三十九个键，
每个数字的来处 Thirty-nine keys,
where every number comes from

1.1一个键的解剖Anatomy of a key

1.2键住在哪里Where the keys live

1.3为什么每条论断都要指向一个键Why every claim points at a key

2.1奠基与构造：这个码是什么Foundations and construction: what the code is

• Alexei Yu. Kitaev, "Fault-tolerant quantum computation by anyons" (2003)
  kitaev2003fault · 环面码及其任意子图像的原始出处（任意子：把警报看成可以在格子上移动的「粒子」，星报警与面报警是两种任意子）。被关系 same_bulk_phase_as → toric 引用（见关系导读第 3 章）。The original source for the toric code and its anyon picture (anyons: alarms viewed as “particles” that move on the lattice; star alarms and plaquette alarms are two kinds). Cited by the relation same_bulk_phase_as → toric (see the relations guide, Chapter 3).
• Sergey B. Bravyi and Alexei Yu. Kitaev, "Quantum codes on a lattice with boundary" (1998)
  bravyi1998quantum · 毛 / 光边界的首次构造，平面表面码由此诞生。表示阶梯（QEC-Bank 用 R1 到 R5 五级台阶，把同一个码写成越来越抽象的几种语言，见表示阶梯导读）R2「同调」视图的 curated 来源；同调视图把检查看成「取边界」。平面码里，一条 Z 串（Z 放在一串边上）要不触发任何星检查，端点只能落在毛边上（横边悬空、没有顶点的那两侧）；论文把这种端点全在边界上的链叫相对闭链。这样的串若等于若干面检查的乘积，例如绕一块面的小环，就是平凡的，不改变存下的信息；从一条毛边连到对面毛边的串不是检查的乘积，它才是逻辑算符。相差若干检查乘积的两条串算同一类（图 2.1）。First construction of rough/smooth boundaries; the planar surface code is born here. Curated origin of the R2 “homological” view on the representation ladder (the QEC-Bank's five rungs R1 to R5 that write the same code in increasingly abstract languages; see the representation-ladder guide); the homological view reads checks as “taking a boundary”. On the planar code a Z string (Z on a chain of edges) trips no star check only if its endpoints lie on the rough edges (the two sides where horizontal edges end free, with no vertex); the paper calls such a chain, all of whose endpoints are on the boundary, a relative cycle. If such a string equals a product of face checks, like a small loop around one face, it is trivial and leaves the stored information alone; a string from one rough edge to the opposite one is not a product of checks, and it is the logical operator. Two strings that differ by a product of checks belong to the same class (Figure 2.1).
• Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill, "Topological quantum memory" (2002)
  dennis2002topological · 引用最多的键：分类标签「topological」的来源、码容量译码器 mwpm 的 10.9%（最优译码）、R5「统计力学」视图（随机键 Ising 模型：把「最优译码能否成功」换算成一个带随机正负耦合的磁性模型是否有序，阈值就是它的相变点）、以及与环面码的关系。The most-cited key: source of the “topological” category tag, of the code-capacity mwpm record's 10.9% (optimal decoding), of the R5 “statistical-mechanics” view (random-bond Ising model: whether optimal decoding succeeds is translated into whether a magnet with random positive and negative couplings orders, and the threshold is its phase transition), and of the toric relation.
• Héctor Bombín and Miguel A. Martín-Delgado, "Homological Error Correction: Classical and Quantum Codes" (2007)
  bombin2007homological · 把「检查 = 取边界；端点只落在相应类型边界上的串叫相对闭链，相差检查乘积的串算同一类；逻辑串是其中不是检查乘积的那些，即非平凡的相对同调类（检查乘积本身是平凡类，什么也不改）」讲成一般理论。与 bravyi1998quantum 共同支撑 R2 同调视图。Turns “checks = taking boundaries; strings whose endpoints lie only on the boundary of the matching type are relative cycles, and strings that differ by products of checks count as one class; the logical strings are the relative cycles that are not products of checks, i.e. the nontrivial relative-homology classes (a product of checks is the trivial class and changes nothing)” into a general theory. Supports the R2 homological view together with bravyi1998quantum.
• Jean-Pierre Tillich and Gilles Zémor, "Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the blocklength" (2009)
  tillich2009quantum · 定义超图积。是分类标签「hypergraph-product」、R1 校验矩阵视图（derived）、R2 超图积视图（canonical）、两条关系与导览多步的来源——本条目构造器的理论依据（qLDPC 教程 2.1 节）。Defines the hypergraph product. Source of the “hypergraph-product” category tag, the R1 check-matrix view (derived), the R2 HGP view (canonical), both relations and several tour steps: the theoretical basis of this entry's constructor (qLDPC guide, §2.1).
• Dorit Aharonov and Michael Ben-Or, "Fault Tolerant Quantum Computation with Constant Error" (1997)
  aharonov1997fault · 阈值定理：足够弱的局域噪声可以被压到任意小。直觉文「设计问题」一节用它说明表面码要解决的不是「能不能」而是「以什么交互契约」。仅条目级。The threshold theorem: sufficiently weak local noise can be suppressed arbitrarily. The intuition essay's “design problem” uses it to say the surface code's question is not whether but under what interaction contract. Entry-level only.

2.2禁区与代价定理：它做不到什么No-go and tradeoff theorems: what it cannot do

• Sergey Bravyi and Barbara M. Terhal, "A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes" (2009)
  bravyi2009nogo · 二维局域稳定子哈密顿量不能自纠错。把检查当能量项，考虑所有「从一个逻辑态出发、每步只加一个局部错误、最后到达另一个逻辑态」的路径：每条路径途中都有一个能量最高点，能量壁垒是这个最高点（比基态高多少）在所有路径里的最小值。论文证明二维里这个壁垒被一个与系统大小无关的常数封顶：造出一条串的两个端点要花能量，但把串拉长不再加价，所以热涨落只需越过一个固定高度就能造成逻辑错误。直觉文「失效模式」里「没有被动热记忆」的依据。仅条目级。2D local stabilizer Hamiltonians cannot self-correct. Take the checks as energy terms and consider every path that starts in one logical state, adds one local error per step, and ends in another logical state: each path has a highest-energy point on the way, and the energy barrier is the smallest such height (above the ground state) over all paths. The paper proves that in 2D this barrier is capped by a constant independent of the system size: creating the two endpoints of a string costs energy, but stretching the string costs nothing more, so thermal fluctuations need only clear a fixed height to cause a logical error. Basis of “no passive thermal memory” in the intuition essay's failure modes. Entry-level only.
• Sergey Bravyi, David Poulin, and Barbara Terhal, "Tradeoffs for reliable quantum information storage in 2D systems" (2010)
  bravyi2010tradeoffs · k·d² ≤ c·n（k 逻辑比特、d 码距、n 物理比特），前提是二维格点、每个格点只放有限个比特（比特密度有界）、每条检查只作用在固定大小的一块区域内；常数 c 只依赖这两个局域性参数。表面码 k = 1、d ∝ √n，使 k·d² 与 n 同阶，达到了这个标度，并不是说等号成立。BB 码教程 1.1 节从这里出发。仅条目级。k·d² ≤ c·n (k logical qubits, d distance, n physical qubits), assuming a 2D lattice with a bounded number of qubits per site (bounded qubit density) and every check acting within a region of fixed size; the constant c depends only on these two locality parameters. The surface code with k = 1, d ∝ √n makes k·d² of the same order as n: it saturates the scaling, which does not mean equality holds. The BB-code guide, §1.1 starts here. Entry-level only.
• Sergey Bravyi and Jeongwan Haah, "Energy landscape of 3D spin Hamiltonians with topological order" (2011)
  bravyi2011energy · 三维拓扑序的能量壁垒分析（能量壁垒：见上一条 bravyi2009nogo 的定义），二维禁区在三维如何松动的对照。仅条目级。Energy-barrier analysis for 3D topological order (the energy barrier: defined in the bravyi2009nogo item above), the contrast case for how the 2D no-go loosens in 3D. Entry-level only.
• Sergey Bravyi and Robert König, "Classification of topologically protected gates for local stabilizer codes" (2013)
  bravyi2013classification · Bravyi–König 定理：对二维拓扑稳定子码（检查几何局域、码距足够大），用深度固定、每个门只作用于相邻比特的电路实现的逻辑门都在 Clifford 群里；所以在这些前提下，没有这样实现的逻辑 T 门（格点手术教程 1.3 节）。仅条目级。The Bravyi–König theorem: for a 2D topological stabilizer code (geometrically local checks, large enough distance), every logical gate implemented by a constant-depth circuit of gates acting on nearby qubits lies in the Clifford group; so under these assumptions no logical T gate is implemented this way (lattice-surgery guide, §1.3). Entry-level only.
• Jeongwan Haah, "Local stabilizer codes in three dimensions without string logical operators" (2011)
  haah2011local · 立方码：三维里没有串状逻辑算符的码（它的逻辑算符不能沿一条线铺开），是「逻辑信息一定住在串上」在三维不再成立的反例。仅条目级。The cubic code: a 3D code with no string-like logical operators (its logical operators cannot be laid out along a line), the counterexample to “logical data must live on strings” beyond 2D. Entry-level only.

