用费米子的语言写电路 Writing circuits in the language of fermions

记号：n = 物理量子比特数，k = 逻辑量子比特数，L = 环面每行每列的格数，c = 碰撞数。CNOT = 受控翻转：控制比特为 1 时翻转目标比特。

Notation: n = number of physical qubits, k = number of logical qubits, L = cells per row and per column of the torus, c = collision count. CNOT = controlled flip: flips the target qubit when the control qubit is 1.

想象一排座位，编号 1、2、3、…。每个座位要么空着（记 0）要么有人（记 1）。一个费米子fermion：一类粒子（电子是最熟悉的例子），同一个「座位」（量子态）最多坐一个，而且交换两个费米子会给整体状态乘上 −1。就是「一个人」。到这里它和量子比特一模一样：座位 i 的占据数 n_i 是 0 或 1，和一个比特的 0/1 没有区别。差别只在一个动作上：把座位 i 和座位 j 的内容互换。对量子比特（盒子），互换就是互换，状态不变号。对费米子，规则是：

Picture a row of seats numbered 1, 2, 3, …. Each seat is either empty (write 0) or taken (write 1). A fermionA kind of particle (the electron is the familiar example) of which at most one can occupy a “seat” (quantum state), and such that exchanging two fermions multiplies the whole state by −1. is “one person”. So far it is identical to a qubit: the occupation n_i of seat i is 0 or 1, no different from a bit's 0/1. The difference lies in one action only: swap the contents of seat i and seat j. For qubits (boxes) a swap is a swap; the state keeps its sign. For fermions the rule is:

记号：n_i = 座位 i 的占据数（0 空、1 满）；|n_i n_j⟩ = 「座位 i 是 n_i、座位 j 是 n_j」这个状态（尖括号是 Toric Code 教程 3.1 节的态记号）；(−1)^n_i·n_j 只在 n_i = n_j = 1 时等于 −1。这条公式只对相邻的两个座位成立，隔着别的座位互换见表下的一般规则。

Notation: n_i = occupation of seat i (0 empty, 1 taken); |n_i n_j⟩ = the state “seat i holds n_i, seat j holds n_j” (the angle-bracket state notation of section 3.1 of the toric-code guide); (−1)^n_i·n_j equals −1 only when n_i = n_j = 1. The formula holds for two adjacent seats; for seats with others in between, see the general rule below the table.

四格里只有一格是 −1。这一个负号就是费米子与量子比特的全部区别，后面每一个困难都从它长出来。第一个后果：列出「谁坐着」的顺序有意义。三个人坐在 1、2、3 号，按 (1,2,3) 记和按 (2,1,3) 记差一次互换，差一个 −1；按 (3,2,1) 记要三次相邻互换，差 (−1)³ = −1；按 (2,3,1) 记要两次，差 (−1)² = +1。量子比特没有这本「符号账」。人话：费米子的语言里，每句话都附带一个正负号，取决于你按什么顺序说。

Only one of the four cells is −1. That single sign is the entire difference between fermions and qubits, and every difficulty below grows out of it. First consequence: the order in which you list “who is seated” matters. Three people in seats 1, 2, 3: written as (1,2,3) versus (2,1,3) differs by one swap, a factor −1; as (3,2,1) it takes three adjacent swaps, (−1)³ = −1; as (2,3,1) two swaps, (−1)² = +1. Qubits keep no such “sign ledger”. In plain words: in the language of fermions every sentence carries a sign that depends on the order you say it in.

隔着别的座位互换怎么办？把它想成两位「旅行者」各走一趟：座位 i 的内容一格一格往右挪到 j，座位 j 的内容一格一格往左挪到 i。每挪一格是一次相邻互换，按上表，只有挪动的人越过另一个人时才乘 −1。于是负号有三个来源：座位 i 的人（如果有）越过中间每一个有人的座位，一次一个 −1；座位 j 的人（如果有）也一样；两位旅行者在路上彼此越过一次（两人都在时才算）。记 m = i 与 j 之间有人的座位数，合起来：

What about swapping seats with others in between? Think of two “travellers”: the content of seat i moves right one seat at a time until it reaches j, and the content of seat j moves left one seat at a time until it reaches i. Each one-seat move is an adjacent swap, and by the table it costs −1 only when the mover passes another person. So the minus signs have three sources: the person from seat i (if any) passes every occupied seat in between, one −1 each; the person from seat j (if any) does the same; and the two travellers pass each other once (only if both exist). With m = the number of occupied seats strictly between i and j, altogether:

两个可以手算的例子。|110⟩ 互换 1 号与 3 号：n₁ = 1、n₃ = 0、m = 1（2 号有人），指数 = 0 + 1·1 = 1，结果 −|011⟩，因为 1 号的人越过了 2 号的人。|1101⟩ 互换 1 号与 4 号：n₁ = n₄ = 1、m = 1，指数 = 1 + 2·1 = 3，结果乘 −1。第二个例子里中间那个人被两位旅行者各越过一次，两个负号抵消，剩下的 −1 来自两位旅行者彼此越过。所以两端都有人时，中间有几个人都不影响符号；只有一端有人时，中间每多一个人符号就翻一次。下面的部件可以摆出任意四座位的例子，逐项看指数怎样凑出来。

Two examples to check by hand. |110⟩, swap seats 1 and 3: n₁ = 1, n₃ = 0, m = 1 (seat 2 is taken), exponent = 0 + 1·1 = 1, result −|011⟩, because the person from seat 1 passed the person in seat 2. |1101⟩, swap seats 1 and 4: n₁ = n₄ = 1, m = 1, exponent = 1 + 2·1 = 3, so the state picks up −1. In the second example the person in the middle is passed once by each traveller; those two signs cancel, and the remaining −1 comes from the travellers passing each other. So when both end seats are taken, the number of people in between does not affect the sign; when only one end is taken, each extra person in between flips it. The widget below sets up any four-seat example and shows how the exponent is assembled term by term.

