proposed：「数值架构提案，不是实验室的逻辑存储器演示」）。A Rydberg layout and gate routing optimised for BB connectivity. Testifies for realizations[1] (optimized Rydberg neutral-atom layout, proposed: “a numerical architecture proposal rather than a laboratory logical-memory demonstration”).
  • Noah Berthusen et al., "Toward a 2D Local Implementation of Quantum LDPC Codes" (arXiv v1 2024; PRX Quantum 6, 010306, 2025)
    berthusen2025local · 待补pending · arXiv 2404.17676 · 用局域搬动 / 通信嵌入 BB 检查，部分长程检查少测。替 realizations[2] 作证（layered two-dimensional hardware，proposed：「改变的节拍与电路噪声必须作为提案的一部分被译码」）。Embeds BB checks with local motion/communication, measuring some long-range checks less often. Testifies for realizations[2] (layered two-dimensional hardware, proposed: “the altered cadence and circuit noise must be decoded as part of the proposal”).
  • Mackenzie H. Shaw and Barbara M. Terhal, "Lowering Connectivity Requirements For Bivariate Bicycle Codes Using Morphing Circuits" (arXiv v1 2024, v3 2025)
    shaw2025morphing · 待补pending · arXiv 2407.16336 · 为一个新定义的 BB 码家族设计变形电路（周期中途比特处在另一个「中途码」里），每个比特只需连 5 个邻居而不是 6 个、每周期 6 层 CNOT；换了家族与探测器模型，不是给 gross 的度 5 电路。替两条记录作证：realizations[3]（degree-five morphing-circuit layout，proposed）与 R4 tanner-circuit 的 refs。关系导读 4.4 节、表示阶梯导读第 5 章。Morphing circuits (midway through a cycle the qubits sit in a different “mid-cycle code”) designed for a newly defined family of BB codes: 5 neighbours per qubit instead of 6, 6 CNOT layers per cycle; family and detector model change, and it is not a degree-5 circuit for gross. Testifies for two records: realizations[3] (degree-five morphing-circuit layout, proposed) and the refs of R4 tanner-circuit. Relations guide §4.4, representation-ladder guide, Chapter 5.
  • Susan X. Chen, Matthias C. Löbl, Ming Lai Chan, Anders S. Sørensen, and Stefano Paesani, "Fusion-based implementation of qLDPC codes with quantum emitters" (2025)
    chen2025fusion · 待补pending · arXiv 2509.17223 · 把任意 CSS qLDPC 检查映射到发射器产生的光子资源态，在损耗与 Pauli 噪声下研究小的 BB 例子。仅条目级；扫描报告标注「New beyond EC Zoo」（Error Correction Zoo 这个公开的在线纠错码目录里还没有收录）。Maps arbitrary CSS qLDPC checks to emitter-generated photonic resource states and studies small BB examples under loss and Pauli noise. Entry-level only; the scout report marks it “New beyond EC Zoo” (not yet listed in the Error Correction Zoo, the public online catalogue of codes).
  • Nitish Kumar Chandra, Eneet Kaur, Reza Nejabati, and Kaushik P. Seshadreesan, "Distributed Quantum Error Correction with Bivariate Bicycle Codes in a Modular Architecture" (2026)
    chandra2026distributed · 待补pending · arXiv 2605.04663 · 把 [[144,12,12]] 分到 4、6、12 台由 Bell 对连接的处理器上，模拟非局域噪声变化下的 BP-OSD 电路性能。仅条目级；「New beyond EC Zoo」。Partitions [[144,12,12]] over 4, 6 and 12 processors linked by Bell pairs and simulates BP-OSD circuit performance as nonlocal noise varies. Entry-level only; “New beyond EC Zoo”.

3.5变体：四个只在 Level-2 页作证的键Variants: four keys that testify only on the Level-2 page

  • Lukas Voss, Sim Jian Xian, Tobias Haug, and Kishor Bharti, "Multivariate Bicycle Codes" (arXiv v1 2024, v4 2025)
    voss2025multivariate · 待补pending · arXiv 2406.19151 · 把多项式扩展到两个变量之外；「一旦用了第三个独立变量，就是可交换 2BGA 的亲戚，不再是字面上的双变量码」。仅条目级：变体轴「变量数」（关系导读 4.1 节）。Extends the polynomial ansatz beyond two variables; “Abelian 2BGA relatives, not literally bivariate codes once a third independent variable is used”. Entry-level only: the “number of variables” axis (relations guide §4.1).
  • Ming Wang and Frank Mueller, "Coprime Bivariate Bicycle Codes and Their Layouts on Cold Atoms" (arXiv v1 2024, v7 2026)
    wang2026coprime · 待补pending · arXiv 2408.10001 · 要求环面两边长 l、m 互素，并只用 π = xy 这一个量写多项式；于是 k = 2 · deg gcd(a(π), b(π), πlm + 1) 在搜索前就能算出，并共同设计搬动原子的版图（互素 = 最大公约数是 1；deg、gcd 与模 2 系数的意思、π 为什么能代替 x 和 y、一个算到 k 的例子，都在下面的补课 3b；gross 的 12 与 6 不互素，不在这个子类里；变体轴见关系导读 4.1 节）；「子类加架构提案，不是硬件实验」。仅条目级：变体轴「多项式限制」。Requires the torus sides l, m to be coprime and writes the polynomials in the single quantity π = xy, so k = 2 · deg gcd(a(π), b(π), πlm + 1) is known before any search, and co-designs a moving-atom layout (coprime = greatest common divisor 1; what deg, gcd and coefficients mod 2 mean, why π can replace x and y, and an example worked through to k are in catch-up 3b below; gross's 12 and 6 are not coprime, so gross is outside this subclass; the variant axis is in relations guide §4.1); “a subclass plus architecture proposal, not a hardware experiment”. Entry-level only: the “polynomial restrictions” axis.
  • Jens Niklas Eberhardt, Francisco Revson F. Pereira, and Vincent Steffan, "Pruning qLDPC codes: Towards bivariate bicycle codes with open boundary conditions" (2024)
    eberhardt2024pruning · 待补pending · arXiv 2412.04181 · 从周期 BB 码删去检查 / 比特得到开边界后代；「剪枝在循环超图积子情形有保证，在此之外是演示而非普适证明」。仅条目级：变体轴「边界」（关系导读 4.2 节）。Removes checks/qubits from periodic BB codes to obtain open-boundary descendants; “pruning is guaranteed for the cyclic-HGP subcase and demonstrated, not proved universal, beyond it”. Entry-level only: the “boundary” axis (relations guide §4.2).
  • Zijian Liang, Jens Niklas Eberhardt, and Yu-An Chen, "Planar quantum low-density parity-check codes with open boundaries" (2025)
    liang2025planar · 待补pending · arXiv 2504.08887 · 构造与 BB 内部图案相关的平面开边界码，区分有限平面局域与周期环面近局域。仅条目级：变体轴「边界」。Constructs planar open-boundary codes related to BB bulk patterns and distinguishes finite planar locality from periodic-torus near locality. Entry-level only: the “boundary” axis.
补课 · 互素 BB 的维数公式 k = 2 · deg gcd(a(π), b(π), πlm + 1)：每个符号，和一个算到 k 的例子Catch-up · The coprime-BB dimension formula k = 2 · deg gcd(a(π), b(π), πlm + 1): every symbol, and one example worked through to k
模 2 系数、deg、gcdCoefficients mod 2, deg, gcd

