The object. Put one qubit at every site of a rows × columns rectangle, using row-major labels q(r,c). Generate a subsystem gauge group from
XX on every vertical nearest-neighbour pair, andZZ on every horizontal nearest-neighbour pair.A vertical XX and a horizontal ZZ that meet at one site anticommute. That is not a defect: gauge generators are allowed to disturb an unprotected gauge subsystem. The protected codespace is fixed only by the commuting center of the gauge group. Multiplying all vertical XX links across one gap between rows produces an X-type center strip; multiplying all horizontal ZZ links across one gap between columns produces a Z-type center strip. The gauge-center derivation proves their centrality and counts every protected and gauge degree of freedom.
For an r × c rectangle the formal ledger is
n = rc, k = 1, gauge qubits = (r-1)(c-1), d_X = c, d_Z = r, and d = min(r,c).
Here d_X names the lightest X-type dressed logical and d_Z the lightest Z-type dressed logical in the entry's Pauli convention. The shared Atlas referee now constructs C(S)/G, enumerates complete Pauli-weight layers, and recomputes the total dressed distance of every shipped finite instance. Those three d claims are therefore verified. The named sector formulas remain literature-attributed because the shared API does not separately certify d_X and d_Z. Beyond the exact-search ceiling, a seeded search may establish an achieved upper bound only; scalable lower bounds and equivalence-safe presentation changes remain on the subsystem dressed-distance gap.
The trick. Bacon–Shor does not make Shor's long repetition constraints disappear. It factorizes each long center measurement into local weight-two gauge measurements whose outcomes can be multiplied afterward. Locality is gained at the measurement interface; the stabilizers in the center still span complete two-row or two-column strips.
One sentence. Bacon–Shor protects one qubit by reducing X and Z errors, modulo local gauge motion, to two perpendicular classical repetition problems.
Shor's nested repetition code has a clean decoding story, but some of its stabilizer checks cover two complete length-c or length-r blocks. Directly coupling an ancilla to such a growing support invites correlated faults and a large circuit. The subsystem move asks a sharper question:
Which local observables contain the same protected syndrome if we agree not to preserve every degree of freedom inside the encoded block?
The answer is the pair-gauge grammar above. Individual gauge outcomes are not simultaneous eigenvalues because perpendicular links can anticommute. Only products in the center define persistent stabilizer syndromes. An error and that error multiplied by a gauge operator act identically on the protected logical qubit, so recovery only needs the correct gauge coset.
This trade has four costs that the word "local" can hide:
(r-1)(c-1), so n-rank(S) is not the protected logical-qubit count;The measurement and fault-tolerance ledger keeps perfect gauge readout, repeated circuits, encoded-ancilla extraction, self-concatenated surgery, and the Floquet descendant separate.
1. Fix the orientation. Qubits are ordered by rows. X-type gauge edges are vertical and Z-type gauge edges are horizontal. Transposing the rectangle and swapping X with Z gives an equivalent convention, but formulas such as d_X=c and d_Z=r depend on declaring this orientation first.
2. Multiply local edges into center strips. For a fixed gap between rows, multiply its c vertical XX edges. Interior cancellations do not occur; instead the product is X on every qubit in the two adjacent rows. