目录Contents
用例与契合：这套码是为哪种硬件写的Use case & fit: which hardware the code was written for
一个纠错码落到任何芯片上都一样好用？不是。它对硬件递出一份「契约」：比特怎样排、谁能和谁说话、多久测一次。这一章写清表面码的契约，再看 QEC-Bank 卡片上的两个平台标签和两项任务从哪里来。（QEC-Bank 是本系列所属的量子纠错码数据库；卡片是它为每个码建的档案，本页 3.1 节时间轴和 3.2 节主表里的硬件数字都抄自表面码卡片的 realizations 字段，即「实验实现」一栏；正文和折叠框里另外补充的数字（循环时长、所用译码器、各变体的 Λ 等）取自原论文，旁边注明「论文」或「摘要」。） Does an error-correcting code work equally well on any chip? No. It hands the hardware a “contract”: how qubits are arranged, who may talk to whom, how often to measure. This chapter spells out the surface code's contract, then traces the two platform tags and two tasks on the QEC-Bank card. (The QEC-Bank is the database of quantum error-correcting codes this series belongs to; a card is its file on one code, and the hardware numbers in the §3.1 timeline and the §3.2 main table are copied from the realizations field, the “experimental realizations” section, of the surface-code card; the extra numbers added in the text and fold-outs (cycle time, which decoder was used, the Λ of each variant, and so on) come from the original papers and are marked “paper” or “abstract” where they appear.)
1.1契约的四条款The four clauses of the contract
直觉教程第 1 章剪开环面，正是因为真实芯片是平板、比特只能和邻居说话——契约第一条：二维、只连最近邻。第二条来自检查的形状：每颗星、每块面最多碰四条边，每条检查是重量 ≤ 4 的小操作。第三条来自4.2 节打开的黑盒：每条检查配一个帮手（辅助）比特。第四条来自4.3 节：测量本身会错，所以必须反复、快速地测量。QEC-Bank 直觉文稿把第四条说得很硬。设想一个偷懒的办法：把所有检查当成能量项写进一个哈密顿量（稳定子哈密顿量），码空间就是能量最低的状态，再把芯片冷却，指望它像磁铁记住南北极那样自己待在码空间里，不用测量。这种「自己纠错」的存储器叫自纠错存储器。Bravyi 与 Terhal 证明：检查只连近邻的二维稳定子码做不到这一点，热涨落造成的错误不会被能量自动压住。所以「持续的主动测量是存储器的一部分，不是可选的实现细节」。
Chapter 1 of the intuition guide cut the torus open precisely because a real chip is flat and its qubits talk only to neighbours — clause one: two-dimensional, nearest-neighbour only. Clause two follows from the checks' shape: each star or face touches at most four edges, so every check is a small weight ≤ 4 operation. Clause three comes from the box opened in §4.2: one helper (ancilla) qubit per check. Clause four comes from §4.3: measurements fail, so the machine must measure repeatedly and fast. The QEC-Bank intuition essay states clause four bluntly. Picture a lazy alternative: write every check into a Hamiltonian as an energy term (a stabilizer Hamiltonian), so that the code space is the lowest-energy state, cool the chip, and hope it stays in the code space by itself, the way a magnet remembers its north pole, with no measurements at all. A memory that corrects itself this way is called self-correcting. Bravyi and Terhal proved that two-dimensional stabilizer codes with nearest-neighbour checks cannot do this: errors from thermal fluctuations are not held back by the energy. Hence “continuous active measurement is part of the memory, not an optional implementation detail”.
四条里没有一条写着「必须是超导」或「必须是原子」。契约只谈几何与节拍；平台若在某条上违约（比如芯片上每个比特只和三个邻居有耦合，排成六边形），要么改造测量电路的排程，要么多做 SWAP 门把比特的状态挪到能相互作用的位置，付出额外的门和错误。第 3 章会看到 Willow 上恰好做了这种改造实验。
None of the clauses says “must be superconducting” or “must be atoms”. The contract is about geometry and tempo; when a platform breaks a clause (say, each qubit is coupled to only three neighbours in a hexagonal pattern) one either reschedules the measurement circuit or adds SWAP gates to move qubit states where they can interact, paying in extra gates and errors. Chapter 3 shows exactly such an experiment on Willow.
1.2两个平台标签，两项任务Two platform tags, two tasks
卡片的 fit 字段只登记两个平台和两项任务——不是「所有可能的平台」，而是有 demonstrated 记录支撑的那些。两个标签的差别：固定超导方格上比特不能动，两块码之间的门靠贴边缝合（《格点手术 · 高中数学版》第 5 章用两次合并做一个 CNOT）；可重构中性原子阵列里原子能被光镊（聚焦的激光束）搬动，把两块码对齐叠放、每一对对应比特同时做一个物理 CNOT，得到横向 CNOTtransversal CNOT：两块同样的码块逐对比特做物理 CNOT。它把第一块的逻辑 X̄ 复制到两块上（X̄⊗I → X̄⊗X̄），把第二块的逻辑 Z̄ 复制到两块上（I⊗Z̄ → Z̄⊗Z̄），这正是逻辑 CNOT 的作用；条件是对应比特能相互作用。（图 1.1）。所有物理 CNOT 同时进行，只占一层时间（深度一），而且每个物理比特只和另一块里的一个比特作用，一个物理错误不会在块内扩散。
The card's fit field registers two platforms and two tasks — not “every conceivable platform” but those backed by a demonstrated record. The difference between the tags: on a fixed superconducting grid qubits cannot move, so gates between patches go through seam merging (Chapter 5 of “Lattice Surgery · High-School Edition” builds a CNOT from two merges); in a reconfigurable neutral-atom array, optical tweezers (focused laser beams) can stack two patches in alignment and apply one physical CNOT to every pair of corresponding qubits at once, giving a transversal CNOTPairwise physical CNOTs between two identical blocks. It copies the first block's logical X̄ onto both blocks (X̄⊗I → X̄⊗X̄) and the second block's logical Z̄ onto both (I⊗Z̄ → Z̄⊗Z̄), which is exactly what a logical CNOT does; it requires corresponding qubits to be able to interact. (Figure 1.1). All the physical CNOTs run simultaneously and take one time layer (depth one), and each physical qubit interacts with a single qubit of the other block, so one physical error cannot spread inside a block.
同一份文稿提醒：「每一个固定版图上的两比特门都是手术」这句话是错的，缺陷编织同样能在固定硬件上工作。做法见 Fowler 等 2012 年综述第 XIV 节：在贴片里关掉几条检查，留下一个「洞」；一个逻辑比特用一对洞来存。洞分两种：关掉 Z 型检查（面）得到 Z 切口洞，关掉 X 型检查（星）得到 X 切口洞。把 Z 切口比特的一个洞沿闭合路线绕 X 切口比特的一个洞走一圈，两个比特的逻辑算符按 X̄₁ → X̄₁X̄₂、Z̄₂ → Z̄₁Z̄₂ 变化（图 1.2），这正是以 Z 切口比特为控制、X 切口比特为目标的逻辑 CNOT。两个同类的洞互相绕，得不到 CNOT：两个 Z 切口比特之间的 CNOT 要借一个 X 切口比特做中介，做三次编织再加两次逻辑测量（同一综述第 XIV 节 D 小节）。
The same essay warns that “every fixed-layout two-qubit gate is surgery” would be false: defect braiding also works on fixed hardware. The recipe is in Section XIV of the 2012 review by Fowler et al.: switch off a few checks in the patch to leave a “hole”, and store one logical qubit in a pair of holes. Holes come in two kinds: switching off Z-type checks (faces) gives a Z-cut hole, switching off X-type checks (stars) an X-cut hole. Move one hole of a Z-cut qubit along a closed route around one hole of an X-cut qubit, and the logical operators change as X̄₁ → X̄₁X̄₂ and Z̄₂ → Z̄₁Z̄₂ (Figure 1.2): exactly a logical CNOT with the Z-cut qubit as control and the X-cut qubit as target. Braiding two holes of the same kind gives no CNOT: a CNOT between two Z-cut qubits borrows an X-cut qubit as an intermediary, with three braids and two logical measurements (Section XIV D of the same review).
| 卡片标签Card tag | 含义Meaning | 支撑记录（第 3 章）Backing records (Chapter 3) |
|---|---|---|
| superconducting-2d-grid | 固定超导方格；帮手就地读出；两比特门靠缝合或缺陷。Fixed superconducting grid; helpers read out in place; two-qubit gates by surgery or defects. | zhao2022realization, google2023suppressing, google2024below, eickbusch2024dynamic |
| neutral-atom-reconfigurable | 可搬动的原子阵列；连接由搬运决定；对齐的码块可做横向门。Movable atom arrays; connectivity set by transport; aligned patches allow transversal gates. | bluvstein2024logical |
| memory | 存住一个逻辑比特：反复跑四拍循环，报告每轮逻辑错误。Keep one logical qubit alive: repeat the four-beat cycle, report logical error per cycle. | 四条超导记录the four superconducting records |
| computation | 在逻辑比特之间做门，报告门的表现。Perform gates between logical qubits, report gate performance. | bluvstein2024logical |
1.3「存储」任务长什么样What the “memory” task looks like
第 3 章的四条超导记录全是存储实验：准备一个逻辑状态，让四拍循环跑很多轮，读出，数逻辑比特被翻转的频率。下面的动画就是这台机器，只是真实芯片每一拍都可能出错，而且很快：Willow 芯片跑一轮要 1.1 微秒（google2024below 摘要）。动画画的是直觉教程的未旋转版图；实验芯片用的是更省比特的旋转版图（第 4 章），循环的节拍相同。先看熟节拍，第 2 章给它配数字。
All four superconducting records of Chapter 3 are memory experiments: prepare a logical state, run the four-beat cycle for many rounds, read out, count how often the logical bit was flipped. The animation below is that machine, except that on a real chip every beat can fail, and it is fast: the Willow chip takes 1.1 microseconds per round (google2024below abstract). The animation draws the intuition guide's unrotated patch; the experiments use the rotated patch, which needs fewer qubits (Chapter 4), with the same rhythm. Learn the rhythm first; Chapter 2 attaches numbers.