2.3译码器与阈值：怎样纠、纠到哪Decoders and thresholds: how to correct, and how far

• Chenyang Wang, Jim Harrington, and John Preskill, "Confinement-Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory" (2003)
  wang2003confinement · 两个数字的家：匹配译码在码容量下的 10.3%，以及唯象噪声（数据 + 测量都出错）下的 2.9%。被两条 mwpm 记录与 R5 统计力学视图引用。请勿把 2.9% 与 10.9% 混作同一阈值（译码器教程 6.1 节）。Home of two numbers: 10.3% for matching under code capacity, and 2.9% under phenomenological noise (data and measurement errors). Cited by two mwpm records and the R5 statistical-mechanics view. Never merge the 2.9% with the 10.9% (decoders guide, §6.1).
• Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland, "Surface codes: Towards practical large-scale quantum computation" (2012)
  fowler2012surface · 电路级 mwpm 记录的 0.57%/步（八步电路，约 4%/整周期）；同时是四条 claimed 操作的键：打洞编织 CNOT、逻辑 |0⟩/|+⟩ 初始化、逻辑 Z/X 读出、Pauli 框架更新。The circuit-level mwpm record's 0.57% per step (eight-step circuit, about 4% per full cycle); also the key behind four claimed operations: hole-braiding CNOT, logical |0⟩/|+⟩ initialization, logical Z/X readout, Pauli-frame update.
• Oscar Higgott, "PyMatching: A Python package for decoding quantum codes with minimum-weight perfect matching" (2022)
  higgott2022pymatching · 最小重量完美匹配的开源实现，码容量 mwpm 记录引用它作为可复现的工具（译码器教程 2.3 节）。The open-source implementation of minimum-weight perfect matching; the code-capacity mwpm record cites it as the reproducible tool (decoders guide, §2.3).
• Nicolas Delfosse and Naomi H. Nickerson, "Almost-linear time decoding algorithm for topological codes" (2021)
  delfosse2021almost · union-find 记录。做法：每个警报先自成一个团；含奇数个警报的团向外长一圈（加权生长版每次只长边界最短的奇团），两团相碰就合并；含偶数个警报的团停止生长；等所有团都含偶数个警报，在每个团里用剥离（peeling）算法从树的叶子往里一条边一条边地剥，得到一个与警报相符的修复。9.9% 与 2.6% 属于加权生长版：前者是二维环面码、只有独立的 Z 错误、测量完美，后者是唯象噪声（每轮数据比特 Z 错误与测量结果翻转以同一概率发生）；所有团一起均匀生长的版本是 9.2% 与 2.4%（论文第 5 节）。用小幅阈值换近线性时间（译码器教程第 4 章）。The union-find record. How it works: each alarm starts as its own cluster; a cluster holding an odd number of alarms grows by one layer (the weighted-growth version grows only the odd cluster with the shortest boundary each time), and clusters that touch merge; a cluster holding an even number of alarms stops growing; once every cluster is even, a peeling algorithm strips each cluster edge by edge from the leaves of a tree inward and returns a correction matching the alarms. The 9.9% and 2.6% belong to the weighted-growth version: the first for the 2D toric code with independent Z errors and perfect measurements, the second for phenomenological noise (per round, data-qubit Z errors and measurement flips at the same probability); growing all clusters uniformly gives 9.2% and 2.4% (Section 5 of the paper). A little threshold is traded for almost-linear time (decoders guide, Chapter 4).
• Guillaume Duclos-Cianci and David Poulin, "Fast Decoders for Topological Quantum Codes" (2010)
  duclos2010fast · renormalization-group（重整化群）记录：先在小块里局部译码，把结果当成更大一层的输入，一层层往上粗粒化，估计每个逻辑类的概率，而不是立刻押最轻的链。The renormalization-group record: decode locally in small blocks, feed the results to a coarser layer, and repeat, estimating the probability of each logical class instead of committing to the lightest chain.
• Sergey Bravyi, Martin Suchara, and Alexander Vargo, "Efficient Algorithms for Maximum Likelihood Decoding in the Surface Code" (2014)
  bravyi2014efficient · tensor-network-mld 记录：独立 X/Z 噪声下精确的最大似然译码，一般 Pauli 噪声下用矩阵乘积态（一种把大张量写成一串小张量相乘的压缩表示）近似（译码器教程 1.2 节的「最优」）。The tensor-network-mld record: exact maximum-likelihood decoding for independent X/Z noise, matrix-product-state approximation (a compressed representation writing a big tensor as a chain of small ones) for general Pauli noise (the “optimal” of the decoders guide, §1.2).
• Joschka Roffe, David R. White, Simon Burton, and Earl Campbell, "Decoding Across the Quantum Low-Density Parity-Check Code Landscape" (2020)
  roffe2020decoding · bp-osd 记录：置信传播（检查与比特之间反复传递「这个比特出错的概率」）加有序统计后处理，是可移植的 qLDPC 基线译码器，不专为几何而设（qLDPC 教程 6.3 节）。别名 roffe2020bposd。The bp-osd record: belief propagation (checks and qubits repeatedly exchange “probability that this qubit failed”) plus ordered-statistics post-processing, a portable qLDPC baseline, not geometry-specialized (qLDPC guide, §6.3). Alias roffe2020bposd.
• Oscar Higgott, Thomas C. Bohdanowicz, Aleksander Kubica, Steven T. Flammia, and Earl T. Campbell, "Improved Decoding of Circuit Noise and Fragile Boundaries of Tailored Surface Codes" (2023)
  higgott2023beliefmatching · belief-matching 记录：先用置信传播估计哪些故障会同时点亮几个警报，再据此修改匹配图上每条边的权重，最后做匹配（译码器教程 5.4 节的电路级噪声）。The belief-matching record: belief propagation first estimates which faults light several alarms together, the edge weights of the matching graph are adjusted accordingly, and matching runs last (circuit-level noise in the decoders guide, §5.4).

2.4逻辑操作：怎样让它计算Logical operations: how to make it compute

• Clare Horsman, Austin G. Fowler, Simon Devitt, and Rodney Van Meter, "Surface code quantum computing by lattice surgery" (2012)
  horsman2012surface · claimed 操作「CNOT · surgery」的键：每次合并 / 分裂 O(d) 轮（d = 码距）加一块辅助路由贴片。格点手术教程第 3 章逐步推导。Key of the claimed operation “CNOT · surgery”: O(d) rounds per merge/split (d = distance) plus an ancilla routing patch. Derived step by step in the lattice-surgery guide, Chapter 3.
• Benjamin J. Brown, Katharina Laubscher, Markus S. Kesselring, and James R. Wootton, "Poking holes and cutting corners to achieve Clifford gates with the surface code" (2017)
  brown2017poking · claimed 操作「单比特 Clifford · braiding」：在贴片上移动扭结或角落来慢慢改变码的形状（码形变），移动路线保证码距不降。Claimed operation “single-qubit Clifford · braiding”: moving twists or corners across the patch to change the code's shape gradually (code deformation), along routes that keep the distance intact.
• J. E. Moussa, "Transversal Clifford gates on folded surface codes" (2016)
  moussa2016transversal · claimed 操作「H / S · fold-transversal」：把正方形贴片沿一条对角线「对折」：关于对角线对称的两个比特叠成一对，对角线上的比特单独一个（图 2.2）。于是 H̄ = 对角线比特各做 H，每对做「H⊗H 再交换」；S̄ = 对角线比特各做 S 或 S†（按比特所属的类型），每对做一个 CZ。这些门的深度固定，不随码距增长。但把贴片折起来要靠一串局部 SWAP 把比特搬到位，论文给出正方形码折叠操作的深度是 6D − 5（D = 码距），会随码距增长：门的深度与搬运比特的代价要分开算。Claimed operation “H / S · fold-transversal”: fold the square patch along a diagonal: two qubits that are mirror images across the diagonal are stacked into a pair, and each qubit on the diagonal stays alone (Figure 2.2). Then H̄ = H on each diagonal qubit and “H⊗H, then swap” on each pair; S̄ = S or S† on each diagonal qubit (depending on its type) and one CZ on each pair. These gates have fixed depth, independent of the distance. Folding the patch in the first place, though, takes a sequence of local SWAPs to move qubits into place, and the paper gives the depth of the folding operation for the square code as 6D − 5 (D = distance), growing with the distance: gate depth and the cost of moving qubits must be counted separately.
• Sergey Bravyi and Alexei Kitaev, "Universal quantum computation with ideal Clifford gates and noisy ancillas" (2005)
  bravyi2005magic · claimed 操作「T · measurement-based」：先制备许多份带噪声的「魔法态」（一种特殊的单比特态，有了它就能用 Clifford 操作和测量做出 T 门），把它们提纯（蒸馏），再用门传送（测量加纠正）把它消耗掉；专门生产魔法态的「工厂」要额外占用比特和纠错周期；它在总开销里占多大份额，取决于算法需要多少个 T 门、工厂能否并行运行，所以资源估算要分别报告比特数、周期数和两者之积（时空体积）（格点手术教程 1.2 节的第三条路）。Claimed operation “T · measurement-based”: prepare many noisy copies of a “magic state” (a special one-qubit state which, together with Clifford operations and measurements, yields a T gate), purify them (distillation), then consume one by gate teleportation (measurement plus correction); the “factories” producing magic states take extra qubits and correction cycles; their share of the total depends on how many T gates the algorithm needs and whether factories run in parallel, so resource estimates should report qubit count, cycle count and their product (spacetime volume) separately (the third road of the lattice-surgery guide, §1.2).
• Robert Raussendorf, Jim Harrington, and Karl Goyal, "Topological fault-tolerance in cluster state quantum computation" (2007)
  raussendorf2007fault · 表面码的「基于测量」版本：先准备一大块纠缠好的三维簇态（许多比特按格子两两纠缠的状态），然后只靠逐个测量比特完成纠错和计算。仅条目级。The measurement-based version of the surface code: prepare a large entangled 3D cluster state (many qubits entangled pairwise along a lattice) and carry out correction and computation purely by measuring qubits one by one. Entry-level only.
• Héctor Bombín, "Topological Order with a Twist: Ising Anyons from an Abelian Model" (2010)
  bombin2010twist · 扭结缺陷的首次构造；直觉文「费米子与扭结」一节的背景（关系导读 4.3 节）。仅条目级。First construction of twist defects; background for the intuition essay's “fermions and twists” (relations guide, §4.3). Entry-level only.

2.5版图变体：同一内部的其他裁法Layout variants: other cuts of the same bulk

• Héctor Bombín and Miguel A. Martín-Delgado, "Optimal Resources for Topological Two-Dimensional Stabilizer Codes: Comparative Study" (2007)
  bombin2007optimal · 最早证明平面版图可达 n = d²（n 物理比特、d 码距）——旋转表面码省一半比特的数学起源（关系导读 4.1 节）。仅条目级。First proof that a planar layout can reach n = d² (n physical qubits, d distance), the mathematical origin of the rotated code's qubit halving (relations guide, §4.1). Entry-level only.
• J. Pablo Bonilla Ataides, David K. Tuckett, Stephen D. Bartlett, Steven T. Flammia, and Benjamin J. Brown, "The XZZX Surface Code" (2021)
  bonilla2021xzzx · 交错 Hadamard 后的 XZZX 检查；在同一个纯 Z 噪声下，最优译码阈值趋近 50%，而 CSS 表面码约 11%——差别来自局部基变换改变了噪声相对于检查的样子，不是换了噪声模型（关系导读 4.2 节）。仅条目级。XZZX checks after alternating Hadamards; under the same pure-Z noise the optimal-decoding threshold approaches 50% versus about 11% for the CSS surface code, the difference coming from the local basis change altering how that noise looks relative to the checks, not from a different noise model (relations guide, §4.2). Entry-level only.