物理学家还喜欢把一个座位劈成两半。两半各叫一个马约拉纳Majorana（马约拉纳费米子）：一个「半座位」。两个马约拉纳配成一对，才等于一个可以空或满的普通座位；单独一半没有「空/满」可言。，记 γ；一对 (γ_a, γ_b) 合起来才是一个能空能满的座位，「这个座位是空是满」是要把这一对合在一起问才有答案的问题，单独一半没有空满可言。这个问题写作 iγ_aγ_b（前面的 i 让答案是实数），答案只有 ±1，一个表示空，一个表示满（哪个号对应空取决于约定）。为什么要劈：劈开之后，可以把不同座位的两半配成新的对，像把两把椅子各锯一半再拼成一把新椅子；而「问新配出的一对是空是满」，下面会看到，正是蜂窝码那种只碰两个比特的检查的核心。QEC-Bank 蜂窝码条目的卡片这样记录这件事（codes/honeycomb/card.yaml，views 字段，type fermionic，原文）：

Physicists also like to split a seat in two. Each half is called a MajoranaA Majorana fermion: “half a seat”. Two Majoranas paired together make one ordinary seat that can be empty or taken; a lone half has no “empty/taken” of its own., written γ; only a pair (γ_a, γ_b) together makes a seat that can be empty or taken, and “is this seat empty or taken” is a question that has an answer only when the pair is asked together; a lone half has no occupation. The question is written iγ_aγ_b (the i in front makes the answer real) and its answer is ±1, one value meaning empty and the other taken (which sign means empty is a convention). Why split: once split, halves from different seats can be paired anew, like sawing two chairs in half and joining them into a new chair; and “is this newly formed pair empty or taken”, as shown below, is the core of the kind of check touching only two qubits that the honeycomb code uses. The QEC-Bank honeycomb card records this as follows (codes/honeycomb/card.yaml, views field, type fermionic, verbatim):

逐词。four-Majorana representation：每个量子比特（自旋）用四个半座位表示，而不是两个——多出的两个是为了让 X、Y、Z 三种泡利都能写成「两个半座位的乘积」。directional Pauli edge：蜂窝码的每条检查是相邻两个比特上的 XX、YY 或 ZZ，由边的方向决定（卡片 description 原文：“Edge operators are fixed as XX, YY, or ZZ by direction”）。factors … into a Z2 gauge variable and a matter-fermion pairing：一条这样的边算符拆成两个因子——一个只取 ±1 的「规范变量」（一个取值可以当成固定的 ±1 的记账量，见下面的补课），和一个「两个半座位配对」。rotating pairing pattern：蜂窝码每一轮测同一种颜色标签的边（每一轮里 XX、YY、ZZ 三种方向都有），就是不断换着配对。plaquette fluxes：绕一个六边形面把规范变量乘起来得到的 ±1，是不随轮次变的守恒量。蜂窝码条目的推导页 pedagogy/representation-dictionary.md 把那个拆分写成一行，原文说是「示意地」（schematically）拆成：

Word by word. four-Majorana representation: each qubit (spin) is written with four half-seats rather than two; the extra two let all three Paulis X, Y, Z be written as “a product of two half-seats”. directional Pauli edge: each honeycomb check is XX, YY or ZZ on two neighbouring qubits, fixed by the edge's direction (card description verbatim: “Edge operators are fixed as XX, YY, or ZZ by direction”). factors … into a Z2 gauge variable and a matter-fermion pairing: such an edge operator splits into two factors, a “gauge variable” taking only ±1 (a bookkeeping quantity whose value can be treated as a fixed ±1; see the catch-up below) and a “pairing of two half-seats”. rotating pairing pattern: the honeycomb code measures the edges of one colour label each round (every round contains XX, YY and ZZ edges alike), that is, keeps re-pairing. plaquette fluxes: the ±1 obtained by multiplying the gauge variables around a hexagonal face, conserved from round to round. The entry's derivation page pedagogy/representation-dictionary.md writes that split in one line, in its own words “schematically”:

记号（只为读出这一行，不推导）：σ_u^a = 比特 u 上的泡利算符，a ∈ {x, y, z} 说明是 X、Y 还是 Z；u、v = 一条边的两个端点比特；b_u^x, b_u^y, b_u^z, c_u = 比特 u 的四个半座位（四个马约拉纳）；i = 虚数单位（i² = −1，Toric Code 教程 1.2 节），保证每个括号取值 ±1。第一个括号是「规范变量」，第二个是「物质配对」。原文见 codes/honeycomb/pedagogy/representation-dictionary.md。「示意」指的是整体符号：按 Kitaev 的约定 σ^a = i b^a c，精确的等式是 σ_u^a σ_v^a = −(i b_u^a b_v^a)(i c_u c_v)（本篇用八个半座位的矩阵模型数值核对过；来源见下面的补课）。这个负号不影响下面的读法。

Notation (to read the line only, not to derive it): σ_u^a = the Pauli operator on qubit u, with a ∈ {x, y, z} saying whether it is X, Y or Z; u, v = the two endpoint qubits of an edge; b_u^x, b_u^y, b_u^z, c_u = the four half-seats (four Majoranas) of qubit u; i = the imaginary unit (i² = −1, section 1.2 of the toric-code guide), ensuring each bracket takes the value ±1. The first bracket is the “gauge variable”, the second the “matter pairing”. Source: codes/honeycomb/pedagogy/representation-dictionary.md. “Schematically” concerns the overall sign: with Kitaev's convention σ^a = i b^a c the exact identity is σ_u^a σ_v^a = −(i b_u^a b_v^a)(i c_u c_v) (checked numerically here in a matrix model of eight half-seats; where it comes from is in the catch-up below). The sign does not affect the reading that follows.

这张图就是那一行公式的全部内容：一条两比特检查 = 一个规范对 × 一个物质对。规范对的值可以当成固定的 ±1（为什么可以，见补课），所以测这条检查，实际上就是在问「c_u 和 c_v 配成的这一对是空还是满」，再乘上一个已知的 ±1。蜂窝码每一轮测一组边，就是把所有 c 按这一组边两两配对来问；下一轮换一组边，就换一种配对。卡片说的 “rotating pairing pattern” 就是这件事。

This figure is the whole content of that one line: one two-qubit check = one gauge pair × one matter pair. The gauge pair's value can be treated as a fixed ±1 (why, see the catch-up), so measuring the check really asks “is the pair formed by c_u and c_v empty or taken”, times a known ±1. Each honeycomb round measures one set of edges, which pairs up all the c's along those edges and asks each pair; the next round uses another set of edges, hence another pairing. That is the card's “rotating pairing pattern”.

现在把「算一次两个比特的奇偶」这份配方用两种语言各写一遍。量子比特语言（第 0 章事实 1）：准备一个辅助比特置 0；CNOT（比特 1 → 辅助）；CNOT（比特 2 → 辅助）；测辅助比特，读出 n₁ + n₂ 的奇偶。三个比特、两个两比特门、一次测量。这是假定芯片只提供单比特门、CNOT 和单比特测量时的一种实现，不是量子比特本身的要求。费米子语言：座位 1 和座位 2「合计有几个人」的奇偶，本来就是一个可以直接问的问题——在能直接测「一对座位的合计」的硬件上，这份配方就是一次测量，不需要辅助比特也不需要 CNOT。QEC-Bank 蜂窝码卡片的 realizations 字段登记了这样一种硬件设想（原文）：platform “Majorana-tetron parity-measurement architecture”，status proposed，“Native pair-parity measurements and planar layouts were simulated with Majorana-specific faults. This is an architecture and decoder study, not a laboratory realization of a logical honeycomb memory.”——proposed（设想），不是 demonstrated（已演示）。卡片为这一条引用的两篇论文说的也是设想：Karzig 等人（arXiv:1610.05289）提出由四个或更多马约拉纳零模组成比特、只靠测量操作的架构设计；Paetznick 等人（arXiv:2202.11829）说明 Floquet 码只用两比特测量，适合「能直接实现两比特测量」的平台，例如基于马约拉纳零模的只测量拓扑比特，并在这种架构上给出不用辅助比特的测量序列和数值模拟。所以「一次测量」是这种硬件的能力，前提是硬件造得出来，不是费米子写法自带的。