「契合」是双向的：码向硬件要几何与节拍，硬件向码要可改造的排程。卡片的平台标签只登记已用实验兑现契约的那一边。
“Fit” runs both ways: the code asks for geometry and tempo, the hardware asks for adaptable schedules. The card's platform tags record only the side where experiments have honoured the contract.
怎样诚实地读一篇实验How to read an experiment honestly
实验论文报告的不是「阈值」，而是三样东西：每轮逻辑错误率、它随码距怎么变、一个叫 Λ（压制因子）的倍数。三样各有确切定义，也各有一种流行的误读。 An experiment does not report “a threshold”. It reports three things: logical error per cycle, how it changes with distance, and a factor called Λ (the suppression factor). Each has a precise definition and a popular misreading.
2.1每轮逻辑错误率Logical error per cycle
存储实验的原始数据是一条衰减曲线：跑 N 轮后，读出的逻辑比特与准备的不一致的概率。把它拟合成「每轮独立地以概率 εL每轮逻辑错误率（logical error per cycle）：一轮纠错循环后逻辑比特被翻转的概率，由多轮衰减曲线拟合得到。 翻转一次」，εL 就是论文报告的每轮逻辑错误率。实验对很多个不同的 N 各重复成千上万次，得到一串失败频率，再找一个 εL 让模型曲线最贴近这串点；所以 εL 是由整条曲线拟合出来的数，总带一个 ± 的拟合误差。它还依赖一轮多长：同样的 εL，一轮 1 微秒和一轮 1 毫秒的机器每秒丢失的信息差一千倍，跨平台比较前先问循环时长。
A memory experiment's raw data is a decay curve: after N cycles, the probability that the logical bit read out disagrees with the one prepared. Fit it as “each cycle independently flips the logical bit with probability εLLogical error per cycle: the probability that the logical bit is flipped over one correction cycle, fitted from a multi-cycle decay curve.”, and εL is the logical error per cycle the paper reports. The experiment repeats the run thousands of times for each of many values of N, obtains a series of failure frequencies, and picks the εL whose model curve fits those points best; εL is therefore fitted from a whole curve and always carries a ± fitting error. It also depends on how long a cycle is: at the same εL, a machine with 1-microsecond cycles loses information a thousand times faster per second than one with 1-millisecond cycles, so ask for the cycle time before comparing platforms.
例：εL = 0.143%（Willow d = 7），N = 10 轮，P失败 ≈ 10 × 0.143% ≈ 1.4%。N·εL 变大以后这个近似会高估，因为同一个逻辑比特被翻两次等于没翻。字母表：N = 循环轮数；εL = 每轮逻辑错误率（上文定义）；P失败(N) = 跑完 N 轮后逻辑比特被翻转的概率。Example: εL = 0.143% (Willow, d = 7) and N = 10 cycles give Pfail ≈ 10 × 0.143% ≈ 1.4%. Once N·εL grows the approximation overestimates, because a logical bit flipped twice is not flipped at all. Legend: N = number of cycles; εL = logical error per cycle (defined above); Pfail(N) = probability that the logical bit is flipped after N cycles.
2.2码距缩放与 ΛDistance scaling and Λ
直觉教程 6.1 节的经验律 PL ≈ A·(p/pth)⌈d/2⌉（PL = 逻辑错误率；A = 常数；p = 每个物理比特的错误概率，见《从 1+1 到 Toric Code》2.2 节；pth = 阈值；d = 码距，见同一教程 5.8 节；⌈d/2⌉ = d/2 向上取整）说：码距 d 每加 2，指数 ⌈d/2⌉ 加 1，逻辑错误率就多乘一次 p/pth。例：若 p/pth = 1/2，d = 3 时 PL ≈ A·(1/2)²，d = 5 时 ≈ A·(1/2)³，两者之比是 2 = pth/p。实验测不到 pth，但能直接测那个因子：同一芯片上做 d=3、5、7，相邻两个 εL 相除，得到Λ压制因子（suppression factor）：Λ = ε_L(d)/ε_L(d+2)。Λ > 1 表示加大码距确实有利；它是从实测数据拟合出的倍数。。Λ > 1 就是「加大真的更好」在这台机器上成立。
The empirical law of §6.1 of the intuition guide, PL ≈ A·(p/pth)⌈d/2⌉ (PL = logical error rate; A = a constant; p = the error probability per physical qubit, see §2.2 of “From 1+1 to the Toric Code”; pth = the threshold; d = the code distance, see §5.8 of the same guide; ⌈d/2⌉ = d/2 rounded up), says that each +2 in d raises the exponent ⌈d/2⌉ by one, multiplying the logical error rate by one more factor p/pth. Example: if p/pth = 1/2, then PL ≈ A·(1/2)² at d = 3 and A·(1/2)³ at d = 5, a ratio of 2 = pth/p. An experiment cannot measure pth, but it can measure that factor: build d = 3, 5, 7 on one chip, divide neighbouring εL values, and you get ΛSuppression factor: Λ = ε_L(d)/ε_L(d+2). Λ > 1 means growing the distance really helps; it is a factor fitted from measured data.. Λ > 1 is “bigger really is better” holding on this machine.
字母表：Λ = 压制因子，码距每加 2 时每轮逻辑错误率缩小的倍数；εL(d) = 码距为 d 时的每轮逻辑错误率；d = 码距；p = 每比特物理错误率；pth = 阈值。Legend: Λ = the suppression factor, how many times the logical error per cycle shrinks when the distance grows by 2; εL(d) = logical error per cycle at distance d; d = code distance; p = physical error rate per qubit; pth = threshold.
这里最容易滑倒：括号里是 p/pth，于是想把 Λ 读成 pth/p。QEC-Bank 直觉文稿明确否决：「Λ 是一个设备加译码器的拟合，不是 pth/p 的直接测量。」真实芯片上各个比特的错误率不一样（非均匀噪声），有时一个事件同时弄坏好几个比特（相关错误），有时比特跑出 0/1 两个能级、落到更高的能级上（泄漏），译码器也不是最优的；这些全揉进了 Λ，而经验律一条也没考虑。《译码器 · 高中数学版》6.4 节整节讲这一点；同一教程的阈值账本把 Λ=2.14 与 10.9%、10.3%、2.9%、0.57% 各放一行，并注明 Λ 那一行不是阈值——不同实验，不能互相换算。
This is the slippery spot: the bracket holds p/pth, so one is tempted to read Λ as pth/p. The QEC-Bank intuition essay vetoes it: “Λ is a device-and-decoder fit, not a direct measurement of pth/p.” On a real chip the qubits have different error rates (heterogeneous noise), one event sometimes damages several qubits at once (correlated errors), a qubit sometimes escapes its two levels 0 and 1 into a higher level (leakage), and the decoder is not optimal; all of this is folded into Λ, and the empirical law accounts for none of it. §6.4 of “Decoders · High-School Edition” spends a section on this; the same guide's threshold ledger gives Λ = 2.14 its own row beside 10.9%, 10.3%, 2.9% and 0.57%, noting that the Λ row is not a threshold — different experiments, no conversion.
为了把「Λ 是拟合值」看具体，下面再跑一次直觉教程里的阈值实验。它的模型是完美测量、只有 Z 错误——码容量模型code-capacity model：只有数据比特出错、检查读数完美的噪声模型。阈值账本的前三行（10.9%、10.3%、≈9.9%）属于它。，不是任何真实芯片的模型。它的译码器是混合的：一轮里警报不超过 14 个时，做精确的最小权配对（把警报两两连起来或连到边界，使修复链的总边数最少）；超过 14 个时改用贪心配对（每次先连最近的一对），这时译码变差，所以测出的阈值偏低。取 p 很小处两条相邻曲线相除得到的「浏览器 Λ」，与 Willow 的 2.14 没有任何可比性；它只是「Λ 依赖整套设定」的一个能动手的例子。
To make “Λ is a fit” concrete, the bench below reruns the intuition guide's threshold experiment. Its model is perfect measurements, Z errors only — the code-capacity modelA noise model with data errors only and perfect check readings. The first three ledger rows (10.9%, 10.3%, ≈9.9%) belong to it., not the model of any real chip. Its decoder is a hybrid: when a round has at most 14 alarms it computes an exact minimum-weight matching (pair the alarms with each other or with the boundary so that the repair chains use the fewest edges in total); with more than 14 alarms it falls back to greedy pairing (always join the closest pair first), which decodes worse and pulls the measured threshold down. The “browser Λ” you get by dividing two neighbouring curves at small p is in no way comparable with Willow's 2.14; it is only a hands-on example of “Λ depends on the whole setup”.
2.3三句不同的话Three different sentences
新闻常把三句合成一句，实验里它们是三个独立判定。「越大越好」只要求相邻码距的 εL 下降，哪怕一点点：2023 年 Google 的记录从 d=3 的 3.028% 到 d=5 的 2.914%（d=3 的数是同一芯片上一组 d=3 贴片的平均，摘要原话「an ensemble of distance-3 logical qubits」），两数之比约 1.04（论文原话「相对错误降低约 4%」），所以卡片用词是「降低」。这个 1.04 不是 Willow 的 Λ = 2.14；论文另外用模拟算了一个 Λ3/5 = 1.10（见 3.3 节），卡片没有登记任何 Λ。「低于阈值」更强：多个码距（3→5→7）上稳定的、明显大于 1 的 Λ——2024 年 Willow 的内容。「逻辑寿命超过最好的物理比特」（英文叫 beyond break-even）是另一条坐标：编码比特与同芯片最好的单个裸比特比谁存得久，与码距缩放无关；Willow 的记录同时含有这句，摘要给出的倍数是 2.4 ± 0.3。
Headlines fuse three sentences into one; experiments keep three verdicts. “Bigger is better” asks only that εL fall between neighbouring distances, however slightly: Google's 2023 record went from 3.028% at d = 3 to 2.914% at d = 5 (the d = 3 figure is an average over a set of d = 3 patches on the same chip, in the abstract's words “an ensemble of distance-3 logical qubits”), a ratio of about 1.04 (the paper: “a relative error reduction of about 4%”), so the card says “reduced”. This 1.04 is not Willow's Λ = 2.14; the paper separately computes a simulated Λ3/5 = 1.10 (see §3.3), and the card registers no Λ at all. “Below threshold” is stronger: a stable Λ well above 1 across several distances (3→5→7) — the content of Willow 2024. “Logical lifetime beyond the best physical qubit” (“beyond break-even”) is a different axis: the encoded qubit against the chip's best single bare qubit, which one keeps its state longer, independent of distance scaling; Willow's record contains this sentence too, with a factor of 2.4 ± 0.3 in the abstract.
| 这句话Sentence | 需要的证据Evidence required | 哪条记录说了它Record that says it |
|---|---|---|
| 越大越好bigger is better | εL(d+2) < εL(d) | google2023suppressing (d=3→5) |
| 低于阈值below threshold | 多个码距上稳定的 Λ > 1stable Λ > 1 over several distances | google2024below (Λ = 2.14 ± 0.02) |
| 超过最好的物理比特beyond the best physical qubit | 编码寿命 > 最好单比特寿命encoded lifetime > best single-qubit lifetime | google2024below |
字母表：εL(d) = 码距 d 下的每轮逻辑错误率；d = 码距；Λ = 压制因子。Legend: εL(d) = logical error per cycle at distance d; d = code distance; Λ = suppression factor.
2.4存储实验与计算实验Memory versus computation experiments
存储实验的被测对象是一轮循环，报告 εL；计算实验的被测对象是一个逻辑门，报告门后逻辑状态的质量。计算实验可以只让门这一步容错，而用非容错方式准备输入。卡片上唯一的计算类记录（bluvstein2024logical）正是如此：横向门容错，但 d=5、7（d = 码距）表面码块的逻辑输入态准备不容错。所以卡片写道：这是逻辑操作演示，不是低于阈值的表面码存储声明。
A memory experiment measures a cycle and reports εL; a computation experiment measures a logical gate and reports the quality of the logical state after it. A computation experiment may make only the gate fault-tolerant while preparing inputs non-fault-tolerantly. The card's only computation record (bluvstein2024logical) is exactly that: the transversal gate was fault-tolerant, the preparation of the d = 5 and d = 7 (d = code distance) surface-code inputs was not. Hence the card's wording: a logical-operations demonstration, not a below-threshold surface-memory claim.
① 每轮 εL 是多少，一轮多长？② 几个码距，Λ 多少、误差多大？③ 说的是「越大越好」「低于阈值」还是「超过物理比特」？④ 存储还是计算，输入准备容错吗？带着四问翻到下一章的表格。
① What is εL per cycle, and how long is a cycle? ② How many distances, what Λ, what error bar? ③ “Bigger is better”, “below threshold” or “beyond the physical qubit”? ④ Memory or computation, and was the input prepared fault-tolerantly? Take these into the next chapter's table.
时间线：卡片上的五条 demonstrated 记录Timeline: the five demonstrated records on the card
卡片的 realizations 字段有五条记录，每条带 demonstrated 标签和一个文献键。先排上 2022→2024 的时间轴，再用一张表抄下精确数字与误差，最后逐条展开。 The card's realizations field holds five records, each labelled demonstrated with a bibkey. First a 2022→2024 axis, then one table with the exact numbers and error bars, then each record unfolded.
3.1时间轴The axis
3.2精确数字表The table of exact numbers
逐字抄自卡片的 realizations 字段：数字、± 误差、平台措辞、状态标签均不改动。「±」读「正负」，是拟合误差的一个标准差范围。「—」与「卡片未列」表示卡片没有给出该项，不是零，也不等于论文没有。每个数都依赖所用的译码器（从警报推断错误的经典算法）：同一批数据换一个译码器，数字会变，3.3 节逐条写明。
Copied verbatim from the card's realizations field: numbers, ± error bars, platform wording and status label unchanged. “±” is read “plus or minus”, a one-standard-deviation fitting range. “—” and “not on card” mean the card does not give that item: not zero, and not necessarily absent from the paper. Every number depends on the decoder used (the classical algorithm that infers errors from the alarms): the same data run through another decoder give different numbers, and §3.3 names the decoder for each record.
| 年份Year | 平台（卡片措辞）Platform (card wording) | d | εL 每轮per cycle | Λ | 任务Task | 状态Status | 文献键Bibkey |
|---|---|---|---|---|---|---|---|
| 2022 | USTC 祖冲之 2.1 超导处理器USTC Zuchongzhi 2.1 superconducting processor | 3 (Surface-17) | — | — | memory | demonstrated | zhao2022realization |
| 2023 | Google 超导方格处理器Google superconducting square-grid processors | 3 → 5 | 3.028% ± 0.023% → 2.914% ± 0.016% | 卡片未列not on card | memory | demonstrated | google2023suppressing |
| 2024 | Google Willow 超导处理器Google Willow superconducting processor | 7 (3, 5, 7) | 0.143% ± 0.003% | 2.14 ± 0.02 | memory | demonstrated | google2024below |
| 2024 | Google Willow 动态超导版图Google Willow dynamic superconducting layouts | 随 d 压制suppression with d | — | — | memory | demonstrated | eickbusch2024dynamic |
| 2024 | 可重构中性原子阵列reconfigurable neutral-atom arrays | 5, 7 | — | — | computation | demonstrated | bluvstein2024logical |
字母表（本表与下方折叠框通用）：d = 码距；εL = 每轮逻辑错误率；Λ = 压制因子，码距每加 2 时 εL 缩小的倍数；± 后的数 = 拟合误差。Legend (for this table and the fold-outs below): d = code distance; εL = logical error per cycle; Λ = suppression factor, how many times εL shrinks per +2 in distance; the number after ± = fitting error.
3.3逐条展开One record at a time
2022 · 祖冲之 2.1 · Surface-172022 · Zuchongzhi 2.1 · Surface-17
Y. Zhao 等，“Realization of an Error-Correcting Surface Code with Superconducting Qubits”（2022），zhao2022realization。中国科学技术大学（USTC）团队在他们的超导处理器「祖冲之 2.1」上，用 17 个比特（9 数据 + 8 帮手，正是第 4 章旋转版 d=3 的比特账）组成 d=3 贴片，连续执行多轮纠错循环。卡片定位：Google 之外第一个独立的平面表面码存储超导实现。卡片没有登记每轮错误率，本页不补。
Y. Zhao et al., “Realization of an Error-Correcting Surface Code with Superconducting Qubits” (2022), zhao2022realization. A team at the University of Science and Technology of China (USTC) used seventeen qubits of their superconducting processor Zuchongzhi 2.1 (9 data + 8 helpers, exactly the rotated d = 3 count of Chapter 4) to form a d = 3 patch running several consecutive correction cycles. The card's framing: the first independent superconducting realization of a planar surface-code memory outside Google. No per-cycle error rate is registered, and this page adds none.