2.6表示阶梯：同一个码的几种语言The representation ladder: several languages for one code

• Aleks Kissinger, "Phase-free ZX diagrams are CSS codes (...or how to graphically grok the surface code)" (2022)
  kissinger2022phasefree · R5「ZX」视图的 curated 来源。ZX 演算把量子线路画成由两种节点（Z 蜘蛛和 X 蜘蛛）连成的图，再用几条画图规则化简；这篇说明：对极大 CSS 码（稳定子多到只固定一个量子态，不存逻辑比特），它的稳定子与一个不带相位的 ZX 图的标准形由同一个二元线性子空间 S 描述，二者给出同一个态；存有逻辑比特的一般 CSS 码，要再用论文的「弯线」扩展，把逻辑输入也画进图里。格点手术于是成了图的改写。Curated origin of the R5 “ZX” view. The ZX calculus draws quantum circuits as graphs of two node types (Z spiders and X spiders) and simplifies them by a few drawing rules; this paper shows that for a maximal CSS code (so many stabilizers that they fix a single quantum state and store no logical qubit), its stabilizers and the normal form of a phase-free ZX diagram are described by the same binary linear subspace S and give the same state; a general CSS code that stores logical qubits needs the paper's “bending wires” extension, which draws the logical inputs into the diagram too. Lattice surgery then becomes rewriting a diagram.
• Niel de Beaudrap and Dominic Horsman, "The ZX calculus is a language for surface code lattice surgery" (2020)
  debeaudrap2020zx · ZX 视图的第二个键：合并 / 分裂对应蜘蛛映射（格点手术教程第 5 章用两次合并做 CNOT）。Second key of the ZX view: merge/split as spider maps (the lattice-surgery guide, Chapter 5 builds a CNOT from two merges).
• Héctor Bombín, Daniel Litinski, Naomi Nickerson, Fernando Pastawski, and Sam Roberts, "Unifying flavors of fault tolerance with the ZX calculus" (2023)
  bombin2023unifying · ZX 视图的第三个键：把电路、簇态、Floquet（检查随时间周期性轮换的码）等容错方案统一成一种图语言。别名 bombin2024zx。Third key of the ZX view: circuit, cluster-state and Floquet (codes whose checks rotate periodically in time) fault tolerance unified in one diagrammatic language. Alias bombin2024zx.
• Zhengwei Liu, Alex Wozniakowski, and Arthur M. Jaffe, "Quon 3D language for quantum information" (2017)
  liu2017quon · R5「费米子 / quon」视图的键之一：quon 是一种把量子比特画成三维空间里带「电荷」的弦的图语言；这种图显示星报警 e 与面报警 m 合在一起的复合粒子是费米子（交换两个这样的粒子，态多一个负号）。该视图挂着一个缺口 wanted:fermionic-compiler。One key of the R5 “fermionic / quon” view: quons are a diagram language drawing qubits as charged strings in three-dimensional space; such diagrams show that the composite of a star alarm e and a plaquette alarm m is a fermion (exchanging two of them adds a minus sign to the state). The view carries the gap wanted:fermionic-compiler.
• Zhengwei Liu, "Quantized Graphs and Quantum Error Correction" (2019)
  liu2019quantized · 费米子视图的 curated 来源：星与面统一为圈算符，e/m 对偶是图的自对偶。Curated origin of the fermionic view: stars and plaquettes unified as cycle operators, e/m duality as graph self-duality.
• Norbert Schuch, J. Ignacio Cirac, and David Pérez-García, "PEPS as ground states: Degeneracy and topology" (2010)
  schuch2010peps · R5「张量网络」视图的 curated 来源：把检查当能量项时的最低能态可以写成 PEPS（每个格点放一个小张量、相邻张量沿连线收缩的二维张量网络），每个小张量在内部连线上带一个 Z₂ 对称；拓扑序，即逻辑信息藏在全局、局部看不见的性质，就体现为对局部张量的这个约束。Curated origin of the R5 “tensor-network” view: with the checks as energy terms, the lowest-energy states can be written as a PEPS (a two-dimensional tensor network with one small tensor per site, contracted along the links), each small tensor carrying a Z₂ symmetry on its internal links; topological order, the property that logical information lives globally and is invisible locally, shows up as this constraint on the local tensor.