Now write the recipe “compute the parity of two qubits” in each language. Qubit language (fact 1 of Chapter 0): prepare a helper qubit at 0; CNOT (qubit 1 → helper); CNOT (qubit 2 → helper); measure the helper, read the parity of n₁ + n₂. Three qubits, two two-qubit gates, one measurement. This is one implementation, assuming a chip that offers only single-qubit gates, CNOTs and single-qubit measurement; it is not something qubits as such require. Fermion language: the parity of “how many people in seats 1 and 2 together” is a question that can be asked directly; on hardware that can measure “the total of a pair of seats” natively, the recipe is one measurement, no helper, no CNOT. The realizations field of the QEC-Bank honeycomb card registers such a hardware proposal (verbatim): platform “Majorana-tetron parity-measurement architecture”, status proposed, “Native pair-parity measurements and planar layouts were simulated with Majorana-specific faults. This is an architecture and decoder study, not a laboratory realization of a logical honeycomb memory.”: proposed, not demonstrated. The two papers the card cites for this entry also describe proposals: Karzig et al. (arXiv:1610.05289) present architecture designs whose qubits are built from four or more Majorana zero modes and are operated by measurements only; Paetznick et al. (arXiv:2202.11829) note that Floquet codes use only two-qubit measurements, which suits platforms “where two-qubit measurements can be implemented directly, such as measurement-only topological qubits based on Majorana zero modes”, and give measurement sequences without auxiliary qubits plus simulations for that architecture. So “one measurement” is a capability of that hardware, if it can be built, and not something the fermionic notation brings with it.

同一份配方，两种写法，两种成本。把一种写法自动翻成另一种、并且保证意思不变、还能数出成本的程序，叫编译器compiler：把用一种语言写的程序翻译成另一种语言、保证语义不变的程序。日常例子：把人读的高级语言翻成机器指令。这里是把量子比特的电路翻成费米子/马约拉纳的操作，或反过来。。「费米子编译器」就是：输入一份量子比特语言的电路（或 QEC-Bank 的一张测量时间表），输出等价的费米子语言的操作序列——或者反过来。人话：QEC-Bank 已经会用量子比特的语言描述每个码；这道题要它学会第二种语言，并且会翻译。

One recipe, two spellings, two costs. A program that translates one spelling into the other automatically, guarantees the meaning is unchanged, and can count the cost, is a compilerA program that translates a program written in one language into another while preserving its meaning. Everyday example: turning a human-readable high-level language into machine instructions. Here: turning a qubit circuit into fermionic/Majorana operations, or back.. A “fermionic compiler” is: input a circuit in the qubit language (or a QEC-Bank measurement schedule), output the equivalent sequence of operations in the fermion language, or the reverse. In plain words: the QEC-Bank already describes every code in the language of qubits; this item asks it to learn a second language, and to translate.

Wanted 板的 statement 说这台编译器「很可能」和另一种东西有相同的计算内容：ZX 图ZX diagram / ZX-calculus：一种画电路的图形语言。线是量子比特，点分两种颜色（Z 型与 X 型），点上可以标一个角度；配有一小组改写规则，规则改写前后的图表示同一个操作。。ZX 图是一种画电路的图形语言：一根线是一个量子比特随时间的轨迹；线上的点分两种颜色——Z 型（本篇按 QEC-Bank 的配色画成蓝）和 X 型（橙）——一个点可以连任意多根线，还可以标一个角度；然后有一小组改写规则，规则两边的图代表同一个操作。最基本的一条叫「融合」：两个同色的点用线连着，可以合成一个点，角度相加。下图画了两个例子，不做任何推导：

The Wanted statement says this compiler “plausibly” has the same computational content as something else: ZX diagramsZX diagram / ZX-calculus: a graphical language for drawing circuits. Wires are qubits, dots come in two colours (Z type and X type) and may carry an angle; a small set of rewrite rules says which diagrams denote the same operation.. A ZX diagram is a graphical language for circuits: a wire is one qubit's track through time; the dots on wires come in two colours, Z type (drawn blue here, following the QEC-Bank palette) and X type (amber); a dot may join any number of wires and carry an angle; and there is a small set of rewrite rules whose two sides denote the same operation. The most basic is “fusion”: two dots of the same colour joined by a wire merge into one, angles adding. The figure shows two examples, deriving nothing:

记号：t = 时间，线从左到右；α、β = 点上标的角度（希腊字母，读「阿尔法、贝塔」），本篇不用它们的具体值。ZX 教科书里 Z 点是绿色、X 点是红色，本篇按 QEC-Bank 的配色用蓝与橙。

Notation: t = time, wires run left to right; α, β = angles on the dots (Greek letters, read “alpha, beta”), whose values this guide never uses. ZX textbooks draw Z dots green and X dots red; this guide follows the QEC-Bank palette with blue and amber.

一个点做什么？只看最简单的情形（角度为 0），在 0/1 上逐个试就能读出来。Z 点（蓝）进一根线、出两根线：进来 0，两根出线都是 0；进来 1，两根都是 1。它把 0/1 的值复制一份。X 点（橙）进两根线、出一根线：出来的是两根进线的奇偶，a + b mod 2（0+0→0，0+1→1，1+0→1，1+1→0），也就是一次奇偶检查。现在读左图：控制比特的值 a 在蓝点被复制一份，顺着竖线送到橙点；橙点把它和目标比特的值 b 加起来，输出 a + b mod 2。控制比特不变、目标比特翻转当且仅当 a = 1，这正是 CNOT（整张图比 CNOT 多一个不影响结果的常数因子 √2，四种输入都在本页配套的矩阵核对里算过）。点上的角度 α 只在「进来 1」时给状态乘上一个相位 e^iα，本篇不需要它。

What does a dot do? Take the simplest case (angle 0) and try 0 and 1 one at a time. A Z dot (blue) with one wire in and two out: 0 in gives 0 on both outputs, 1 in gives 1 on both. It copies the 0/1 value. An X dot (amber) with two wires in and one out outputs the parity of its inputs, a + b mod 2 (0+0→0, 0+1→1, 1+0→1, 1+1→0): a parity check. Now read the left figure: the control's value a is copied at the blue dot and sent down the vertical wire to the amber dot, which adds it to the target's value b and outputs a + b mod 2. The control is unchanged and the target flips exactly when a = 1: that is the CNOT (the diagram equals the CNOT times a harmless constant factor √2; all four inputs were checked in a matrix computation for this page). An angle α on a dot only multiplies the state by a phase e^iα when the input is 1; this guide does not need it.