2023 与 2024 · Google · d=3→5，然后 Willow d=72023 and 2024 · Google · d=3→5, then Willow d=7
2023：Google Quantum AI，“Suppressing quantum errors by scaling a surface code logical qubit”，google2023suppressing——d=3 平均 3.028% ± 0.023%，d=5 为 2.914% ± 0.016%；卡片加注「没有报告后来 Willow 的压制因子」。论文补充（不在卡片里）：这两个数用张量网络译码器得到，它在所有与警报相符的错误组合里比较「翻转逻辑比特」与「不翻转」两类的总概率，挑大的一类，慢但接近最优；换用较快的信念匹配译码器，同一数据给出 3.118% ± 0.025% → 3.056% ± 0.015%（降约 2%，而不是 4%）。论文也定义了压制因子 Λ3/5 = ε3/ε5，用模拟估算每种物理错误对 1/Λ3/5 贡献多少，模拟值 Λ3/5 = 1.10（补充材料第 XVII 节）。所以准确的说法是：2023 年的论文不是 Willow 的 Λ = 2.14，卡片也没有登记它的 Λ。2024：“Quantum error correction below the surface code threshold”，google2024below——d=7 每轮 0.143% ± 0.003%，Λ = 2.14 ± 0.02，逻辑寿命超过芯片上最好的物理比特。论文摘要把范围说得更细：它做了两个存储器，一个 d=7（上面的 0.143% 和 Λ 属于它，共 101 个比特，寿命是最好物理比特的 2.4 ± 0.3 倍），另一个 d=5 接上了实时译码器（译码跟得上每 1.1 微秒一轮的节拍，平均延迟 63 微秒），同样低于阈值。译码器（论文正文）：0.143% 和 Λ = 2.14 是用离线神经网络译码器得到的，即实验结束后在经典电脑上慢慢译码（约每轮 24 微秒，跟不上 1.1 微秒的节拍），并对 X 基、Z 基两种初始化取平均。同一批数据换用集成匹配译码器，得到 Λ = 2.04 ± 0.02、d=7 每轮 0.171% ± 0.003%。实时译码那个 d=5 存储器上，实时译码器给出每轮 0.35% ± 0.01%、Λ = 2.0 ± 0.1；同样的数据离线交给神经网络，得到 0.269% ± 0.008%、Λ = 2.18 ± 0.09。换一个译码器，数字就变。QEC-Bank 阈值账本把后者单列为「硬件契合」行，写作「Willow 旋转版存储器 + 设备实测的非均匀噪声 + 实时译码器」；按摘要，实时译码对应的是 d=5 那个存储器，d=7 的数字不应读作实时译码的结果。文献笔记补充（不在卡片里）：d=7 的 101 个比特 = 49 数据 + 48 测量 + 4 个用于清除泄漏的比特。
2023: Google Quantum AI, “Suppressing quantum errors by scaling a surface code logical qubit”, google2023suppressing — d = 3 average 3.028% ± 0.023%, d = 5 at 2.914% ± 0.016%; the card notes it “did not report the later Willow suppression factor”. From the paper (not on the card): these two numbers come from a tensor-network decoder, which compares the total probability of the two classes of error patterns consistent with the alarms, “flips the logical qubit” and “does not”, and picks the larger; it is slow but close to optimal. With the faster belief-matching decoder the same data give 3.118% ± 0.025% → 3.056% ± 0.015% (a drop of about 2%, not 4%). The paper also defines the suppression factor Λ3/5 = ε3/ε5 and uses simulations to budget how much each physical error type contributes to 1/Λ3/5, with a simulated value Λ3/5 = 1.10 (Supplement §XVII). So the accurate statement is: the 2023 paper is not Willow's Λ = 2.14, and the card registers no Λ for it. 2024: “Quantum error correction below the surface code threshold”, google2024below — d = 7 at 0.143% ± 0.003% per cycle, Λ = 2.14 ± 0.02, logical lifetime beyond the chip's best physical qubit. The paper's abstract draws the scope more finely: it ran two memories, one at d = 7 (the 0.143% and Λ above belong to it; 101 qubits; lifetime 2.4 ± 0.3 times the best physical qubit), and one at d = 5 connected to a real-time decoder (decoding keeps pace with the 1.1-microsecond cycle, average latency 63 microseconds), also below threshold. Decoders (paper, main text): the 0.143% and Λ = 2.14 come from an offline neural-network decoder, run on a classical computer after the experiment (about 24 microseconds per cycle, too slow for the 1.1-microsecond cycle), and are averaged over X-basis and Z-basis initializations. The same data through an ensembled matching decoder give Λ = 2.04 ± 0.02 and 0.171% ± 0.003% per cycle at d = 7. On the real-time d = 5 memory, the real-time decoder gives 0.35% ± 0.01% per cycle and Λ = 2.0 ± 0.1; the same data decoded offline by the neural network give 0.269% ± 0.008% and Λ = 2.18 ± 0.09. Change the decoder and the numbers change. The QEC-Bank threshold ledger files the latter paper as its “hardware fit” row, written “Willow rotated memories + the device's measured heterogeneous noise + real-time decoder”; per the abstract, the real-time decoding belongs to the d = 5 memory, and the d = 7 numbers should not be read as real-time-decoded results. From the literature notes (not on the card): the 101 qubits at d = 7 are 49 data + 48 measure + 4 qubits used to remove leakage.
2024 · Willow · 动态版图：六边形、行走、iSWAP2024 · Willow · dynamic layouts: hexagonal, walking, iSWAP
A. Eickbusch 等，“Demonstrating dynamic surface codes”（2024），eickbusch2024dynamic。三种改造排程各解一条契约违约：六边形连接把每比特需要的耦合器从 4 个降到 3 个；行走版每轮交换数据比特与测量比特的角色，贴片像在芯片上「走动」，每个比特每两轮被测量并复位一次（图 3.2），积累的泄漏因此被清掉；iSWAP 版改用 iSWAP 门代替 CNOT，论文说不增加额外开销。iSWAP 是芯片原生的两比特门：|00⟩ 和 |11⟩ 不变，|01⟩ → i|10⟩，|10⟩ → i|01⟩。这里 i 是虚数单位（i² = −1），乘上 i 等于在复平面上转 90°。只有 |01⟩、|10⟩ 两个分量乘上 i，|00⟩、|11⟩ 不乘，所以这是相对相位：它会改变叠加态后续的干涉结果。若四个分量都乘上同一个数，那是整体相位，任何测量都看不出来。卡片只登记定性结论：三者都随码距压制错误，证明码「可以适应连接、泄漏与原生门的约束」。论文摘要给出从 d=3 到 d=5 的压制因子 Λ：六边形 2.15(2)、行走 1.69(6)、iSWAP 1.56(2)（括号里是末位数字的误差，2.15(2) 即 2.15 ± 0.02）；这些数不在卡片里。它们都用 Harmony 译码器（把多个匹配译码器的结果合成一个判决）加上一张用强化学习（一种按反馈反复试错调参的机器学习方法）训练出的噪声先验（译码器假定的各类错误概率表）得到；每个 ε 都是 X 基与 Z 基两种实验的平均，d=3 的数还是同一芯片上几块 d=3 贴片的平均。论文附录 F 强调 Λ「强烈依赖所用译码器」：例如六边形版改用稀疏 blossom 匹配译码器加通用噪声先验，得到 2.09(2)。
A. Eickbusch et al., “Demonstrating dynamic surface codes” (2024), eickbusch2024dynamic. Three re-engineered schedules, each fixing one contract breach: hexagonal connectivity cuts the couplers each qubit needs from 4 to 3; the walking code swaps the roles of data and measure qubits every round, so the patch “walks” across the chip and every qubit is measured and reset once every two rounds (Figure 3.2), which removes accumulated leakage; the iSWAP version uses iSWAP gates instead of CNOTs, according to the paper without extra overhead. iSWAP is a native two-qubit gate of the chip: |00⟩ and |11⟩ are unchanged, |01⟩ → i|10⟩ and |10⟩ → i|01⟩. Here i is the imaginary unit (i² = −1), and multiplying by i is a 90° turn in the complex plane. Only the |01⟩ and |10⟩ components pick up the i, while |00⟩ and |11⟩ do not, so this is a relative phase: it changes how a superposition interferes later. Multiplying all four components by the same number would be a global phase, which no measurement can see. The card records only the qualitative conclusion: all three suppress error with distance, showing the code “can be adapted to connectivity, leakage, and native-gate constraints”. The paper's abstract gives the suppression factor Λ from d = 3 to d = 5: hexagonal 2.15(2), walking 1.69(6), iSWAP 1.56(2) (the bracket is the error in the last digit, 2.15(2) meaning 2.15 ± 0.02); these values are not on the card. All three use the Harmony decoder (which merges the verdicts of several matching decoders into one) with a noise prior (the table of error probabilities the decoder assumes) trained by reinforcement learning (a machine-learning method that tunes parameters by repeated trial and feedback); each ε is the average of an X-basis and a Z-basis experiment, and the d = 3 values are further averaged over several d = 3 patches on the same chip. Appendix F of the paper stresses that Λ “is strongly dependant on the decoder used”: the hexagonal code decoded with a sparse-blossom matching decoder and a generic noise prior gives 2.09(2), for example.
为什么清掉泄漏的是复位，不是「换个名字」：泄漏的比特落在 |2⟩ 这样的高能级上。X、Z 这类 Pauli 纠正只在 |0⟩ 与 |1⟩ 之间翻转或加符号，作用在 |2⟩ 上不会把它带回 0/1，所以译码器算出的任何 Pauli 修复都救不回它。复位是另一种操作：论文用的多能级复位不管比特处在 |0⟩、|1⟩ 还是 |2⟩，都把它拉回 |0⟩。标准排程只复位测量比特，数据比特上的泄漏会一直留着（论文的对照实验里，不加泄漏清除的标准排程出现了持续近 50 轮的警报相关尾巴，要另加一步叫 DQLR 的数据比特泄漏清除操作才能消掉）；行走排程让每个比特每两轮轮到一次测量加复位，泄漏的比特最多等两轮就会被复位。论文实测：每次复位后，那一组比特的泄漏比例下降约一个数量级，行走版里没有再加任何专门的泄漏清除步骤。
Why it is the reset that removes leakage, not the relabelling: a leaked qubit sits in a higher level such as |2⟩. Pauli corrections such as X and Z only flip or sign-change between |0⟩ and |1⟩; applied to |2⟩ they do not bring it back into 0/1, so no Pauli repair the decoder computes can rescue it. A reset is a different operation: the multi-level reset used in the paper returns the qubit to |0⟩ whether it was in |0⟩, |1⟩ or |2⟩. The standard schedule resets only the measure qubits, so leakage on data qubits stays (in the paper's comparison, the standard schedule without leakage removal showed a tail of correlated alarms lasting nearly 50 rounds, which a separate data-qubit leakage-removal step, DQLR, was needed to remove); the walking schedule gives every qubit a measurement and reset every two rounds, so a leaked qubit waits at most two rounds for a reset. Measured in the paper: after each reset, the leaked fraction of that group drops by about an order of magnitude, and the walking code used no dedicated leakage-removal step at all.