这里的多项式系数只取 0 或 1，并且 1 + 1 = 0，所以两项相同就抵消。例：(1 + π)(1 + π + π²) = 1 + π + π² + π + π² + π³ = 1 + π³，中间的 π 与 π、π² 与 π² 各自抵消。deg（degree，次数）是多项式里最高的幂：deg(1 + π + π²) = 2。gcd（greatest common divisor，最大公因式）是能同时整除几条多项式的、次数最高的那条公因式，和整数的最大公约数一样可以用辗转相除求出来。

The polynomials here have coefficients 0 or 1 only, and 1 + 1 = 0, so equal terms cancel. Example: (1 + π)(1 + π + π²) = 1 + π + π² + π + π² + π³ = 1 + π³; the two π’s and the two π²’s cancel. deg (degree) is the highest power in a polynomial: deg(1 + π + π²) = 2. gcd (greatest common divisor) is the common factor of highest degree that divides all the given polynomials; as for integers, it can be found by repeated division with remainder (Euclid’s algorithm).

为什么斜着走 π = xy 能走遍环面Why diagonal steps π = xy visit the whole torus

π = xy 是「向右一步再向上一步」。走 t 步后停在格子 (t mod l, t mod m)。第一次回到 (0,0)，要 t 同时是 l 和 m 的倍数；l、m 互素时，最小的这种 t 就是 lm。于是 π⁰, π¹, …, πlm−1 这 lm 步停在互不相同的格子上：若第 s 步与第 t 步（s < t）停在同一格，那么 t − s 同时是 l 和 m 的倍数，至少是 lm，与 t − s < lm 矛盾。环面一共只有 lm 格，所以每格恰好走到一次，每个单项式 xayb 都等于 π 的某个幂，x、y 的多项式都能改写成只含 π 的多项式，而且 πlm = 1。l、m 不互素时就不行：l = m = 2 时第 2 步已回到原点，只到 2 格（下图右）。gross 的 l = 12、m = 6 最大公约数是 6，π 只能到 72 格中的 12 格。

π = xy is “one step right, then one step up”. After t steps you stand on the cell (t mod l, t mod m). To get back to (0,0) for the first time, t must be a multiple of both l and m; when l and m are coprime the smallest such t is lm. So the lm steps π⁰, π¹, …, πlm−1 land on different cells: if steps s and t (s < t) landed on the same cell, t − s would be a multiple of both l and m, hence at least lm, contradicting t − s < lm. The torus has only lm cells, so each is visited exactly once; every monomial xayb equals some power of π, every polynomial in x and y can be rewritten in π alone, and πlm = 1. Without coprimality this fails: for l = m = 2 step 2 is already back at the start and only 2 cells are reached (right of the figure). For gross, l = 12 and m = 6 have greatest common divisor 6, and π reaches only 12 of the 72 cells.

l = 2, m = 3（互素） l = 2, m = 3 (coprime) 6 步走遍 6 格 6 steps reach all 6 cells π⁰ π⁴ π² π³ π¹ π⁵ 0 1 0 1 2 a → b ↑ π⁰ … π⁶： π⁰ … π⁶: (0,0) (1,1) (0,2) (1,0) (0,1) (1,2) (0,0) l = m = 2（不互素） l = m = 2 (not coprime) 只到 2 格 only 2 cells reached π⁰,π² — — π¹,π³ 0 1 0 1 a → b ↑ π⁰ … π⁴： π⁰ … π⁴: (0,0) (1,1) (0,0) (1,1) (0,0)
图 3.3Fig. 3.3斜着走 π = xy：格子里写的是第几步停在这里（πt 停在 (t mod l, t mod m)）。左：l = 2、m = 3 互素，6 步各停一格，第 6 步回到 (0,0)。右：l = m = 2 不互素，第 2 步就回到 (0,0)，(1,0)、(0,1) 永远走不到（「—」）。Diagonal steps π = xy: each cell shows which steps land on it (πt lands on (t mod l, t mod m)). Left: l = 2, m = 3 are coprime, the 6 steps land on 6 different cells, and step 6 returns to (0,0). Right: l = m = 2 are not coprime; step 2 is already back at (0,0), and (1,0), (0,1) are never reached (“—”).
公式，与一个算到 k 的例子The formula, and an example worked through to k

既然只剩一个变量 π 且 πlm = 1，Wang 与 Mueller 证明（论文式 (14)）：k = 2 · deg g，其中 g = gcd(a(π), b(π), πlm + 1)。例：l = 2、m = 3（互素），a = 1 + π + π²，b = 1 + π³。先换回 x、y：π² = x²y² = y²（x² = 1），π³ = x³y³ = x（x² = 1、y³ = 1），所以 a = 1 + xy + y²，b = 1 + x，正是补课 3a 的那个小码。再求 g：上面算过 b = (1 + π)·a，所以 a 整除 b，gcd(a, b) = a；π⁶ + 1 = (π³ + 1)²（模 2 下交叉项 2π³ = 0），也能被 a 整除。于是 g = 1 + π + π²，deg g = 2，k = 2 × 2 = 4。补课 3a 用 Gröbner 基数出 dim R/(a, b) = 2，也是 k = 4；本页脚本按矩阵秩算，n = 12、k = 4。这个公式换来的是：在搜索之前挑好 g，k 就定了，搜索只需要找码距好的 a、b。

With a single variable π left and πlm = 1, Wang and Mueller prove (their Eq. (14)) that k = 2 · deg g with g = gcd(a(π), b(π), πlm + 1). Example: l = 2, m = 3 (coprime), a = 1 + π + π², b = 1 + π³. Back in x, y: π² = x²y² = y² (x² = 1) and π³ = x³y³ = x (x² = 1, y³ = 1), so a = 1 + xy + y² and b = 1 + x, exactly the small code of catch-up 3a. Now g: we computed above that b = (1 + π)·a, so a divides b and gcd(a, b) = a; π⁶ + 1 = (π³ + 1)² (mod 2 the cross term 2π³ vanishes), which a also divides. So g = 1 + π + π², deg g = 2 and k = 2 × 2 = 4. Catch-up 3a counts dim R/(a, b) = 2 with a Gröbner basis, again k = 4; this page’s script computes the matrix ranks and gets n = 12, k = 4. What the formula buys: pick g before the search and k is fixed; the search only has to find a and b with good distance.