为什么 QEC-Bank 关心 ZX：设计文档 3.2 节把它定为「跨模型的桥」（原文：“ZX diagram — the cross-model bridge: circuit-based, measurement-based, fusion-based, and Floquet pictures are local-equivalence flavors of one underlying ZX object (Bombin–Litinski–Nickerson–Pastawski–Roberts, arXiv:2303.08829). CSS codes auto-generate phase-free ZX; long-term target semantics for schedule-equivalence checking.”）。也就是说，QEC-Bank 打算让各种「写法」——电路、测量、蜂窝码那样一轮轮的时间表——都翻成 ZX 图，再用改写规则判断两张图是不是同一回事（本系列另一篇 《两张测量时间表是不是同一回事》专讲这件事）。Wanted 板这道题的猜想是：费米子语言与 ZX 语言表达力相当——凡是能用 ZX 图画的，都能用半座位配对写出来，反之亦然。如果猜想成立，一部「ZX ↔ 费米子」的词典就把费米子语言也接上了这座桥。

Why the QEC-Bank cares about ZX: the design document, section 3.2, fixes it as “the cross-model bridge” (verbatim: “ZX diagram — the cross-model bridge: circuit-based, measurement-based, fusion-based, and Floquet pictures are local-equivalence flavors of one underlying ZX object (Bombin–Litinski–Nickerson–Pastawski–Roberts, arXiv:2303.08829). CSS codes auto-generate phase-free ZX; long-term target semantics for schedule-equivalence checking.”). That is, the QEC-Bank intends to translate every “spelling”, circuits, measurements, round-by-round schedules like the honeycomb's, into ZX diagrams and then use the rewrite rules to decide whether two diagrams are the same thing (another guide in this series, “Are two measurement schedules the same thing?”, is devoted to it). The conjecture behind this Wanted item: the fermion language and the ZX language are equally expressive; whatever can be drawn as a ZX diagram can be written as half-seat pairings, and vice versa. If so, a “ZX ↔ fermionic” dictionary connects the fermion language to that bridge too.

逐词翻译。Compilation：1.3 节的翻译。the unified Majorana/parton/quon representation：三个名字、一种表示。majorana-pairs方言一：直接用「半座位配对」写；Floquet 码（蜂窝码一类）的常用写法。 是 1.2 节的半座位配对；parton方言二：把每个自旋拆成若干「部分子」（parton）——就是四个半座位加规范变量的写法；蜂窝码卡片用的正是这一种。yaml 引用 arXiv:1703.02973。（部分子）是把每个自旋拆成若干零件的写法，蜂窝码卡片的 dialect 字段填的正是它；quon方言三：把码画成「量子化的图」，检查变成图上的圈算符。yaml 引用 arXiv:1910.12065（QEC-Bank 登记键 liu2019quantized，“Quantized Graphs and Quantum Error Correction”）。 是把码画成一张「量子化的图」的写法。设计文档 3.2 节的决定原文：“the Majorana-pair picture of Floquet codes, the parton construction (Hsieh–Halász, arXiv:1703.02973), and the quon/quantized-graph language (Liu, arXiv:1910.12065) are the same underlying representation under different names — recorded as one type fermionic with dialect: majorana-pairs | parton | quon”。spec 3.2：就是这份设计文档的 3.2 节。Plausibly the same computational content as ZX compilation：1.4 节末尾的猜想，注意「plausibly」（很可能）——这是一个尚未证明的判断。First deliverable: a formal ZX <-> fermionic dictionary：第一份交付物不是编译器本身，而是一部形式化的（机器可核对的）双向词典：ZX 的每个基本图元对应费米子语言的什么，改写规则对应什么恒等式。人话：先别造翻译机，先把词典编出来，而且词典的每一条都要能被机器核对。

Word by word. Compilation: the translation of 1.3. the unified Majorana/parton/quon representation: three names, one representation. majorana-pairsDialect one: write directly in “half-seat pairings”; the usual spelling for Floquet codes such as the honeycomb. is the half-seat pairing of 1.2; partonDialect two: split each spin into several “partons”, which is the four-half-seats-plus-gauge-variable spelling; the honeycomb card uses exactly this one. The yaml cites arXiv:1703.02973. splits each spin into several parts, and is what the honeycomb card's dialect field says; quonDialect three: draw the code as a “quantized graph”, checks becoming cycle operators on the graph. The yaml cites arXiv:1910.12065 (QEC-Bank key liu2019quantized, “Quantized Graphs and Quantum Error Correction”). draws the code as a “quantized graph”. The design document's decision, section 3.2, verbatim: “the Majorana-pair picture of Floquet codes, the parton construction (Hsieh–Halász, arXiv:1703.02973), and the quon/quantized-graph language (Liu, arXiv:1910.12065) are the same underlying representation under different names — recorded as one type fermionic with dialect: majorana-pairs | parton | quon”. spec 3.2: section 3.2 of that design document. Plausibly the same computational content as ZX compilation: the conjecture at the end of 1.4; note “plausibly”, an unproven judgement. First deliverable: a formal ZX <-> fermionic dictionary: the first deliverable is not the compiler itself but a formal (machine-checkable) two-way dictionary: what each basic ZX element corresponds to in the fermion language, and which identities the rewrite rules correspond to. In plain words: do not build the translating machine yet; write the dictionary first, and make every entry machine-checkable.

逐词。Curated：人工整理——由人读论文、写成一段话；QEC-Bank 证据词表里与它相对的是 derived（机器从数据算出）。per-entry：一个条目一段，条目之间互不相通。fermionic notes：卡片 views 字段里 type 为 fermionic 的那条记录（1.2 节引过蜂窝码的那条）。only：除此之外什么都没有——没有程序、没有统一语法、没有翻译。refs：两篇论文，1910.12065 已登记为 liu2019quantized，1703.02973 尚未登记（据条目记载，登记待补）。人话：今天 QEC-Bank 里的「费米子视图」是四段人写的笔记，机器读不懂、也不会检查它们是否一致。

Word by word. Curated: assembled by hand, a person read the papers and wrote a paragraph; in the QEC-Bank vocabulary its opposite is derived (computed by the machine from data). per-entry: one paragraph per entry, with no connection between entries. fermionic notes: the record of type fermionic in the card's views field (the honeycomb one quoted in 1.2). only: nothing else exists, no program, no common syntax, no translation. refs: two papers, 1910.12065 registered as liu2019quantized, 1703.02973 not yet registered (cited by the entry, registration pending). In plain words: today the QEC-Bank “fermionic view” is four hand-written notes the machine can neither read nor check for consistency.

先认识主角。honeycomb 是这道题 related 里唯一的条目，卡片 description 原文：“A period-three dynamical stabilizer memory made only from weight-two Pauli measurements on a honeycomb torus. Edge operators are fixed as XX, YY, or ZZ by direction, while a plaquette three-coloring partitions them into three perfect-matching rounds. The code is the ordered schedule and its outcome-dependent instantaneous stabilizer group, not any one static set of mutually commuting edge checks; after startup that group carries two toric-code logical qubits.” 人话：它只测两比特的检查（XX、YY 或 ZZ，由边的方向决定），分三轮轮流测；「码」是这张时间表，不是某一份固定的检查清单；跑起来之后存 2 个逻辑比特，和环面码一样（第 0 章事实 4）。instances.yaml 登记两个实例：L3x3（n = 18 verified，k = 2 verified）与 L6x6（n = 72 verified，k = 2 verified）；两者的 d 在条目页上都显示 d = ?（那是本系列另一篇 《检查每一轮都在变，码距该怎么数》的题目）。

Meet the protagonist first. honeycomb is the only entry in this item's related list; its card description verbatim: “A period-three dynamical stabilizer memory made only from weight-two Pauli measurements on a honeycomb torus. Edge operators are fixed as XX, YY, or ZZ by direction, while a plaquette three-coloring partitions them into three perfect-matching rounds. The code is the ordered schedule and its outcome-dependent instantaneous stabilizer group, not any one static set of mutually commuting edge checks; after startup that group carries two toric-code logical qubits.” In plain words: it measures only two-qubit checks (XX, YY or ZZ, fixed by edge direction), in three rounds taken in turn; the “code” is that schedule, not any fixed check list; once running it stores 2 logical qubits, like the toric code (fact 4 of Chapter 0). instances.yaml registers two instances: L3x3 (n = 18 verified, k = 2 verified) and L6x6 (n = 72 verified, k = 2 verified); both show d = ? on the entry page (the subject of another guide in this series, “When the checks change every round, how do you count distance?”).

卡片的 views 字段登记了八种表示，每条带一个 rung（阶梯层级）、一个 type 和一个 origin。下表按卡片顺序逐字列出 type 与 origin 的 kind：

The card's views field registers eight representations, each with a rung, a type and an origin. The table lists type and origin kind verbatim, in card order:

数一数：8 条里 1 条 derived（机器算的，附程序名 honeycomb.build 与版本 v1）、1 条 canonical（定义本身）、6 条 curated（人写的）。R5 这一层——设计文档叫「跨模型视图」——的 4 条全部是 curated，其中费米子那条还是全表唯一带 gap 字段的：卡片自己承认这里有个洞，并把洞指向本题。QEC-Bank 的条目页照此渲染：在蜂窝码「表示阶梯」页的 R5 一栏，fermionic 记录旁边是一个 “curated from kitaev2006anyons” 的标签和一个 “Open machine-derivation problem →” 的链接，点过去就是 Wanted 板的这一项。人话：QEC-Bank 会算「这一轮的检查清单」，不会算「这个码的费米子写法」；后者是抄来的，而且卡片上写明了是抄来的。

Count: of 8 records, 1 derived (machine-computed, with program name honeycomb.build and version v1), 1 canonical (the definition itself), 6 curated (hand-written). All 4 records on rung R5, which the design document calls “cross-model views”, are curated, and the fermionic one is the only record in the whole table with a gap field: the card admits a hole here and points the hole at this item. The QEC-Bank entry pages render it accordingly: in the R5 row of the honeycomb “Representation ladder” page, the fermionic record carries a “curated from kitaev2006anyons” chip and an “Open machine-derivation problem →” link that leads to this item on the Wanted board. In plain words: the QEC-Bank can compute “this round's check list”; it cannot compute “this code's fermionic spelling”; the latter is copied, and the card says so.

蜂窝码不是唯一带费米子视图的条目。对 codes/*/card.yaml 搜索 “type: fermionic”（2026-09-12），恰有四个条目命中，三种方言都出现了：

The honeycomb is not the only entry with a fermionic view. A search of codes/*/card.yaml for “type: fermionic” (2026-09-12) hits exactly four entries, and all three dialects appear:

四条记录，四个 curated，四个 gap 都指向本题；四个条目的「表示阶梯」页上各有一个 “Open machine-derivation problem →” 链接。注意表里的四段笔记说的不是同一件事：一段讲怎样把自旋拆成半座位（honeycomb），一段讲怎样把整个码翻成马约拉纳码（steane），一段讲怎样画成图（surface），一段讲怎样加速模拟（toric）。它们用三种方言写成，彼此之间没有任何机器能核对的对应关系——这正是 statement 里「unified」（统一）这个词今天还只是一个决定而不是一个事实的原因（4.3 节）。

Four records, four curated, four gaps all pointing at this item; each of the four entries' “Representation ladder” pages carries an “Open machine-derivation problem →” link. Note that the four notes are not about the same thing: one on splitting spins into half-seats (honeycomb), one on translating a whole code into a Majorana code (steane), one on drawing it as a graph (surface), one on speeding up simulation (toric). They are written in three dialects with no machine-checkable correspondence among them, which is exactly why “unified” in the statement is today a decision rather than a fact (4.3).

蜂窝码条目的推导页 codes/honeycomb/pedagogy/representation-dictionary.md 在「Majorana / parton gauge theory」一节的 QEC-Bank status 里写（原文）：“There is no generic fermionic compiler in the repository, so this view is not machine-derived.” 同一页末尾有一张「来源 → 目标 → 状态」的账本，其中一行：“spin Paulis | Majorana gauge variables | curated dictionary”，并解释这张账本的目的是 “prevents a derived rank from lending machine certainty to a curated conversion farther down the chain” —— 防止上游机器算出的秩（R1 的 derived）把「机器的确定性」借给下游人写的换算。人话：条目自己知道费米子那一栏是人写的，并且专门立了一张账本，防止读者把它误当成机器算的。

The honeycomb entry's derivation page codes/honeycomb/pedagogy/representation-dictionary.md writes under QEC-Bank status in its “Majorana / parton gauge theory” section (verbatim): “There is no generic fermionic compiler in the repository, so this view is not machine-derived.” The same page ends with a “source → target → status” ledger containing the row “spin Paulis | Majorana gauge variables | curated dictionary”, and explains the ledger's purpose as “prevents a derived rank from lending machine certainty to a curated conversion farther down the chain”: it stops a machine-computed rank upstream (the derived R1) from lending “machine certainty” to a hand-written conversion downstream. In plain words: the entry knows its fermionic column is hand-written, and keeps a ledger precisely so readers do not mistake it for machine output.

2.1 节的表里已经有一个榜样：R1 那条 steady-isg-checks，origin 是 kind: derived, converter: "honeycomb.build Schedule -> ISGTracker steady snapshot", version: "v1"——机器从时间表算出检查清单，并记下用的是哪个程序的哪个版本。一台费米子编译器会让 R5 的费米子记录长成同样的样子：

The table of 2.1 already contains a model: the R1 record steady-isg-checks, whose origin is kind: derived, converter: "honeycomb.build Schedule -> ISGTracker steady snapshot", version: "v1"; the machine computed the check list from the schedule and recorded which program and version it used. A fermionic compiler would let the R5 fermionic record take the same shape:

Floquet 码Floquet code：检查清单按固定周期一轮轮变化的码；蜂窝码是最早的例子（周期三轮）。名字来自研究周期性系统的数学家 Floquet。是检查清单按周期变化的码，蜂窝码是它的代表。它只用两比特的检查，而 1.3 节说过，「问一对座位的合计」恰好是马约拉纳硬件设想中原生（native，硬件直接提供）的操作。蜂窝码卡片的 fit.platforms 字段把 majorana-parity-measurement 列在第一位，realizations 字段里那条马约拉纳架构的状态是 proposed。今天缺的一环：QEC-Bank 存的是量子比特语言的时间表（R3 canonical），要在马约拉纳硬件上跑，得有人手工把每一轮的 XX/YY/ZZ 翻成「测哪几个半座位」（tetron 是设想中的一种马约拉纳量子比特，一个比特由四个半座位组成；在这种硬件上，一次 XX/YY/ZZ 测量是一次同时涉及四个半座位的奇偶测量），再手工数成本。编译器把这一步变成机器的事：输入 R3 的时间表，输出马约拉纳语言的操作序列和成本。这样卡片上才能出现「同一个码，两种硬件语言，各自的成本」这样可比的数字——而不是一个 proposed 加一段文字。人话：蜂窝码天生适合费米子硬件，但 QEC-Bank 今天没有工具把它翻成那种硬件听得懂的话。

A Floquet codeA code whose check list changes round by round with a fixed period; the honeycomb code is the earliest example (period three). Named after Floquet, the mathematician of periodic systems. is a code whose check list changes periodically, and the honeycomb is its representative. It uses only two-qubit checks, and as 1.3 said, “asking a pair of seats for their total” is exactly the operation that Majorana hardware proposals offer natively (directly provided by the hardware). The honeycomb card's fit.platforms field lists majorana-parity-measurement first, and the Majorana architecture in its realizations field has status proposed. The missing link today: the QEC-Bank stores the schedule in the qubit language (R3 canonical); to run it on Majorana hardware someone must hand-translate each round's XX/YY/ZZ into “which half-seats to measure” (a tetron is a proposed Majorana qubit made of four half-seats; on such hardware one XX/YY/ZZ measurement is a single parity measurement involving four half-seats) and count the cost by hand. A compiler makes that the machine's job: input the R3 schedule, output the Majorana-language operation sequence and its cost. Only then can the card show comparable numbers, “one code, two hardware languages, each with its cost”, instead of one proposed and a paragraph. In plain words: the honeycomb code is born for fermionic hardware, but the QEC-Bank today has no tool to translate it into words that hardware understands.

三件事。第一，要求 QEC-Bank 输出任何一个码的费米子写法。只有四个条目有人写过笔记；其余条目连笔记也没有，而且笔记不是写法。第二，核对两段方言不同的笔记是否说的是同一件事。steane 的 majorana-pairs 笔记与 honeycomb 的 parton 笔记之间没有任何可执行的对应；设计文档说三种方言「是同一种表示」，但这句话今天没有机器能验证。第三，在费米子语言里比较成本。「蜂窝码在马约拉纳硬件上一个周期要几次配对测量」这样的数字，卡片上没有，因为没有人算过、也没有程序能算。人话：QEC-Bank 今天能用一种语言完整描述蜂窝码，第二种语言只有几句人写的注释。

Three things. First, ask the QEC-Bank to output the fermionic spelling of any code. Only four entries have hand-written notes; the rest have none, and a note is not a spelling. Second, check whether two notes in different dialects say the same thing. There is no executable correspondence between steane's majorana-pairs note and honeycomb's parton note; the design document says the three dialects “are the same representation”, but no machine can verify that sentence today. Third, compare costs in the fermion language. A number like “how many pair-parity measurements per period for the honeycomb on Majorana hardware” is not on the card, because nobody has computed it and no program can. In plain words: the QEC-Bank can fully describe the honeycomb code in one language today; in the second language it has only a few hand-written remarks.

日常词典的一条词条是「一个词 → 另一种语言里的对应词 + 用法」。这部词典的一条词条是：ZX 图的一个基本图元（一根线、一个带角度的 Z 点、一个 X 点、一个把 Z 点和 X 点互换角色的「换颜色」小方块，即哈达玛门 H）→ 费米子语言里的对应物（哪几个半座位、怎样配对、带什么符号）+ 一份证明，证明两边表示同一个操作。此外还要有规则的词条：ZX 的每条改写规则（比如 1.4 节的融合）对应费米子语言里的哪条恒等式，同样附证明。formal（形式化）的意思是：这些证明写成机器能逐行核对的形式，而不是论文里的一段话——就像本系列 《一张机器能核对的「没有更轻逻辑算符」证书》里「声明」与「证书」的区别。<->（双向）：从 ZX 到费米子、从费米子到 ZX 都要能查。人话：合格的答案是一张两栏对照表，每行附一份机器能核对的证明，两个方向都能查。编译器（把整张图翻过去）是第二步，statement 没有把它列为第一份交付物。

An entry in an everyday dictionary is “one word → its counterpart in the other language + usage”. An entry in this dictionary is: one basic element of a ZX diagram (a wire, a Z dot with an angle, an X dot, a small “change colour” box that swaps the roles of Z and X dots, i.e. the Hadamard gate H) → its counterpart in the fermion language (which half-seats, how paired, with which sign) + a proof that both sides denote the same operation. There must also be rule entries: which identity in the fermion language each ZX rewrite rule (such as the fusion of 1.4) corresponds to, again with proof. formal means these proofs are written in a form a machine can check line by line, not as a paragraph in a paper, the distinction between “claim” and “certificate” of this series' “A machine-checkable certificate that nothing lighter exists”. <-> (two-way): lookups from ZX to fermionic and from fermionic to ZX must both work. In plain words: a qualifying answer is a two-column table, each row with a machine-checkable proof, consultable in both directions. The compiler (translating whole diagrams) is step two; the statement does not list it as the first deliverable.

Wanted 板给难度只有两个词。hard-engineering = 「知道怎么做，但要做很多」；open-research = 「没人知道怎么做」。这道题是 open-research，关键词是 statement 里的 plausibly：「ZX 编译与费米子编译有相同的计算内容」是一个很可能成立的判断，不是定理。一部双向词典存在的前提是两种语言表达力相当；如果某种 ZX 图元在费米子语言里没有对应物（或反过来），词典就有空行，编译器就翻不过去。今天没有人证明过两边相当，也没有人找到反例。人话：先要知道两种语言能不能互译，才谈得上怎么编词典；第一个问题今天还没有答案。

The Wanted board grades difficulty with two words only. hard-engineering = “known how, but much to do”; open-research = “nobody knows how”. This item is open-research, and the key word is plausibly in the statement: “ZX compilation and fermionic compilation have the same computational content” is a judgement that probably holds, not a theorem. A two-way dictionary presupposes that the two languages are equally expressive; if some ZX element has no counterpart in the fermion language (or the reverse), the dictionary has blank rows and the compiler cannot translate. Nobody has proved the two sides equivalent, and nobody has found a counterexample. In plain words: one must first know whether the two languages are inter-translatable before discussing how to write the dictionary; the first question has no answer today.