2.7实验：硬件上做到了什么Experiments: what hardware has done

• Google Quantum AI, "Suppressing quantum errors by scaling a surface code logical qubit" (2023)
  google2023suppressing · 实现记录 0（demonstrated）：从 d=3 到 d=5，每周期逻辑错误率由 3.028% ± 0.023%（d=3 集合平均）降到 2.914% ± 0.016%。记录原文特别注明：它没有报告后来 Willow 的压制因子。Realization record 0 (demonstrated): from d=3 to d=5, logical error per cycle fell from 3.028% ± 0.023% (d=3 ensemble average) to 2.914% ± 0.016%. The record itself notes that it did not report the later Willow suppression factor.
• Google Quantum AI, "Quantum error correction below the surface code threshold" (2024)
  google2024below · 实现记录 1（demonstrated）：d=7 贴片每周期 0.143% ± 0.003%，码距每加 2 错误被压 Λ = 2.14 ± 0.02 倍（Λ 是器件 + 译码器的拟合量，不是 p_th/p，见译码器教程 6.4 节）。Realization record 1 (demonstrated): a d=7 patch at 0.143% ± 0.003% per cycle, each +2 in distance suppressing error by Λ = 2.14 ± 0.02 (Λ is a device-plus-decoder fit, not p_th/p; decoders guide, §6.4).
• A. Eickbusch et al., "Demonstrating dynamic surface codes" (2024)
  eickbusch2024dynamic · 实现记录 2（demonstrated）：六角连接、数据 / 测量角色行走、iSWAP 三种时序都表现出随码距的错误压制——码能适应连接度、泄漏与本征门约束。Realization record 2 (demonstrated): hexagonal-connectivity, walking data/measure-role and iSWAP schedules all showed distance-dependent suppression; the code adapts to connectivity, leakage and native-gate constraints.
• Y. Zhao et al., "Realization of an Error-Correcting Surface Code with Superconducting Qubits" (2022)
  zhao2022realization · 实现记录 3（demonstrated）：USTC 祖冲之 2.1 上 17 比特 d=3 的 Surface-17 重复纠错周期，Google 之外的独立超导实现。Realization record 3 (demonstrated): a 17-qubit d=3 Surface-17 running repeated correction cycles on USTC's Zuchongzhi 2.1, an independent superconducting realization.
• Dolev Bluvstein et al., "Logical quantum processor based on reconfigurable atom arrays" (2024)
  bluvstein2024logical · 实现记录 4（demonstrated）与 claimed 操作「CNOT · transversal」的键：可移动中性原子对齐两块贴片做横向纠缠门。记录原文强调：这是逻辑操作演示，不是低于阈值的存储器结论；d=5、d=7 的逻辑输入态制备本身不容错。别名 bluvstein2023logical。Realization record 4 (demonstrated) and key of the claimed operation “CNOT · transversal”: movable neutral atoms align two patches for transversal entangling gates. The record stresses this is a logical-operations demonstration, not a below-threshold memory claim; the d=5 and d=7 logical input preparation was not itself fault-tolerant. Alias bluvstein2023logical.

2.8通向更省的码Toward thriftier codes

• 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 · 双变量 bicycle 码：表面码「k=1 太贵」问题的最著名回答，BB 码教程整本讲它。仅条目级；别名 bravyi2024high。Bivariate bicycle codes: the best-known answer to the surface code's “k=1 is expensive” problem; the BB-code guide is devoted to it. Entry-level only; alias bravyi2024high.

3.1从 0.143% 走到 arXivFrom 0.143% to arXiv

摘录一：codes/surface/card.yaml，realizations 列表第 1 项（从 0 数）；文字逐字复制，只改了换行和缩进Excerpt 1: codes/surface/card.yaml, item 1 (counting from 0) of the realizations list; words copied verbatim, only line breaks and indentation changed

{
  "description": "A distance-7 patch reached 0.143% ± 0.003% logical error per cycle and each
                  distance increase of two suppressed error by Λ = 2.14 ± 0.02, with logical
                  lifetime beyond the best physical qubit.",
  "platform": "Google Willow superconducting processor",
  "refs": ["google2024below"],
  "status": "demonstrated"
}

摘录二：registry/refs.yaml 中键 google2024below 的条目，逐字复制Excerpt 2: the entry for the key google2024below in registry/refs.yaml, copied verbatim

- id: google2024below
  title: "Quantum error correction below the surface code threshold"
  authors: "Google Quantum AI"
  year: 2024
  url: "https://arxiv.org/abs/2408.13687"

3.2一个键如何钉住一条关系How a key pins a relation

3.3一个被防住的误引A misattribution that got caught

符号与读法Symbols & how to read them