数目核对Count check

5 + 3 + 7 + 6 + 4 = 25；加上第 2 章的 7 条，共 32，与 QEC-Bank 参考文献页逐条对应，无增无减。25 条里绑定到记录的 12 个：wang2023abelian、lin2023twoblock、postema2025existence、chen2025anyon、sahay2026matching、eberhardt2024fold、yoder2025tour、liang2025selfdual、wang2026lowoverhead、poole2025rydberg、berthusen2025local、shaw2025morphing；仅条目级 13 个：其余全部。分组依据是「引用它的记录属于哪个侧面」，仅条目级的键则按它来自哪份扫描报告归组。

5 + 3 + 7 + 6 + 4 = 25; with the 7 of Chapter 2, 32 in all, matching the QEC-Bank references page item for item. Of the 25, twelve are record-bound: wang2023abelian, lin2023twoblock, postema2025existence, chen2025anyon, sahay2026matching, eberhardt2024fold, yoder2025tour, liang2025selfdual, wang2026lowoverhead, poole2025rydberg, berthusen2025local, shaw2025morphing; thirteen are entry-level only: all the rest. Grouping follows the facet of the citing record; entry-level-only keys are grouped by the scout report they come from.

上一章Back2 · 七条已登记的文献2 · Seven registered references 下一章Next4 · 追踪一个键4 · Tracing one key
第 4 章Chapter 4

追踪一个键：sahay2026matching 从导出页走回 card.yaml，再走到扫描报告Tracing one key: sahay2026matching from the exported page back to card.yaml and on to the scout report

第 3 章说了每个键「替谁作证」；这一章把其中一个键的整条证据链亲手走一遍，看每一步在哪个文件、写着什么、哈希是多少，以及链条在哪一步断了——因为这个键还没登记。然后用 wang2026lowoverhead 再走一遍短的，最后看卡片如何用「不是……」的句子防住三种常见误引。 Chapter 3 said whose witness each key is; this chapter walks one key's whole evidence chain by hand, seeing at every step which file, what text, which hash, and where the chain breaks, because this key is not yet registered. Then a shorter walk with wang2026lowoverhead, and finally how the card uses “not …” clauses to block three common misattributions.

4.1六步：从「mwpm · research-grade」到 arXiv 2602.22770Six steps: from “mwpm · research-grade” to arXiv 2602.22770

第一步，导出页。打开 exports/site/entries/bivariate_bicycle/decoders.html，第三条记录写着：「Minimum-weight perfect matching · mwpm · research-grade · Noise setting: BB symmetry reduction to toric-code-like cylinders · For this code: A cylinder construction exposes matching subproblems and can be combined with BP and over-matching. Its certificate is the specific symmetry reduction; it does not turn every BB instance into independent toric codes. · References: higgott2022pymatching, sahay2026matching」。第二步，源文件。在 codes/bivariate_bicycle/card.yaml 的 legacy_card.decoders[2] 找到同一条：id: mwpm、status: research-grade、noise 与 notes 逐字相同，refs: [sahay2026matching]——只有一个键。导出页多出来的 higgott2022pymatching 来自译码器目录 registry/decoders.yaml：那里 mwpm 这种译码器类型自带默认 refs [higgott2022pymatching]（PyMatching 的实现论文），导出时和卡片的 refs 合并显示。所以读导出页的 References 行要记得：它是「译码器类型的通用出处 + 这个码的专门出处」两份的并集，前者不在 BB 条目的 32 个键里。第三步，断言层。同一条记录在 card.yaml 的 assertions 里还有一份带元数据的副本：id: atlas-assertion:bivariate_bicycle:decoders:2，applies_to: eczoo-entity:qcga#collection（说的是整个 BB 家族），source_locator: bivariate_bicycle/card.yaml#/decoders/2，source_sha256: 26aee9ae…。哈希保证你在导出页上看到的那句话就是登记时的那句话。第四步，同一个键的第二个住处。views[7]，R2 的 toric-cylinder：origin: {kind: curated, ref: sahay2026matching}，refs 也是它；描述说这个约化「是一本可操作的译码词典，不是全局的相等价」。断言 id 是 atlas-assertion:bivariate_bicycle:views:7。两条记录、一个键、一句共同的边界（表示阶梯导读 3.2 节）。第五步，登记表——链条在这里断了。registry/refs.yaml 里这个键只有一行：- {id: sahay2026matching}。没有 title、authors、year、url。于是参考文献页只能显示「sahay2026matching · metadata pending」，读者没有链接可以点。第六步，扫描报告。要找到论文，得离开网站，打开 docs/lit/bivariate_bicycle/bivariate_bicycle-decoders.md（或 -representations.md）：bibkey: sahay2026matching，title「A Matching Decoder for Bivariate Bicycle Codes」，authors「Kaavya Sahay, Dominic J. Williamson, and Benjamin J. Brown」，year 2026，arxiv 2602.22770，arxiv_verified: true，检索日期 2026-07-26。五个字段都在，只是没被抄进登记表。

Step one, the exported page. Open exports/site/entries/bivariate_bicycle/decoders.html; the third record reads: “Minimum-weight perfect matching · mwpm · research-grade · Noise setting: BB symmetry reduction to toric-code-like cylinders · For this code: A cylinder construction exposes matching subproblems and can be combined with BP and over-matching. Its certificate is the specific symmetry reduction; it does not turn every BB instance into independent toric codes. · References: higgott2022pymatching, sahay2026matching”. Step two, the source. In codes/bivariate_bicycle/card.yaml under legacy_card.decoders[2] the same record has id: mwpm, status: research-grade, identical noise and notes, and refs: [sahay2026matching], one key only. The extra higgott2022pymatching on the exported page comes from the decoder catalog registry/decoders.yaml, where the decoder type mwpm carries default refs [higgott2022pymatching] (the PyMatching implementation paper), merged with the card's refs at export. So when reading a References line on an exported page remember: it is the union of “generic sources for the decoder type” and “sources specific to this code”, and the former are not among the BB entry's 32 keys. Step three, the assertion layer. The same record has a metadata-bearing copy under assertions in card.yaml: id: atlas-assertion:bivariate_bicycle:decoders:2, applies_to: eczoo-entity:qcga#collection (the whole BB family), source_locator: bivariate_bicycle/card.yaml#/decoders/2, source_sha256: 26aee9ae…. The hash guarantees that the sentence on the exported page is the sentence that was registered. Step four, the key's second home. views[7], the R2 toric-cylinder: origin: {kind: curated, ref: sahay2026matching}, refs likewise; its description calls the reduction “an operational decoder dictionary, not a global phase equivalence”. Assertion id atlas-assertion:bivariate_bicycle:views:7. Two records, one key, one shared boundary (representation-ladder guide §3.2). Step five, the registry, where the chain breaks. In registry/refs.yaml this key has one line: - {id: sahay2026matching}. No title, authors, year or url. So the references page can only show “sahay2026matching · metadata pending”, and the reader has no link to click. Step six, the scout report. To reach the paper you must leave the site and open docs/lit/bivariate_bicycle/bivariate_bicycle-decoders.md (or -representations.md): bibkey: sahay2026matching, title “A Matching Decoder for Bivariate Bicycle Codes”, authors “Kaavya Sahay, Dominic J. Williamson, and Benjamin J. Brown”, year 2026, arxiv 2602.22770, arxiv_verified: true, fetched 2026-07-26. All five fields are there; they were simply never copied into the registry.

导出页exported page → card.yaml decoders[2] (refs)
→ 断言assertion decoders:2 (sha256 26aee9ae…)
→ views[7] (origin.ref)
→ registry/refs.yaml: { id } 仅此一行only → 页面：metadata pendingpage: metadata pending
→ docs/lit/…-decoders.md: title, authors, 2026, arXiv 2602.22770
对照已登记的键（如 bravyi2024bicycle），第五步直接给出 url，
第六步不需要for a registered key (e.g. bravyi2024bicycle) step five yields the url directly
and step six is unnecessary

记号：decoders[2]、views[7] = card.yaml 里译码器列表的第 3 条、表示阶梯列表的第 8 条（从 0 数起）；sha256 = 内容哈希，写前 8 位；origin.ref = 记录「出处」里指向的键；higgott2022pymatching = 表面码条目也在用的 PyMatching 论文键（《Surface Code · 参考文献》2.3 节），已登记。

Notation: decoders[2], views[7] = the 3rd decoder record and 8th ladder record in card.yaml (counting from 0); sha256 = the content hash, first 8 characters shown; origin.ref = the key a record's provenance points at; higgott2022pymatching = the PyMatching paper key also used by the surface-code entry (“Surface Code · References”, §2.3), registered.

4.2再走一遍短的：(8.91 ± 0.17)% 的键A shorter walk: the key behind (8.91 ± 0.17)%

你在实现导读 3.1 节读到「每个逻辑比特每轮 (8.91 ± 0.17)%」。记录：card.yaml 的 realizations[0]，platform「Kunlun long-range-coupled superconducting processor」，status demonstrated，description 原句含「(8.91 +/- 0.17)%」与「this establishes feasibility, not below-threshold scaling and not a gross-code run」，refs [wang2026lowoverhead]。断言：atlas-assertion:bivariate_bicycle:realizations:0，位置 bivariate_bicycle/card.yaml#/realizations/0。登记表：- {id: wang2026lowoverhead}，裸行——参考文献页显示「metadata pending」。扫描报告：bivariate_bicycle-realizations.md「Literal experiment」一节：Ke Wang et al.，「Demonstration of Low-Overhead Quantum Error Correction Codes」，2026，arXiv 2505.09684，platform superconducting_kunlun，status experiment，contribution 里同样写着「a feasibility result, not a below-threshold scaling demonstration」。说人话：这个数字有记录、有哈希、有键、有扫描报告抓到的 arXiv 号，唯一缺的是登记表里的那五个字段——所以它是「有依据但没链接」，不是「没依据」。引用它时要连同条件一起引：丢掉泄漏轮次后、BP-OSD 按电路级模型译码、对 1 到 6 轮拟合、X 与 Z 两基平均，± 是一个标准差（3.4 节）。

In §3.1 of the realizations guide you read “(8.91 ± 0.17)% per logical qubit per cycle”. Record: realizations[0] in card.yaml, platform “Kunlun long-range-coupled superconducting processor”, status demonstrated, description containing “(8.91 +/- 0.17)%” and “this establishes feasibility, not below-threshold scaling and not a gross-code run”, refs [wang2026lowoverhead]. Assertion: atlas-assertion:bivariate_bicycle:realizations:0, locator bivariate_bicycle/card.yaml#/realizations/0. Registry: - {id: wang2026lowoverhead}, a bare row, so the references page shows “metadata pending”. Scout report: the “Literal experiment” section of bivariate_bicycle-realizations.md: Ke Wang et al., “Demonstration of Low-Overhead Quantum Error Correction Codes”, 2026, arXiv 2505.09684, platform superconducting_kunlun, status experiment, its contribution likewise saying “a feasibility result, not a below-threshold scaling demonstration”. In plain words: the number has a record, a hash, a key and an arXiv id fetched by the sweep; the only thing missing is the five fields in the registry, so it is “sourced but unlinked”, not “unsourced”. Cite it together with its conditions: after discarding leakage runs, BP-OSD decoding with a circuit-level model, a fit over 1 to 6 cycles, the X and Z bases averaged, ± being one standard deviation (§3.4).

记号：± 读「正负」，后面的 0.17 个百分点是论文给的一个标准差（统计加拟合的不确定度），条件见 3.4 节那一条：丢掉泄漏轮次、BP-OSD 按电路级模型译码、对 1 到 6 轮拟合、X 与 Z 两基平均；「+/-」是卡片里的 ASCII 写法，与 ± 同义；[[18,4,4]] = 18 个数据比特、4 个逻辑比特、码距 4。

Notation: ± reads “plus or minus”; the 0.17 percentage points after it are one standard deviation (statistical plus fit uncertainty) as the paper states, under the conditions listed in the §3.4 item: leakage runs discarded, BP-OSD decoding with a circuit-level model, a fit over 1 to 6 cycles, the X and Z bases averaged; “+/-” is the card's ASCII spelling of the same sign; [[18,4,4]] = 18 data qubits, 4 logical qubits, distance 4.

4.3三个被防住的误引Three misattributions that got caught

防误引的手段不是要读者小心，而是把否定句写进记录本身。误引一：「BB 码的阈值是 0.8%」。先看这个数在论文里怎么来。Bravyi 等人（arXiv 2308.07915）用的是标准电路级噪声症状提取电路里的每个操作（每个 CNOT、每次制备、每次测量、每个空闲步）各自以同一个概率 p 出错，彼此独立。：提取电路里每个操作各自以同一个概率 p 出错、互相独立；译码器是他们扩展到这种噪声下的 BP-OSD（min-sum 置信传播，最多 10,000 轮迭代，OSD 用「组合扫描」版本，程序是 Roffe 等人公开的 BP-OSD 软件）。对每一个码，先模拟出 pL(p)：整块码每轮提取后失败的概率，k 个逻辑比特里任何一个出错都算失败；再解盈亏平衡方程 pL(p) = kp，右边的 kp 近似「k 个不编码的裸比特里至少一个出错」的概率。解出的 p 叫这个码的伪阈值pseudo-threshold：一个固定的码上，编码后的失败率恰好等于不编码时的失败率的那个物理错误率 p₀。它属于单个码，不是几种尺寸的曲线的交叉点。 p0：物理错误率低于它，这块编码的码就比 k 个裸比特更可靠。手算一例：gross 的 k = 12，v2 给它的 p0 = 0.0065，于是在这一点上 pL = 12 × 0.0065 ≈ 0.078，即每轮约 7.8%。pL 只在 p ≥ 10−3 模拟，更低处用拟合公式外推。所以它既不是几种尺寸的码的曲线交叉点，也不是 gross 一个码的性质。各版本的数也不同：v1（2023）表 1 列五个码，p0 从 0.007(3) 到 0.008(3)（括号里那位数论文没有解释），正文说「接近 0.008」；v2 表 1 改成 0.0048 到 0.0069（gross 为 0.0065），正文说 144 与 288 比特的两个码「接近 0.007」；v1 和 v2 的摘要都写 0.8%，Nature 发表版（2024）的摘要写 0.7%。QEC-Bank 卡片用的是摘要里的 0.8%，并在 decoders[1] 的 notes 里写明：「It is not a decoder-independent threshold of every BB construction: changing the family, CNOT order, detector model, noise normalization, or postprocessing defines a different threshold experiment.」这个数字挂在 bravyi2024bicycle 上（换了就算另一个实验的五项见《BB 码 · 高中数学版》6.4 节）。把它挂到 roffe2020decoding（BP-OSD 的基线论文）或 sahay2026matching（匹配约化）上都是错的：前者没有测 BB 的电路级阈值，后者是另一种译码器。

The defence against misattribution is not reader vigilance but negative clauses written into the records. Misattribution one: “the BB threshold is 0.8%”. First, how the paper gets this number. Bravyi et al. (arXiv 2308.07915) use standard circuit-level noiseEvery operation of the syndrome-extraction circuit (each CNOT, preparation, measurement and idle step) fails independently with the same probability p.: every operation of the extraction circuit fails independently with the same probability p. The decoder is BP-OSD extended to this noise (min-sum belief propagation, at most 10,000 iterations, the “combination sweep” version of OSD, run with the public BP-OSD software of Roffe et al.). For each code they first simulate pL(p), the probability that the whole block fails in one extraction round, where a failure of any of its k logical qubits counts. Then they solve the break-even equation pL(p) = kp, whose right side kp approximates the chance that at least one of k bare, unencoded qubits fails. The solution is that code’s pseudo-thresholdFor one fixed code, the physical error rate p₀ at which the encoded failure rate equals the unencoded one. It belongs to a single code; it is not where the curves of several sizes cross. p0: below it, the encoded block is more reliable than k bare qubits. A check by hand: gross has k = 12 and v2 gives it p0 = 0.0065, so at that point pL = 12 × 0.0065 ≈ 0.078, about 7.8% per round. pL was simulated only for p ≥ 10−3 and extrapolated below with a fitting formula. So the number is neither a crossing of curves for several code sizes nor a property of gross alone. It also changes between versions: v1 (2023), Table 1, lists five codes with p0 from 0.007(3) to 0.008(3) (the paper does not explain the digit in parentheses), and the text says “close to 0.008”; v2, Table 1, has 0.0048 to 0.0069 (gross 0.0065), and the text says “close to 0.007” for the 144- and 288-qubit codes; the v1 and v2 abstracts both say 0.8%, and the abstract of the Nature version (2024) says 0.7%. The QEC-Bank card uses the abstracts’ 0.8% and its decoders[1] notes say: “It is not a decoder-independent threshold of every BB construction: changing the family, CNOT order, detector model, noise normalization, or postprocessing defines a different threshold experiment.” The number hangs on bravyi2024bicycle (the five settings that make a different experiment when changed: §6.4 of “BB Codes · High-School Edition”). Hanging it on roffe2020decoding (the BP-OSD baseline paper) or sahay2026matching (the matching reduction) is wrong either way: the former measured no BB circuit-level threshold, the latter is a different decoder.

误引二：「BB 码已在真机上低于阈值 / 跑了 gross」。realizations[0] 的原文：「not below-threshold scaling and not a gross-code run」。误引三：「BB 码有横向的 H 和 S 门」。operations[4] 的 cost 原文：「constant-depth physical layers on blocks satisfying the extra self-duality and twisted-torus conditions; not a gross-code claim」——那是 liang2025selfdual 那个重量 8 的自对偶子族的门，不是 gross 的。看到一个数字带着「不是……」的句子，那是 QEC-Bank 在替你挡子弹；引用时把那半句一起抄上。

Misattribution two: “BB codes have run below threshold / run gross on hardware”. realizations[0]: “not below-threshold scaling and not a gross-code run”. Misattribution three: “BB codes have transversal H and S”. The cost of operations[4]: “constant-depth physical layers on blocks satisfying the extra self-duality and twisted-torus conditions; not a gross-code claim”, the gates of the weight-8 self-dual subfamily of liang2025selfdual, not of gross. When a number arrives with a “not …” clause, that is the QEC-Bank taking the bullet for you; copy that half-sentence along when you cite.

检查站Checkpoint第 4 章Chapter 4
QEC-Bank 参考文献页上 sahay2026matching 旁边写着「metadata pending」。按第 1 章与本章，这四个字精确地表示什么？On the QEC-Bank references page sahay2026matching carries the chip “metadata pending”. By Chapter 1 and this chapter, what exactly does that mean?

对。构建脚本只检查四个元数据字段是否齐全；记录（decoders[2]、views[7]）带哈希且完整，论文在扫描报告里 arxiv_verified: true（2602.22770，2026-07-26 抓取）。缺的是登记表里的五个字段，第 5 章讲怎么补。

Right. The builder checks only whether the four metadata fields are complete; the records (decoders[2], views[7]) are hashed and complete, and the paper is arxiv_verified: true in the scout report (2602.22770, fetched 2026-07-26). What is missing is the five fields in the registry; Chapter 5 says how to add them.

回看 1.1 节的规则：「metadata pending」是对登记表那一行的判断，不是对论文或记录的判断。

Look again at the rule in §1.1: “metadata pending” is a verdict on the registry row, not on the paper or the record.

上一章Back3 · 二十五条待登记3 · Twenty-five pending 下一章Next5 · 最大的缺口5 · The largest gap
第 5 章Chapter 5

最大的缺口：登记——它意味着什么、不意味着什么、补起来要做什么The largest gap: registration, what it means, what it does not, and what closing it takes

25 / 32 ≈ 78% 的键没有元数据。这是 BB 条目目前最大的一处未完成，比任何一个 bounded 或 claimed 都大——因为它影响的不是某一个数字，而是读者核对任何一个数字的能力。这一章把它说清楚，不缩小也不夸大。 25 of 32 keys, about 78%, have no metadata. It is the BB entry's largest piece of unfinished work, larger than any single bounded or claimed, because it affects not one number but the reader's ability to check any number. This chapter states it plainly, neither minimised nor inflated.