一、「统一」是一个决定，不是一个已证的事实。设计文档 3.2 节把三种方言记为同一种表示，用的词是 “(unified; decided)”——「决定」。同一段接着说这需要 “a curated correspondence note documenting the translation. (That unification note is itself a small research artifact of the QEC-Bank.)” —— 那份说明三种方言如何互译的对照笔记「本身就是一件小的研究成果」，也就是还没写出来。而设计文档 4.1 节的对照表对这一行的现状描述是：“Papers only (arXiv:1703.02973, 1910.12065; Floquet-Majorana folklore) — no tooling, no agreed common syntax”。没有工具，没有公认的统一语法。一部形式化词典需要两边各有一套固定的语法；ZX 有，费米子这边今天有三种方言、零套语法。

One: “unified” is a decision, not a proven fact. The design document, section 3.2, records the three dialects as one representation with the words “(unified; decided)”: decided. The same passage continues that this needs “a curated correspondence note documenting the translation. (That unification note is itself a small research artifact of the QEC-Bank.)”: the note explaining how the three dialects translate into one another “is itself a small research artifact”, that is, not yet written. And the comparison table in section 4.1 of the design document describes the state of this row as “Papers only (arXiv:1703.02973, 1910.12065; Floquet-Majorana folklore) — no tooling, no agreed common syntax”. No tooling, no agreed common syntax. A formal dictionary needs a fixed syntax on each side; ZX has one, the fermionic side today has three dialects and zero syntaxes.

二、那个负号让「局部」变得不局部。回到 1.1 节：费米子的每句话带一个正负号，取决于列举的顺序。常见的做法是固定一个顺序（比如按座位编号），那么「操作座位 j」的符号就会依赖于它前面所有座位的占据数——一个只碰一个座位的费米子操作，翻成量子比特语言会变成一条从 1 号一直拖到 j 号的长泡利串。手算一例：按座位编号的顺序记账，「请一个人坐进 3 号座位」这个操作的符号是 (−1)^{(1、2 号里有人的座位数)}，因为新来的人要排到前面每一位已坐者之后。对 |110⟩ 做，前面有 2 人，符号 +1，得 +|111⟩；对 |100⟩ 做，前面有 1 人，符号 −1，得 −|101⟩。同一个操作，符号由它没碰到的座位决定。翻成量子比特，就要在 1 号和 2 号比特上各放一个 Z 来读出这个符号（Z 在 1 上给 −1），于是变成 Z₁Z₂ 再乘上 3 号比特上的「0 变 1」。这种翻译叫 Jordan–Wigner 变换，座位越靠后，拖的 Z 越长。这就是 1.1 节检查站里「中间有几个人会影响符号」的同一个现象。ZX 图的改写规则是局部的：只看图上相邻的几个点。一条局部的 ZX 规则翻到费米子那边可能带上一长串符号；一条局部的费米子恒等式翻到 ZX 那边可能变成一张跨越许多根线的图。词典的每一条都得处理这本符号账，而且要让机器核对。这是这道题技术上最重的部分，本篇只指出它，不展示任何一条翻译。

Two: that sign makes “local” non-local. Back to 1.1: every fermionic sentence carries a sign depending on the listing order. The usual remedy fixes an order (say by seat number), and then the sign of “act on seat j” depends on the occupations of all seats before it; a fermionic operation touching one seat becomes, in the qubit language, a long Pauli string dragging from seat 1 to seat j. A worked case: with the ledger kept in seat order, the operation “seat a person in seat 3” carries the sign (−1)^{(number of occupied seats among 1 and 2)}, because the newcomer must be listed after every person already seated before it. On |110⟩ there are 2 people before it, sign +1, giving +|111⟩; on |100⟩ there is 1, sign −1, giving −|101⟩. The same operation gets its sign from seats it does not touch. In the qubit language a Z must sit on qubits 1 and 2 to read that sign off (Z gives −1 on a 1), so the operation becomes Z₁Z₂ times “turn 0 into 1” on qubit 3. This translation is called the Jordan–Wigner transformation, and the later the seat, the longer the tail of Z’s. It is the same effect as “the people in between change the sign” in the checkpoint of 1.1. ZX rewrite rules are local: they look at a few adjacent dots. A local ZX rule may carry a long tail of signs on the fermionic side; a local fermionic identity may become a diagram spanning many wires on the ZX side. Every dictionary entry must handle this sign ledger, and let a machine check it. This is the technically heaviest part of the item; this guide points at it and shows no translation.

三、ZX 是量子比特的语言，费米子多一条守恒律。ZX 图里的每根线是一个量子比特，两种颜色对应 X 与 Z 两种基；费米子那边，「座位总人数的奇偶」是一个物理上不能改变的量：自然界允许的每个操作都成对地增减费米子，所以总人数的奇偶永远不变（物理学叫它费米子宇称超选择规则；1.1 节的互换只挪人、不增减人，是一个特例）。ZX 语言没有天生的位置放这条约束——需要为它添加规则，或者只翻译遵守它的那一部分图。词典要说明哪些 ZX 图没有费米子对应物，这本身就是研究内容。

Three: ZX is a qubit language; fermions carry one more conservation law. Every wire in a ZX diagram is a qubit and the two colours are the X and Z bases; on the fermionic side, “the parity of the total number of people” is a quantity physics never changes: every operation nature allows adds or removes fermions in pairs, so the parity of the head count stays fixed (physicists call this the fermion-parity superselection rule; the swaps of 1.1 move people without adding or removing any, a special case). The ZX language has no built-in place for that constraint; rules must be added for it, or only the diagrams that respect it translated. The dictionary must say which ZX diagrams have no fermionic counterpart, and that is research in itself.

和 《大码的码距》不同，这道题的难点不是组合爆炸。1.1 节的符号表只有 2 × 2 = 4 格；ZX 图的基本图元和基本规则各只有个位数到十位数的条目；蜂窝码 L3x3 只有 18 个比特、每轮 9 条两比特检查（18 ÷ 2）。词典的行数不会多，难在每一行该写什么（4.3 节的一、三）和每一行怎样写成机器能核对的证明（4.3 节的二）。人话：这不是一部厚词典，是一部还不知道该用什么字母表来写的词典。

Unlike “Distance for large codes”, the difficulty here is not combinatorial explosion. The sign table of 1.1 has 2 × 2 = 4 cells; ZX has only single- to double-digit numbers of basic elements and rules; the honeycomb L3x3 has 18 qubits and 9 two-qubit checks per round (18 ÷ 2). The dictionary will not have many rows; the difficulty is what to write in each row (points one and three of 4.3) and how to write each row as a machine-checkable proof (point two). In plain words: this is not a thick dictionary; it is a dictionary whose alphabet has not been agreed yet.