2024 · 可重构中性原子 · 横向门，带明确警示2024 · Reconfigurable neutral atoms · transversal gates, with a caveat
Dolev Bluvstein 等，“Logical quantum processor based on reconfigurable atom arrays”（2024），bluvstein2024logical。一台由可搬动原子组成、同时运行几种不同纠错码的逻辑处理器（最多 280 个物理比特）里，实现了表面码贴片，并通过对齐对应数据比特做贴片间横向纠缠门。摘要说，两比特逻辑门的表现随表面码码距从 d=3 增加到 d=7 而改善。卡片警示原文：「这是一次逻辑操作演示，不是低于阈值的表面码存储声明；对 d=5、7 的表面码块，逻辑输入态的准备本身不是容错的。」直觉文稿措辞一致：「这一结果不得被读作端到端的低于阈值存储演示。」
Dolev Bluvstein et al., “Logical quantum processor based on reconfigurable atom arrays” (2024), bluvstein2024logical. A logical processor made of movable atoms and running several different codes (up to 280 physical qubits) realized surface-code patches and performed transversal inter-patch entangling gates by aligning corresponding data qubits. The abstract reports that a two-qubit logical gate improved as the surface-code distance was raised from d = 3 to d = 7. The card's caveat, verbatim: “This is a logical-operations demonstration, not a below-threshold surface-memory claim; for the distance-five and distance-seven surface blocks, logical input-state preparation was not itself fault-tolerant.” The intuition essay agrees: “this result must not be read as an end-to-end below-threshold memory demonstration.”
demonstrated 的意思是「有一篇被引用的实验论文报告了此事」。QEC-Bank 机器不重算任何硬件数字；它重算的是构造器、参数与证书；哪些数字由机器重算、哪些只引自文献，推导页分开列出。文献笔记里还有未进卡片的早期实验（Delft d=2 surface-7，marques2021logical；ETH surface-17 约 3% 每轮，krinner2022realizing），本页不把它们算作条目数据。
demonstrated means “a cited experimental paper reported this”. The QEC-Bank machine does not recompute hardware numbers; it recomputes the constructor, parameters and certificates; the derivations page lists separately which numbers a machine recomputes and which are only cited. The literature notes hold earlier experiments not on the card (Delft d = 2 surface-7, marques2021logical; ETH surface-17 at about 3% per cycle, krinner2022realizing); this page does not count them as entry data.
常见误读，每条钉在一个出处上Common misreadings, each pinned to a source
六条误读都不是稻草人——每一条都在 QEC-Bank 的卡片、直觉文稿或文献笔记里被点名纠正过。格式统一：误读 → 正确读法 → 出处。 None of the six is a straw man — each has been named and corrected in the QEC-Bank card, the intuition essay or the literature notes. Same format throughout: misreading → correct reading → source.
| 误读Misreading | 正确读法Correct reading | 出处Source |
|---|---|---|
| 「2023 年 Google 测得 Λ = 2.14」“Google measured Λ = 2.14 in 2023” | 2023 年报告的是 3.028% ± 0.023% 与 2.914% ± 0.016%；Λ = 2.14 ± 0.02 属于 2024 年的 Willow 论文（arXiv 2408.13687）。2023 reported 3.028% ± 0.023% and 2.914% ± 0.016%; Λ = 2.14 ± 0.02 belongs to the 2024 Willow paper (arXiv 2408.13687). | 卡片记录 google2023suppressing；文献笔记 Nugget 5「common conflation」Card record google2023suppressing; literature notes Nugget 5, “common conflation” |
| 「Λ = 2.14 即 pth/p = 2.14」“Λ = 2.14 means pth/p = 2.14” | Λ 是设备加译码器的拟合倍数，不是任何阈值行除以芯片的 p。Λ is a fitted device-plus-decoder factor, not any threshold row divided by the chip's p. | 直觉文稿「Laboratory memory」段；《译码器 · 高中数学版》6.4 节Intuition essay, “Laboratory memory”; §6.4 of “Decoders · High-School Edition” |
| 「中性原子也做到了低于阈值的表面码存储」“Neutral atoms also achieved below-threshold surface-code memory” | 逻辑操作演示：横向门容错，d=5、7 的输入准备不容错；支撑 computation 标签，不是 memory。A logical-operations demonstration: the gate was fault-tolerant, the d = 5, 7 input preparation was not; it backs computation, not memory. | 卡片记录 bluvstein2024logical；直觉文稿「Operations story」段Card record bluvstein2024logical; intuition essay, “Operations story” |
| 「Willow 的 d=7 就是 QEC-Bank 参数页上的 d=7」“Willow's d=7 is the d=7 on the QEC-Bank parameters page” | QEC-Bank 构造器只生成直觉教程第 2 章那种未旋转版图，边长 L 的一块用 2L²−2L+1 个物理比特存 1 个逻辑比特、码距 L；实验室部署的是旋转贴片（参数页 4.2 节有图）。同一个 d，两本比特账（下式）；两者都不含路由、备用与工厂比特。The QEC-Bank constructor builds only the unrotated layout of Chapter 2 of the intuition guide: a patch of side L stores 1 logical qubit in 2L²−2L+1 physical qubits with distance L; labs deploy the rotated patch (pictured in §4.2 of the parameters page). Same d, two qubit ledgers (below); both exclude routing, spare and factory qubits. | 直觉文稿「Failure modes · Area overhead」；阈值账本「Hardware fit」行Intuition essay, “Failure modes · Area overhead”; threshold ledger, “Hardware fit” row |
| 「固定版图上的两比特门只能靠格点手术」“On a fixed layout the only two-qubit gate is surgery” | 手术是惯用原语，但缺陷编织也能在固定硬件上工作，堆叠或可搬动版图能暴露横向 CNOT。Surgery is the usual primitive, but defect braiding also works on fixed hardware, and stacked or movable layouts expose the transversal CNOT. | 直觉文稿「Operations consume spacetime」段；《格点手术 · 高中数学版》1.2 节Intuition essay, “Operations consume spacetime”; §1.2 of “Lattice Surgery · High-School Edition” |
| 「表面码阈值是 10.3%，Willow 的 p 远低于它」“The surface-code threshold is 10.3% and Willow's p is far below it” | 10.9%（最优，码容量）、10.3%（匹配，码容量）、2.9%（现象学）、0.57% 每步（Fowler 电路级）与 Λ = 2.14（Willow 硬件拟合）是五个不同的实验；门错误率只能与电路级那一行对照。10.9% (optimal, code capacity), 10.3% (matching, code capacity), 2.9% (phenomenological), 0.57% per step (Fowler circuit level) and Λ = 2.14 (Willow hardware fit) are five different experiments; a gate error rate compares only with the circuit-level row. | 《译码器 · 高中数学版》6.1 节与6.2 节账本；QEC-Bank threshold-ledger§6.1 and the §6.2 ledger of “Decoders · High-School Edition”; QEC-Bank threshold-ledger |
字母表（本式与上表通用）：d = 码距；n = 数据比特数；「含帮手」= 数据比特加每条检查一个的帮手比特；Λ = 压制因子；p = 每比特物理错误率；pth = 阈值。公式 2d²−2d+1、4d²−4d+1、2d²−1 均取自 QEC-Bank 直觉文稿。Legend (for this formula and the table above): d = code distance; n = number of data qubits; “with helpers” = data qubits plus one helper per check; Λ = suppression factor; p = physical error rate per qubit; pth = threshold. The formulas 2d²−2d+1, 4d²−4d+1 and 2d²−1 are taken from the QEC-Bank intuition essay.
最后一行的 97 正好对上 Willow 的「49 数据 + 48 测量」。这不是巧合，而是同一个 d 下旋转贴片的比特账；QEC-Bank 的参数与证书描述的是左边那本账。
The 97 on the right matches Willow's “49 data + 48 measure” exactly. Not a coincidence: it is the rotated patch's ledger at the same d, while the QEC-Bank parameters and certificates describe the ledger on the left.
对。3.028/2.914 ≈ 1.04，只能说「降低了」。Λ = 2.14 与「超过物理比特」都属于 2024 年的 google2024below。这就是第 2.3 节三句话必须分开的原因。
Right. 3.028/2.914 ≈ 1.04 supports only “reduced”. Both Λ = 2.14 and “beyond the physical qubit” belong to google2024below (2024). This is why §2.3 keeps the three sentences apart.
回到卡片原文：「This is the 2023 distance-scaling result; it did not report the later Willow suppression factor.」再看第 2.3 节的表。
Back to the card's words: “This is the 2023 distance-scaling result; it did not report the later Willow suppression factor.” Then re-read the table in §2.3.
尚未演示的：只列出处说过的Not yet demonstrated: only what the sources state
诚实的「实现」页面也要说清什么没有做到。这一章不做预测，只抄录卡片、直觉文稿与文献笔记明确写下的空白，并按可信级别分三层。 An honest “realizations” page must also say what has not been done. No forecasts here: only the gaps written down explicitly in the card, the intuition essay and the literature notes, in three tiers of trust.
5.1卡片本身划出的边界Boundaries drawn by the card itself
五条记录合起来只支撑一句强结论：固定超导方格上的表面码存储器可以低于阈值（一台设备，一个团队）。卡片没有登记：容错准备输入、低于阈值、又做逻辑门的端到端计算——中性原子记录明确把自己排除在外；也没有任何非超导平台上的低于阈值存储。fit 字段只有两个平台标签，正是这一空白的直接后果。
Together the five records support one strong conclusion: a surface-code memory on a fixed superconducting grid can run below threshold (one device, one team). The card registers no end-to-end computation that prepares inputs fault-tolerantly, runs below threshold and performs logical gates — the neutral-atom record explicitly excludes itself; and no below-threshold memory on any non-superconducting platform. The two platform tags in fit are the direct consequence.
5.2直觉文稿指出的结构性限制Structural limits named by the intuition essay
这一节的三条性质不同：前两条是带前提的定理，第三条是常用协议的代价，不是定理。
The three items here differ in kind: the first two are theorems with stated assumptions, the third is a cost of the usual protocols, not a theorem.
① 没有被动的热存储。Bravyi–Terhal 证明：检查只连近邻的二维稳定子哈密顿量不能自纠错（1.1 节）。
① No passive thermal memory. Bravyi–Terhal prove that a two-dimensional stabilizer Hamiltonian with nearest-neighbour checks cannot be self-correcting (§1.1).
② 没有「浅电路」做的 T 门。先说门（下方补课 5a 从零讲 H、S、T）：把任意 Pauli 算符「夹」一遍后仍得到 Pauli 算符的门叫 Clifford 门，H、S、CNOT 及它们的组合都是；它们全体（Clifford 群）不够做通用计算，还缺一个 T 门这样的非 Clifford 门。Bravyi 与 König 的定理有三个前提：码是几何局域的稳定子码（每条检查只碰彼此相邻的比特），码距足够大，逻辑门由深度固定（不随码距增长）、每个门只作用于相邻比特的电路实现。这样的电路里一个错误只能扩散到固定大小的邻域，这正是它吸引人的地方。结论：在二维里，这样实现的逻辑门都在 Clifford 群里（见《格点手术 · 高中数学版》1.3 节）。所以在这些前提下，二维表面码上没有这样实现的逻辑 T 门，通用计算要另找资源：用魔术态（补课 5a 的 ①→④），或者临时换到另一种码上做这一步（码切换）；卡片上没有记录演示这一步。
② No T gate from a shallow circuit. First the gates (catch-up 5a below teaches H, S and T from zero): a gate that turns every Pauli operator, when sandwiched by it, into a Pauli operator again is a Clifford gate; H, S, CNOT and all their combinations are Clifford. All of them together (the Clifford group) are not enough for universal computation, which also needs a non-Clifford gate such as T. The Bravyi–König theorem has three assumptions: the code is a geometrically local stabilizer code (each check touches only qubits near each other), its distance is large enough, and the logical gate is implemented by a circuit of fixed depth (not growing with the distance) whose gates act only on nearby qubits. In such a circuit an error can spread only over a neighbourhood of fixed size, which is what makes it attractive. The conclusion: in two dimensions every logical gate implemented this way lies in the Clifford group (see §1.3 of “Lattice Surgery · High-School Edition”). So under these assumptions the 2D surface code has no logical T gate implemented this way, and universal computation needs another resource: magic states (steps ① to ④ of catch-up 5a), or switching temporarily to a different code for that step (code switching); no card record demonstrates that step.
③ 操作消耗时空（协议的代价，不是定理）。格点手术与码变形类的协议（合并、分裂、辫织）每一步要维持 d 轮测量（d = 码距；见同一教程第 6 章）；这是这些协议的设计决定的，换一种协议代价就不同。卡片上另有两类不付这笔钱的逻辑操作。一类是可移动原子上的横向 CNOT：只需一层物理 CNOT，剩下的代价是搬运原子，以及译码时要把两块的症状放在一起解（关联译码）。另一类是 Pauli 框架更新：需要施加的 Pauli 纠正不真去做，只在经典电脑里记下来，之后解读测量结果时把它算进去，不需要任何物理操作。用的哪种协议、多少轮，代价必须单独报告。
③ Operations consume spacetime (a protocol cost, not a theorem). Lattice-surgery and code-deformation protocols (merge, split, braiding) sustain d rounds of measurement per step (d = code distance; see Chapter 6 of the same guide); this follows from how those protocols are designed, and another protocol has another cost. The card also lists two kinds of logical operation that do not pay this price. One is the transversal CNOT on movable atoms: a single layer of physical CNOTs, the remaining costs being atom transport and decoding the syndromes of both blocks together (correlated decoding). The other is the Pauli-frame update: a Pauli correction that should be applied is not performed but recorded in the classical computer and taken into account when later measurement results are interpreted, with no physical operation at all. Which protocol and how many rounds is a cost to report separately.
补课 · H、S、T 门，Clifford 门，以及魔术态怎样换来一个 T 门Catch-up · The H, S and T gates, Clifford gates, and how a magic state buys a T gate
单比特门由它对 |0⟩、|1⟩ 做什么完全决定（叠加态按线性照搬）。下表里 i 是虚数单位（i² = −1），eiπ/4 = (1 + i)/√2 是复平面上转 45° 的单位数。
A one-qubit gate is fixed by what it does to |0⟩ and |1⟩ (superpositions follow linearly). In the table, i is the imaginary unit (i² = −1) and eiπ/4 = (1 + i)/√2 is the unit number at 45° in the complex plane.
| 门Gate | |0⟩ → | |1⟩ → | 白话In words |
|---|---|---|---|
| H | (|0⟩ + |1⟩)/√2 | (|0⟩ − |1⟩)/√2 | 把「0/1」换成「+/−」，互换 X 与 Z 的角色swaps “0/1” for “+/−”, exchanging the roles of X and Z |
| S | |0⟩ | i|1⟩ | 给 |1⟩ 分量转 90°turns the |1⟩ component by 90° |
| T | |0⟩ | eiπ/4|1⟩ | 给 |1⟩ 分量转 45°；做两次就是 S（T² = S）turns the |1⟩ component by 45°; twice gives S (T² = S) |
Clifford 门是什么。纠错时我们追踪的错误是 Pauli 算符（X、Y、Z 及其乘积）。门 U 若把每个 Pauli 算符 P「夹」成另一个 Pauli 算符，即 U P U† 仍是（带 ± 号的）Pauli 算符，就叫 Clifford 门（U† 是 U 的逆）。可以手算验证：H X H = Z，H Z H = X；S X S† = Y；CNOT 把 X⊗I 变成 X⊗X（这就是图 1.1 里 X̄ 被复制的原因）。好处是，Clifford 门之前的一个 Pauli 错误，到门之后仍是一个 Pauli 错误，记账方式不变。T 不行：T X T† = (X + Y)/√2，是两个 Pauli 的和，不再是单个 Pauli 算符。
What a Clifford gate is. The errors error correction tracks are Pauli operators (X, Y, Z and their products). A gate U is Clifford if it “sandwiches” every Pauli operator P into another one, that is, U P U† is again a Pauli operator up to a ± sign (U† is the inverse of U). By hand: H X H = Z and H Z H = X; S X S† = Y; CNOT turns X⊗I into X⊗X (which is why X̄ gets copied in Figure 1.1). The benefit: a Pauli error before a Clifford gate is still a Pauli error after it, so the bookkeeping stays the same. T fails the test: T X T† = (X + Y)/√2, a sum of two Paulis rather than a single Pauli operator.
魔术态的四步。既然二维表面码上不能用浅电路直接做 T，就把「T 的成分」装进一个态里，再用 Clifford 门加测量把它用掉：
The four steps of a magic state. Since the 2D surface code cannot do T with a shallow circuit, the “T ingredient” is packed into a state and then used up with Clifford gates and measurements:
| 步骤Step | 做什么What happens | 质量Quality |
|---|---|---|
| ① 制备① Prepare | 在一个物理比特上做出魔术态 |T⟩ = T|+⟩ = (|0⟩ + eiπ/4|1⟩)/√2。Make the magic state |T⟩ = T|+⟩ = (|0⟩ + eiπ/4|1⟩)/√2 on one physical qubit. | 带物理比特的噪声carries physical-qubit noise |
| ② 注入② Inject | 把这个态编码进一块表面码贴片。Encode that state into a surface-code patch. | 此后受码保护，但注入前的噪声已经带进来，码不会把它消掉protected from now on, but the noise picked up before injection is already inside and the code cannot remove it |
| ③ 蒸馏③ Distil | 拿多份有噪声的编码 |T⟩，只用 Clifford 操作和测量，筛出一份更干净的；可以多轮重复（Bravyi 与 Kitaev，2005）。Take several noisy encoded copies of |T⟩ and, with Clifford operations and measurements only, filter out one cleaner copy; repeat for more rounds if needed (Bravyi and Kitaev, 2005). | 份数换质量copies traded for quality |
| ④ 消耗④ Consume | 用一份干净的 |T⟩ 给数据比特做一次 T（下面的算例）。Spend one clean |T⟩ to apply one T to a data qubit (worked example below). | 每个 T 门用掉一份one copy per T gate |
算例：第 ④ 步为什么给出 T。数据比特 |ψ⟩ = a|0⟩ + b|1⟩，魔术态 |T⟩ = (|0⟩ + w|1⟩)/√2，记 w = eiπ/4。以数据比特为控制、魔术态比特为目标做一个 CNOT（数据是 1 时翻转魔术态比特）：
Worked example: why step ④ yields T. Data qubit |ψ⟩ = a|0⟩ + b|1⟩, magic state |T⟩ = (|0⟩ + w|1⟩)/√2 with w = eiπ/4. Apply a CNOT with the data qubit as control and the magic qubit as target (flip the magic qubit when the data qubit is 1):
→ [ a|0⟩(|0⟩ + w|1⟩) + b|1⟩(|1⟩ + w|0⟩) ]/√2
= [ (a|0⟩ + w b|1⟩)|0⟩ + (w a|0⟩ + b|1⟩)|1⟩ ]/√2
再在 Z 基测量魔术态比特（第二个比特）。读到 0（概率 1/2）：数据比特留在 a|0⟩ + w b|1⟩ = T|ψ⟩，完成。读到 1（概率 1/2）：数据比特是 w a|0⟩ + b|1⟩ = w(a|0⟩ + w̄ b|1⟩)，前面的 w 是整体相位，后面是 T†|ψ⟩（w̄ = e−iπ/4）；再补一个 S：S T† = T，得到 T|ψ⟩。全过程只用了 CNOT、测量和 S 这些 Clifford 操作，非 Clifford 的部分全部来自那份魔术态。
Now measure the magic qubit (the second one) in the Z basis. Outcome 0 (probability 1/2): the data qubit is left in a|0⟩ + w b|1⟩ = T|ψ⟩, done. Outcome 1 (probability 1/2): the data qubit is w a|0⟩ + b|1⟩ = w(a|0⟩ + w̄ b|1⟩), where the leading w is a global phase and the rest is T†|ψ⟩ (w̄ = e−iπ/4); one more S fixes it, since S T† = T, giving T|ψ⟩. The whole procedure used only CNOT, measurement and S, all Clifford operations; the non-Clifford part came entirely from the magic state.
5.3文献笔记里的空白（笔记级）Gaps in the literature notes (note-level)
以下来自 docs/lit/surface/ 的侦察报告与变体笔记：策展人的检索记录，没有进入卡片，可信级别低于上两节，措辞保留原文的否定形式。
From the scout report and variants notes under docs/lit/surface/: the curator's search records, not on the card, below the two sections above in trust, keeping the original negative wording.
| 空白Gap | 笔记原话（缩写）Note wording (abridged) | 出处Source |
|---|---|---|
| IBM | 「没有专门的低于阈值表面码论文」；旗舰 QEC 结果用 heavy-hex 码族，不是表面码本体。（heavy-hex 是 IBM 芯片的连接方式：比特排成六边形，边上再加一个比特，每个比特只连 2 到 3 个邻居。）“NO dedicated below-threshold surface paper”; flagship QEC results use the heavy-hex family, not the surface code itself. (Heavy-hex is the connectivity of IBM chips: qubits on a hexagonal lattice with an extra qubit on each edge, each qubit linked to only 2 or 3 neighbours.) | 变体笔记 Nugget 5；侦察报告variants notes Nugget 5; scout report |
| 离子阱Trapped ions | 「尚无离子阱团队在实验上运行过多轮的、字面意义的表面码存储器」。“No trapped-ion group has yet run a literal multi-round surface-code memory experimentally.” | 侦察报告scout report |
| 光子平台Photonics | 融合式架构（用光子之间的联合测量「融合」小的纠缠态来搭建纠错码）是理论框架；「尚未发表多轮逻辑存储的硬件演示」。Fusion-based architecture (building the code by joint measurements, “fusions”, of small entangled photon states) is a framework; “has not published a multi-round logical-memory hardware demonstration”. | 侦察报告scout report |
| 扭缺陷Twist defects | 「截至 2025 年的检索，没有硬件上的扭缺陷演示」。更正：这句笔记不成立。Google Quantum AI 与合作者 2023 年在超导芯片上把表面码的检查图变形，造出「三度顶点」：一种位错式的缺陷，与扭缺陷同属一类，可以承载非阿贝尔任意子（两个这样的准粒子互换位置两次，状态会被一个矩阵转动，而不只是乘一个相位）。他们移动并编织了这些缺陷，验证了它们的融合规则与编织统计（Nature 618, 264–269 (2023)，“Non-Abelian braiding of graph vertices in a superconducting processor”）。那是编织的物理演示：没有重复的纠错循环，也不是容错、低于阈值的逻辑运算；摘要自己说，要到将来加入纠错才能得到拓扑保护。“No hardware twist-defect demo exists through 2025 searches.” Correction: this note is wrong. In 2023 Google Quantum AI and collaborators deformed the check graph of the surface code on a superconducting chip to create “degree-3 vertices”: dislocation-type defects of the same family as twist defects, which can host non-Abelian anyons (exchange two such quasiparticles twice and the state is rotated by a matrix, not just multiplied by a phase). They moved and braided these defects and verified their fusion rules and braiding statistics (Nature 618, 264–269 (2023), “Non-Abelian braiding of graph vertices in a superconducting processor”). That was a physical braiding demonstration: no repeated error-correction cycles, and no fault-tolerant, below-threshold logical operation; the abstract itself says topological protection awaits the future inclusion of error correction. | 变体笔记 Nugget 6；更正依据 Nature 618, 264 (2023)variants notes Nugget 6; correction from Nature 618, 264 (2023) |
| 更新的候选记录Newer candidates | 报告列出 2025–2026 年候选（含第二个平台的 d=7 低于阈值结果），其中一条注明「arXiv 编号尚未确认」；卡片未采纳。The report lists 2025–2026 candidates (including a second platform's d = 7 below-threshold result), one marked “arXiv id not yet confirmed”; the card has not adopted them. | 侦察报告scout report |
「尚未演示」不是「不可能」，也不是「即将演示」。5.2 节前两条是带前提的定理（前提不满足时结论不适用），第三条是常用协议的代价；5.1 节是卡片当前版本的空白；5.3 节是检索记录，其中一条已被 2023 年的实验推翻。可信级别递减，引用时不要拉平。
“Not yet demonstrated” is neither “impossible” nor “imminent”. The first two items of §5.2 are theorems with assumptions (they say nothing when the assumptions fail) and the third is a cost of the usual protocols; §5.1 is the card's gaps as of its current version; §5.3 is search records, one of them already overturned by a 2023 experiment. Trust decreases in that order; do not flatten it when citing.
复盘，与去处Replay, and where to go next
论证链：四条款契约，卡片只给兑现了契约的平台贴标签（第 1 章）→ 实验报告 εL、码距缩放与 Λ，Λ 是设备加译码器的拟合值，三句话要分开（第 2 章）→ 五条 demonstrated 记录，数字与 ± 原样抄录（第 3 章）→ 六条误读各钉一个出处（第 4 章）→ 空白分三级：定理、卡片、笔记（第 5 章）。
The chain: a four-clause contract, and the card tags only platforms that honoured it (Ch. 1) → an experiment reports εL, distance scaling and Λ, where Λ is a device-plus-decoder fit and three sentences must stay apart (Ch. 2) → five demonstrated records, numbers and ± verbatim (Ch. 3) → six misreadings, each pinned to a source (Ch. 4) → gaps in three tiers: theorem, card, notes (Ch. 5).
三条去处。译码器：Λ 是设备加译码器的拟合值，硬件与译码器各占几分 QEC-Bank 不做切分；《译码器 · 高中数学版》第 5 章的时空译码正是 Willow 实时译码器在做的事。逻辑操作：从存储到计算的距离，就是《格点手术 · 高中数学版》第 1 章「存住不等于计算」。条目本身：QEC-Bank 的实现侧面页是本页的数据源，参考文献侧面给出每个文献键的完整条目。
Three roads. Decoders: Λ is a device-plus-decoder fit, and the QEC-Bank does not apportion it between hardware and decoder; the spacetime decoding of Chapter 5 of “Decoders · High-School Edition” is what Willow's real-time decoder performs. Logical operations: the distance from memory to computation is Chapter 1 of “Lattice Surgery · High-School Edition”, “storing is not computing”. The entry itself: the QEC-Bank realizations facet page is this page's data source; the references facet gives the full entry for each bibkey.
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 |
|---|---|---|
| Λ | 大写希腊字母，读「拉姆达」（lambda）Capital Greek letter, read “LAM-da” | 压制因子 Λ = εL(d)/εL(d+2)：码距加 2 时每轮逻辑错误缩小的倍数；设备加译码器的拟合值，不是 pth/p。Willow：2.14 ± 0.02（离线神经网络译码器）。Suppression factor Λ = εL(d)/εL(d+2): how many times logical error per cycle shrinks when d grows by 2; a device-plus-decoder fit, not pth/p. Willow: 2.14 ± 0.02 (offline neural-network decoder). |
| εL | 希腊字母 epsilon 带下标 L，读「每轮逻辑错误率」Greek epsilon, subscript L, read “logical error per cycle” | 一轮纠错循环后逻辑比特被翻转的概率，由多轮衰减曲线拟合。Probability that the logical bit is flipped over one correction cycle, fitted from a multi-cycle decay curve. |
| pth | 读「p 阈值」Read “p threshold” | 加大码距开始有利的临界物理错误率，定义见直觉教程 6.1 节；数值依赖噪声模型与译码器（账本）。The critical physical error rate at which growing the distance starts paying, defined in §6.1 of the intuition guide; its value depends on noise model and decoder (the ledger). |
| d | 码距（distance）Code distance | 最短逻辑算符的长度，即横跨串的边数，见直觉教程 3.2 节。实验里 d = 3、5、7。Length of the shortest logical operator, the edge count of a spanning string, see §3.2 of the intuition guide. Experiments use d = 3, 5, 7. |
| ± | 读「正负」Read “plus or minus” | 实验数字后的拟合误差，一个标准差：0.143% ± 0.003%。The fitting error after an experimental number, one standard deviation: 0.143% ± 0.003%. |
| [[n, k, d]] | 双方括号，读「n、k、d 码」Double brackets, read “an n, k, d code” | n 个物理比特存 k 个逻辑比特、码距 d。QEC-Bank 构造器的族是 [[2L²−2L+1, 1, L]]。n physical qubits storing k logical qubits at distance d. The QEC-Bank constructor's family is [[2L²−2L+1, 1, L]]. |
| ≈ | 读「约等于」Read “approximately equals” | 两边近似相等，如 P失败(N) ≈ N·εL。The two sides are nearly equal, as in Pfail(N) ≈ N·εL. |
本页新增词汇New terms on this page
| 术语Term | 白话定义Plain definition |
|---|---|
| 硬件契约hardware contract | 码对硬件的几何与节拍要求：二维最近邻、重量 ≤ 4 的检查、每检查一个帮手、反复快速测量。The code's geometry-and-tempo demands: 2D nearest-neighbour, weight ≤ 4 checks, one helper per check, repeated fast measurement. |
| demonstrated | QEC-Bank 状态标签：有被引用的实验论文报告了此事；QEC-Bank 机器不重算硬件数字。QEC-Bank status label: a cited experimental paper reported it; the QEC-Bank machine does not recompute hardware numbers. |
| 码容量模型code-capacity model | 只有数据比特出错、检查读数完美；浏览器实验台与账本前三行属于它。Data errors only, perfect check readings; the browser bench and the ledger's first three rows belong to it. |
| 横向 CNOTtransversal CNOT | 两块对齐的码块逐对做物理 CNOT，得到深度一的逻辑 CNOT。Pairwise physical CNOTs between two aligned blocks, giving a depth-one logical CNOT. |
| 泄漏leakage | 物理比特跑出 0/1 两个能级、落到更高能级（如 |2⟩）。它常引起一串显眼的、在时间上相关的警报；但普通的 Pauli 纠正只在 0 与 1 之间翻转，不能把泄漏的比特带回 0/1，要靠复位或专门的泄漏清除操作。A physical qubit escaping its two levels 0/1 into a higher level (such as |2⟩). It often causes a conspicuous run of alarms correlated in time; but ordinary Pauli corrections only flip between 0 and 1 and cannot bring a leaked qubit back into 0/1, which takes a reset or a dedicated leakage-removal operation. |
| 循环时长cycle time | 跑一轮纠错循环所需的时间；Willow 为 1.1 微秒。比较不同平台的 εL 前要先看它。The time one correction cycle takes; 1.1 microseconds on Willow. Check it before comparing εL across platforms. |
| Clifford 门 / 非 Clifford 门Clifford / non-Clifford gate | 把每个 Pauli 算符夹成 Pauli 算符的门，如 H、S、CNOT 及其组合；它们不够通用，还需要 T 这样的非 Clifford 门（补课 5a）。Gates that sandwich every Pauli operator into a Pauli operator, such as H, S, CNOT and their combinations; they are not universal, which takes a non-Clifford gate such as T (catch-up 5a). |
| 魔术态magic state | |T⟩ = T|+⟩ 这样的态：制备、注入、蒸馏之后，用 CNOT、测量和一个可能的 S 把它消耗掉，就给数据比特做了一次 T（补课 5a）。A state such as |T⟩ = T|+⟩: after preparation, injection and distillation, consuming it with a CNOT, a measurement and possibly an S applies one T to a data qubit (catch-up 5a). |
| Pauli 框架更新Pauli-frame update | 不真去施加 Pauli 纠正，只在经典电脑里记账，解读后续测量时算进去。Instead of applying a Pauli correction, record it classically and account for it when interpreting later measurements. |
| 旋转 / 未旋转贴片rotated / unrotated patch | 同一码距下两本比特账：未旋转 n = 2d²−2d+1（QEC-Bank 构造器），旋转 n = d²（实验室部署）。Two qubit ledgers at the same distance: unrotated n = 2d²−2d+1 (QEC-Bank constructor), rotated n = d² (lab deployments). |
延伸阅读Further reading
- Google Quantum AI, “Suppressing quantum errors by scaling a surface code logical qubit” (2023)
google2023suppressing · d=3→5，3.028% ± 0.023% → 2.914% ± 0.016% 每轮（张量网络译码）；卡片未列 Λ，论文给出模拟值 Λ3/5 = 1.10。arXiv: 2207.06431google2023suppressing · d=3→5, 3.028% ± 0.023% → 2.914% ± 0.016% per cycle (tensor-network decoding); no Λ on the card, a simulated Λ3/5 = 1.10 in the paper. arXiv: 2207.06431 - Google Quantum AI, “Quantum error correction below the surface code threshold” (2024)
google2024below · Willow d=7，0.143% ± 0.003% 每轮，Λ = 2.14 ± 0.02。arXiv: 2408.13687google2024below · Willow d=7, 0.143% ± 0.003% per cycle, Λ = 2.14 ± 0.02. arXiv: 2408.13687 - A. Eickbusch et al., “Demonstrating dynamic surface codes” (2024)
eickbusch2024dynamic · 六边形、行走、iSWAP 排程都随 d 压制。arXiv: 2412.14360eickbusch2024dynamic · hexagonal, walking and iSWAP schedules all suppress with d. arXiv: 2412.14360 - Y. Zhao et al., “Realization of an Error-Correcting Surface Code with Superconducting Qubits” (2022)
zhao2022realization · 祖冲之 2.1 上的 d=3 Surface-17。arXiv: 2112.13505zhao2022realization · d=3 Surface-17 on Zuchongzhi 2.1. arXiv: 2112.13505 - Dolev Bluvstein et al., “Logical quantum processor based on reconfigurable atom arrays” (2024)
bluvstein2024logical · 可重构阵列上的表面码贴片与横向门；输入准备不容错。arXiv: 2312.03982bluvstein2024logical · surface-code patches and transversal gates on reconfigurable arrays; input preparation not fault-tolerant. arXiv: 2312.03982 - A. G. Fowler, M. Mariantoni, J. M. Martinis, A. N. Cleland, “Surface codes: Towards practical large-scale quantum computation” (2012)
fowler2012surface · 电路级阈值 0.57% 每步的出处；「硬件契约」的工程全景。arXiv: 1208.0928fowler2012surface · source of the 0.57%-per-step circuit-level threshold; the engineering panorama behind the “hardware contract”. arXiv: 1208.0928
3.1 节时间轴与 3.2 节主表的硬件数字来自卡片的 realizations 字段，标签一律 demonstrated；QEC-Bank 机器不重算它们。正文与折叠框里标「论文」「摘要」的数字（循环时长、译码器、X/Z 基平均、各变体 Λ、2023 年的模拟 Λ3/5）是本页从 arXiv 原文补的，不在卡片里。折叠框里标注「文献笔记」的细节来自 docs/lit/surface/，不是条目数据。3.028/2.914 ≈ 1.04 是本页对两个卡片数字做的除法，论文把同一件事写成「相对错误降低约 4%」；第 4 章的 85/169 与 49/97 由直觉文稿的公式代入 d=7 算出。浏览器实验台是码容量模型，译码器在警报 ≤ 14 个时精确配对、更多时贪心配对，与芯片数字不可比。QEC-Bank 构造器只覆盖未旋转平面族，所有实验记录用的是旋转贴片。5.3 节的空白是检索记录，可信级别最低。
The hardware numbers in the §3.1 timeline and the §3.2 main table come from the card's realizations field, all labelled demonstrated; the QEC-Bank machine does not recompute them. Numbers marked “paper” or “abstract” in the text and fold-outs (cycle time, decoders, X/Z-basis averaging, the Λ of each variant, the simulated 2023 Λ3/5) were added by this page from the arXiv originals and are not on the card. Details marked “literature notes” in the fold-outs come from docs/lit/surface/ and are not entry data. 3.028/2.914 ≈ 1.04 is this page's division of two card numbers, which the paper states as “a relative error reduction of about 4%”; Chapter 4's 85/169 and 49/97 follow from the essay's formulas at d = 7. The browser bench is a code-capacity model whose decoder matches exactly up to 14 alarms and greedily beyond, not comparable with chip numbers. The QEC-Bank constructor covers only the unrotated planar family; every experimental record uses rotated patches. The gaps of §5.3 are search records and carry the lowest trust.
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（本页） · 关系 · 推导与指南 · 参考文献 · 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 (this page) · Relations · Derivations & guides · References · Open in the QEC-Bank →