5.178% 意味着什么，不意味着什么What 78% means, and what it does not

它意味着：在 QEC-Bank 网站上，25 个键没有可点的链接、没有标题、没有作者，读者看到 chen2025anyon 只能猜它是谁写的什么；「数字 → 键 → 论文」这条链的最后一步要靠仓库里的 docs/lit/ 扫描报告完成，而扫描报告不在网站上渲染。相比之下，表面码条目的 39 个键全部登记齐全，参考文献页上没有一条「metadata pending」（《Surface Code · 参考文献》1.1 节）——同一套机制，两个条目的完成度差别就在这里。它也意味着六个侧面导读页（表示阶梯、译码器、逻辑操作、实现、关系、本页）里每一处「登记待补」的标签都是必要的，而不是啰嗦。它不意味着：引用这 25 个键的论断没有依据。每个键都有扫描报告里 arxiv_verified: true 的记录（2026-07-26 抓取）；25 个键里有 12 个绑定到卡片记录（3.5 节后的数目核对），引用这 12 个键的记录都带 sha256，按第 3 章逐条列出的绑定数，这是 17 条不同的记录（有的键被几条记录引用，比如 yoder2025tour 被 3 条，也有几个键共用一条记录，比如 relations[0]）；卡片对每条记录的边界（「not a gross-code claim」「not below-threshold」）也不依赖登记与否。它也不意味着这 25 篇不重要——BB 的直接父类、唯一的真机实验、四本代数词典的出处都在这 25 个里。

It means: on the QEC-Bank site 25 keys have no clickable link, no title, no authors; a reader seeing chen2025anyon can only guess who wrote what. The last step of the chain “number → key → paper” has to be completed from the docs/lit/ scout reports in the repository, which are not rendered on the site. By contrast, all 39 keys of the surface-code entry are fully registered and its references page shows no “metadata pending” (“Surface Code · References”, §1.1); same mechanism, and that is exactly where the two entries' completeness differs. It also means that every “from the scout report, registration pending” label across the six facet guides (representation ladder, decoders, logical operations, realizations, relations, this page) is necessary rather than verbose. It does not mean: that the claims citing these 25 keys lack a source. Every key has a record with arxiv_verified: true in a scout report (fetched 2026-07-26); 12 of the 25 keys are bound to card records (the count check after §3.5), and the records citing those 12 keys carry sha256 hashes; counted from the bindings Chapter 3 lists item by item, these are 17 distinct records (some keys are cited by several records, e.g. yoder2025tour by 3, and some records cite several keys, e.g. relations[0]); and the card's boundary clauses for each record (“not a gross-code claim”, “not below-threshold”) do not depend on registration. Nor does it mean the 25 papers are unimportant: BB's direct parents, its only hardware experiment and the sources of four algebraic dictionaries are all among them.

25 / 32 ≈ 78%　待登记pending　·　表面码条目surface entry: 0 / 39
影响的是「可点性」，不是「有据性」what suffers is clickability, not sourcing

5.2补起来是机械的，但有两处要人来决定Closing it is mechanical, with two places that need a human decision

六份扫描报告里每一条都是一个 YAML 块：bibkey、title、authors、year、arxiv。登记表需要的五个字段是 id、title、authors、year、url，其中 url 就是 https://arxiv.org/abs/ 加上 arxiv 号。所以补齐 25 条是把裸行 - {id: KEY} 换成五行完整记录的抄写工作，构建脚本会在下一次导出时自动把「metadata pending」换成带链接的书目——和关系导读 1.3 节里 pending 目标条目的情形一样，卡片本身一个字不用改。有两处需要人来决定。其一，键名不一致。译码扫描报告用 roffe2020bposd、卡片用 roffe2020decoding，QEC-Bank 用别名解决；奠基报告的 bravyi2024high 同理。登记时如果发现某个待补的键在别的报告里换了名字，要决定谁是主键、谁是别名。其二，拼写取舍。登记表与扫描报告在已登记的键上已经出现了差别：mackay2004sparse 的标题一处是「Error-Correction」一处是「Error Correction」；steffan2025tile 的第二作者一处是「Shin Ho Choe」一处是「Seung-Hoon Choe」；译码报告把 bravyi2024bicycle 的作者简写成「Sergey Bravyi et al.」而登记表列全六人。本次对照 arXiv 摘要页后，前两处都以登记表为准（arXiv 写法与登记表相同）；反过来也有登记表与 arXiv 不同的：roffe2020decoding 的登记标题把 arXiv 的「LDPC」展开成了「Low-Density Parity-Check」。所以最省事的规则是「以 arXiv 摘要页为准」，但这仍要编者拍板。页面显示的永远是登记表，所以登记时抄哪一种拼法，就是页面上永远显示的那一种。人话：抄是机器活，选是人的活。

Every item in the six scout reports is a YAML block with bibkey, title, authors, year, arxiv. The registry needs five fields, id, title, authors, year, url, and the url is just https://arxiv.org/abs/ followed by the arxiv id. So completing the 25 is the clerical job of replacing each bare row - {id: KEY} by a five-line record; the builder will swap “metadata pending” for a linked bibliography at the next export, and, as with the pending target entry in §1.3 of the relations guide, the card itself changes by not one character. Two places need a human. First, key mismatches. The decoders report uses roffe2020bposd where the card uses roffe2020decoding; the QEC-Bank resolves it by an alias, as with the foundations report's bravyi2024high. If a pending key turns out to have a different name in another report, someone decides which is primary and which the alias. Second, spelling choices. Registry and scout reports already differ on the registered keys: the title of mackay2004sparse is “Error-Correction” in one and “Error Correction” in the other; the second author of steffan2025tile is “Shin Ho Choe” in one and “Seung-Hoon Choe” in the other; the decoders report abbreviates the authors of bravyi2024bicycle to “Sergey Bravyi et al.” while the registry lists all six. Checked against the arXiv abstract pages, the first two go the registry's way (arXiv matches the registry); the opposite also occurs: the registry title of roffe2020decoding expands arXiv's “LDPC” into “Low-Density Parity-Check”. So the simplest rule is “follow the arXiv abstract page”, but that is still the editors' call. The page always shows the registry, so whichever spelling is copied at registration is the one shown forever. In plain words: copying is machine work, choosing is human work.

扫描报告块scout block { bibkey, title, authors, year, arxiv }
→ 登记表行registry row { id, title, authors, year, url }
url = https://arxiv.org/abs/ + arxiv
× 25 次；
每次两问：键名是否与别处一致？拼法取哪一种？× 25;
two questions each time: does the key name agree elsewhere? which spelling?