记号：2 × 2 = 4 与 18 ÷ 2 = 9 是本篇自己算的数；18 来自 instances.yaml 的 L3x3（n = 18 verified），「每轮是一组完美匹配」来自卡片 description 的 “three perfect-matching rounds”（完美匹配 = 每个比特恰好配一对，所以每轮 n/2 条检查）。

Notation: 2 × 2 = 4 and 18 ÷ 2 = 9 are computed for this guide; 18 comes from instances.yaml's L3x3 (n = 18 verified), and “each round is a perfect matching” from the card description's “three perfect-matching rounds” (perfect matching = every qubit in exactly one pair, hence n/2 checks per round).

三个座位 1、2、3 全部有人。把「先说谁」的顺序从 (1,2,3) 换成别的顺序，每一步只允许互换两个座位的内容；按 1.1 节的表，两个有人的座位互换乘 −1。问：到达每一种顺序，总共乘上几个 −1？

Seats 1, 2, 3 are all taken. Change the order of “who is named first” from (1,2,3) to another order, one swap of two seats' contents per step; by the table of 1.1, swapping two taken seats multiplies by −1. Question: reaching each order, how many factors of −1 in total?

六种顺序，三种得 −1、三种得 +1。值得自己验证的一点：到同一个目标顺序可以走不同的路径，路径长短可以不同，但互换次数的奇偶总是一样。请自己试：从 (1,2,3) 出发，用 2 次或 4 次互换到达 (2,3,1) 总能做到，用 1 次或 3 次永远做不到。这条「奇偶不变」叫置换的奇偶性（大学线性代数讲行列式时才正式出现；下面的补课用数数给出证明），它也正是费米子语言里符号有意义的原因：符号只依赖终点，不依赖路径。人话：费米子的符号账记的是「换了奇数次还是偶数次」，一本机器完全能记的账——难的是把它和 ZX 图的每条规则对上。

Six orders, three giving −1 and three giving +1. Worth checking yourself: the same target order can be reached by different paths of different lengths, but the parity of the swap count is always the same; try reaching (2,3,1) from (1,2,3) in 4 swaps, which you will find always possible, and in 3 swaps, which you will find never possible. This “parity invariance” is called the sign of a permutation (formally met in university linear algebra with determinants; the catch-up below proves it by counting), and it is exactly why signs in the fermion language are meaningful: the sign depends on the destination, not on the path. In plain words: the fermionic sign ledger records “odd or even number of swaps”, a ledger a machine can perfectly keep; the hard part is matching it to every ZX rule.

相关条目。yaml 的 related 只有 honeycomb。本教程系列目前没有蜂窝码的独立教程；请直接看 QEC-Bank 的条目页：蜂窝码条目（两个实例、d = ?、三种平台）与它的表示阶梯页（2.1 节那八条记录和 “Open machine-derivation problem →” 链接）。带费米子视图的另外三个条目：steane、surface、toric 的表示阶梯页。表面码与环面码在本系列的教程：Surface Code 条目总览、《从高考数学到 Toric Code》。

Related entries. The yaml's related list has only honeycomb. This tutorial series has no standalone honeycomb guide yet; go straight to the QEC-Bank entry pages: the honeycomb entry (two instances, d = ?, three platforms) and its representation-ladder page (the eight records of 2.1 and the “Open machine-derivation problem →” link). The other three entries with a fermionic view: the ladder pages of steane, surface and toric. Surface and toric codes in this series: Surface Code entry overview, “From Exam Maths to the Toric Code”.

同一系列。ZX 图作为「判断两张时间表是否等价」的工具：《两张测量时间表是不是同一回事》；蜂窝码的 d = ? 从哪里来：《检查每一轮都在变，码距该怎么数》；「formal」（机器可核对）是什么意思：《一张机器能核对的「没有更轻逻辑算符」证书》。这道题的 yaml 没有登记 working_repo。

In this series. ZX diagrams as the tool for “are two schedules equivalent”: “Are two measurement schedules the same thing?”; where the honeycomb's d = ? comes from: “When the checks change every round, how do you count distance?”; what “formal” (machine-checkable) means: “A machine-checkable certificate that nothing lighter exists”. The item's yaml registers no working_repo.

回放一遍。费米子与量子比特只差一个负号：互换两个有人的座位乘 −1，于是每句话带一个由顺序决定的符号。一个座位可以劈成两个半座位（马约拉纳），把不同座位的半座位重新配对、问一对是空是满（再乘一个固定的 ±1 规范变量），就是蜂窝码那种两比特检查——QEC-Bank 卡片用 parton 方言把这件事写成一段 curated 的笔记。同一份「算奇偶」的配方，在只提供 CNOT 与单比特测量的量子比特芯片上常写成辅助比特加两个 CNOT，在设想中能原生测配对奇偶的马约拉纳硬件上是一次配对测量；在两种语言之间翻译并数成本的程序叫编译器。QEC-Bank 今天有四个条目带费米子视图（honeycomb、steane、surface、toric），三种方言，四个 curated，四个 gap 指向本题；条目自己的推导页写着 “There is no generic fermionic compiler in the repository”。Wanted 板要的第一份交付物是一部形式化的 ZX ↔ 费米子词典：每行一个图元或一条规则，附机器能核对的证明。难度 open-research：两种语言是否表达力相当只是「plausibly」；三种方言的「统一」是决定而非证明，没有公认语法；那个负号让局部规则拖出长串符号。

Replay. Fermions differ from qubits by one sign: swapping two taken seats multiplies by −1, so every sentence carries a sign fixed by order. A seat can be split into two half-seats (Majoranas); re-pairing half-seats from different seats and asking whether a pair is empty or taken (times a fixed ±1 gauge variable) is the honeycomb's two-qubit check, which the QEC-Bank card writes, in the parton dialect, as one curated note. The same “compute a parity” recipe is commonly a helper qubit plus two CNOTs on a qubit chip that offers CNOTs and single-qubit measurement, and one pair measurement on proposed Majorana hardware with native pair-parity measurement; a program translating between the two and counting cost is a compiler. Today four QEC-Bank entries carry a fermionic view (honeycomb, steane, surface, toric), three dialects, four curated, four gaps pointing at this item; the entry's own derivation page says “There is no generic fermionic compiler in the repository”. The first deliverable the board asks for is a formal ZX ↔ fermionic dictionary: one element or rule per row, with a machine-checkable proof. Difficulty open-research: whether the two languages are equally expressive is only “plausibly”; the “unification” of three dialects is decided, not proved, with no agreed syntax; and that one sign drags long sign strings out of local rules.