5.3三个在 Level-2 页出现、却不在参考文献页上的键Three keys that appear on a Level-2 page but not on the references page

还有一层更小的缺口。Level-2 页「变体与边界地图」和扫描报告用到三个键：liang2026qudit（素数维 qudit 的 BB，arXiv 2602.20158）、cruzbenito2026evolutionary（LLM 引导的演化搜索，arXiv 2606.02418）、wang2026bbs（BB 表面码的代数与边界，arXiv 2606.08771）。三个键在登记表里都有一行裸 id，但不在本条目的 32 个 refs 里，所以参考文献页上根本不出现——不是「metadata pending」，是不在列表上。一个读者从关系导读 4.3 节或 Level-2 页看到这三个键，回到参考文献页找不到它们。这三个键在扫描报告里的 priority 都是 scout（最低一级），也许这就是它们没进 refs 列表的原因；本页只记录这个事实，不替编者决定它们该不该进。

There is a smaller gap beneath. The Level-2 page “variant and boundary map” and the scout reports use three keys, liang2026qudit (BB over prime-dimensional qudits, arXiv 2602.20158), cruzbenito2026evolutionary (LLM-guided evolutionary search, arXiv 2606.02418) and wang2026bbs (algebra and boundaries of BB surface codes, arXiv 2606.08771). All three have a bare-id row in the registry but are not among this entry's 32 refs, so they do not appear on the references page at all: not “metadata pending”, simply absent from the list. A reader who meets them in §4.3 of the relations guide or on the Level-2 page and turns to the references page will not find them. All three carry priority scout (the lowest) in the reports, which may be why they never entered the refs list; this page records the fact and does not decide for the editors whether they should.

陷阱：两个 pending，三级缺口Pitfall: two pendings, three levels of gap

「metadata pending」（键有、名字没有）与关系页上的「pending」（边有、目标条目没有）是两个不同的机制，都是构建时算出来的，都不是卡片字段。缺口分三级：25 个键没有元数据（本章）；3 个键不在 refs 列表里（5.3 节）；two_bga 目标条目不存在（关系导读）。三者都不改变任何一条记录的可信度，但都让读者少了一次「点过去看看」的机会。

“metadata pending” (key present, name absent) and the relations page's “pending” (edge present, target entry absent) are two different mechanisms, both computed at build time, neither a card field. The gap has three levels: 25 keys without metadata (this chapter); 3 keys not on the refs list (§5.3); the two_bga target entry not existing (relations guide). None changes the trustworthiness of any record, but each takes away one chance for the reader to “click through and look”.

关键想法Key idea

BB 条目的 32 份契约都签了名，25 份没填地址。地址在扫描报告里一个不缺，抄进登记表是机械活；键名冲突与拼法取舍要人来定。补齐之前，任何引用这 25 个键的地方都应带「登记待补」四个字——本系列的六页导读正是这么做的。

All 32 of the BB entry's contracts are signed; 25 lack an address. Every address is in the scout reports, and copying them into the registry is clerical; key conflicts and spelling choices need a person. Until then, every place that cites one of the 25 should carry the words “registration pending”, which is exactly what the six guides in this series do.

上一章Back4 · 追踪一个键4 · Tracing one key 下一章Next尾声 · 词汇表与去处Epilogue · Glossary & onward
尾声Epilogue

一份可以被质问的书单，和它还没填完的地址栏A reading list that can be cross-examined, and its unfilled address column

回放：键 = 姓 + 年 + 词，登记在 registry/refs.yaml，完整行五个字段、裸行一个字段、别名行两个字段；「metadata pending」精确地表示裸行（第 1 章）→ 32 个键在 card.yaml 里有三种住处：条目级列表、记录的 refs、记录的 origin.ref；16 个绑定到记录，16 个仅条目级；被引最多的是 bravyi2024bicycle，7 处（第 1 章）→ 7 条已登记：3 条祖先与奠基、2 条译码器、2 条操作与变体，元数据照抄登记表（第 2 章）→ 25 条待登记按侧面分五组，每条标题与 arXiv 号来自扫描报告并标「登记待补」，5 + 3 + 7 + 6 + 4 = 25（第 3 章）→ 追 sahay2026matching 六步，链条在登记表断开，扫描报告接上；导出页的 References 是「类型默认 + 卡片专属」的并集；三个「不是……」句子防住三种误引（第 4 章）→ 78% 影响的是可点性不是有据性；补齐是抄写活加两个人为决定；另有三个键不在列表上（第 5 章）。

Replay: key = surname + year + word, registered in registry/refs.yaml, a complete row having five fields, a bare row one, an alias row two; “metadata pending” means precisely a bare row (Ch. 1) → the 32 keys have three homes in card.yaml, the entry-level list, record refs and record origin.ref; 16 record-bound, 16 entry-level only; the most cited is bravyi2024bicycle, 7 places (Ch. 1) → 7 registered: 3 ancestry and foundation, 2 decoders, 2 operations and variants, metadata copied from the registry (Ch. 2) → 25 pending in five facet groups, each title and arXiv id from a scout report and labelled “registration pending”, 5 + 3 + 7 + 6 + 4 = 25 (Ch. 3) → six steps tracing sahay2026matching, the chain breaking at the registry and reconnecting at the scout report; an exported References line is the union of “type defaults + card-specific”; three “not …” clauses block three misattributions (Ch. 4) → 78% affects clickability, not sourcing; closing it is copying plus two human decisions; three further keys are off the list (Ch. 5).

符号与读法Symbols & how to read them

符号Symbol名称与读法Name & how to read it在本页中的含义Meaning on this page
bibkey「文献键」“bib-key”姓 + 年 + 助记词（常取自标题，bravyi2024bicycle 取的是码族名）构成的全局唯一名字，如 bravyi2024bicycle、wang2026lowoverhead。A globally unique name, surname + year + mnemonic word (often from the title; bravyi2024bicycle uses the code family's name), e.g. bravyi2024bicycle, wang2026lowoverhead.
registry/refs.yaml登记表The registry受控词表；完整行 { id, title, authors, year, url }，裸行 { id }，别名行 { id, alias_of }。The controlled vocabulary; complete rows { id, title, authors, year, url }, bare rows { id }, alias rows { id, alias_of }.
metadata pending「元数据待补」“metadata pending”构建脚本对裸行的渲染：title、authors、year、url 缺任何一个。BB 条目 25 / 32。The builder's rendering of a bare row: any of title, authors, year, url missing. BB entry: 25 / 32.
alias_of「是……的别名」“alias of”同一论文的第二个键，不算第二篇：bravyi2024high → bravyi2024bicycle，roffe2020bposd → roffe2020decoding。A second key for the same paper, not a second paper: bravyi2024high → bravyi2024bicycle, roffe2020bposd → roffe2020decoding.
refs / origin.ref引用列表 / 出处键refs list / origin key记录的多键列表 / 记录「出处」里的单键（操作 kind: claimed，视图 kind: curated，分类 kind: curated）。A record's multi-key list / the single key in a record's provenance (operations kind: claimed, views kind: curated, categories kind: curated).
decoders[2], views[7], realizations[0]方括号下标，读「第几条」Bracket index, read “record number”card.yaml 里某个列表的第 i 条，从 0 数起；本页追踪的三条记录。The i-th record of a list in card.yaml, counting from 0; the three records traced on this page.
atlas-assertion:…, source_locator, sha256断言 id、位置、哈希Assertion id, locator, hash记录的元数据副本：例如 atlas-assertion:bivariate_bicycle:decoders:2，bivariate_bicycle/card.yaml#/decoders/2，26aee9ae…。A record's metadata copy: e.g. atlas-assertion:bivariate_bicycle:decoders:2, bivariate_bicycle/card.yaml#/decoders/2, 26aee9ae….
eczoo-entity:qcga#collection实体 idEntity id全部 BB 断言的主语：Error Correction Zoo 的 BB 家族条目，整族。The subject of every BB assertion: the Error Correction Zoo BB family entry, the whole family.
[[n, k, d]]「n k d 码」“an n-k-d code”n 数据比特、k 逻辑比特、码距 d（定义、定义）。本页：[[144,12,12]]、[[18,4,4]]、[[32,8,6]]、[[98,6,12]]、[[162,8,12]]。n data qubits, k logical qubits, distance d (definition, definition). Here: [[144,12,12]], [[18,4,4]], [[32,8,6]], [[98,6,12]], [[162,8,12]].
±, +/-读「正负」Read “plus or minus”实验数字后的不确定度；(8.91 ± 0.17)% 的 0.17 按论文是一个标准差（统计加拟合）；卡片用 ASCII 写法 +/-。The uncertainty after an experimental number; in (8.91 ± 0.17)% the paper states that 0.17 is one standard deviation (statistical plus fit); the card uses the ASCII spelling +/-.
≈读「约等于」Read “approximately equals”25 / 32 ≈ 78%。25 / 32 ≈ 78%.
arXiv 2602.22770arXiv 编号，读「年年月月.序号」arXiv id, read “YYMM.number”预印本服务器 arXiv 的文章编号；url = https://arxiv.org/abs/ + 编号。扫描报告里每条都带 arxiv_verified: true。The article id on the preprint server arXiv; url = https://arxiv.org/abs/ + id. Every scout-report item carries arxiv_verified: true.
p「p」，物理错误率“p”, the physical error rate只在 3.2 节 pakhunov2026peeling 的 p = 10−3 出现：每个电路位置出错的概率。Appears only in §3.2 for pakhunov2026peeling's p = 10−3: the probability of a fault per circuit location.
诚实总账The honesty ledger

本页没有引入任何 QEC-Bank 仓库之外的文献。7 条已登记的标题、作者、年份、链接照抄 registry/refs.yaml；25 条待登记加 5.3 节的 3 条，标题、作者、年份、arXiv 号最初取自 docs/lit/bivariate_bicycle/ 六份扫描报告（检索日期 2026-07-26），每处都标「登记待补」；本次修订（2026-09-24）把这 28 条和 7 条已登记的 arXiv 摘要页逐一打开核对，改正了三处标题用词、若干大小写与作者全名、一处子类定义（wang2026coprime）与两处内容描述（sahay2026matching、wang2026lowoverhead），并标出论文自己的 [[32,8,6]] 标签与其构造矛盾（本页脚本 notes/references/verify.py：存在重量 2 的逻辑算符）。每个键「替谁作证」照抄 card.yaml 的 legacy_card 里 refs 与 origin.ref 的绑定，由本页逐路径统计：16 绑定 + 16 仅条目级，7 已登记 + 25 待登记。「仅条目级」是本页的叫法，不是 QEC-Bank 的官方词。「metadata pending 的精确含义」读自 tools/site/build.py 的 _reference_html；「导出页 References 是并集」读自 registry/decoders.yaml 里 mwpm 与 bp-osd 的默认 refs。本页转述的 0.8%、(8.91 ± 0.17)%、330 倍、118 倍等数字各属其原论文与卡片记录，本页未重算、也未互相比较。审阅后（2026-09-24）又按原文补了四处：0.8% 的来历与各版本的数（2308.07915 的 v1、v2 源文件与 Nature 摘要）、330 倍与 118 倍各比的是什么（2604.11352 表 5 与 §VII）、(8.91 ± 0.17)% 的条件（2505.09684 源文件）、圆柱技巧读的是什么（2602.22770 v1 §III.2–III.3）；图 3.1、补课 3a、3b 里的每个数由 notes/references/verify2.py 算出。

This page introduces no reference outside the QEC-Bank repository. The 7 registered titles, authors, years and links copy registry/refs.yaml; the 25 pending plus the 3 of §5.3 originally took their titles, authors, years and arXiv ids from the six scout reports under docs/lit/bivariate_bicycle/ (fetched 2026-07-26), labelled “registration pending” at each use; this revision (2026-09-24) opened the arXiv abstract pages of these 28 and of the 7 registered keys one by one, corrected the wording of three titles, some capitalisation and full author names, one subclass definition (wang2026coprime) and two content descriptions (sahay2026matching, wang2026lowoverhead), and flags that the paper's own [[32,8,6]] label contradicts its construction (this page's script notes/references/verify.py: a weight-2 logical exists). Each key's “whose witness” copies the refs and origin.ref bindings in the legacy_card of card.yaml, tallied path by path on this page: 16 bound + 16 entry-level only, 7 registered + 25 pending. “Entry-level only” is this page's term, not an official QEC-Bank one. “The precise meaning of metadata pending” was read from _reference_html in tools/site/build.py; “an exported References line is a union” from the default refs of mwpm and bp-osd in registry/decoders.yaml. The numbers relayed here, 0.8%, (8.91 ± 0.17)%, 330×, 118× and the rest, each belong to their own paper and card record; this page neither recomputes nor compares them. After review (2026-09-24) four points were added from the papers themselves: where 0.8% comes from and the numbers of each version (the v1 and v2 sources of 2308.07915 and the Nature abstract), what 330× and 118× each compare (2604.11352, Table 5 and §VII), the conditions behind (8.91 ± 0.17)% (the 2505.09684 source), and what the cylinder trick reads (2602.22770 v1, §§III.2–III.3); every number in Fig. 3.1 and catch-ups 3a and 3b is computed by notes/references/verify2.py.

本条目的其他侧面Other facets of this entry

条目总览 · 直觉 · 导览 · 表示阶梯 · 参数与实例 · 形式化 IR 与证书 · 译码器 · 逻辑操作 · 用例与实现 · 关系 · 推导与指南 · 参考文献 —— 表面码的同一侧面：《Surface Code · 参考文献》 —— QEC-Bank 原页：在 QEC-Bank 中查看此条目 →

Entry overview · Intuition · Guided tour · Representation ladder · Parameters & instances · Formal IR & certificates · Decoders · Logical operations · Use cases & realizations · Relations · Derivations & guides · References — the same facet for the surface code: “Surface Code · References” — QEC-Bank page: Open in the QEC-Bank →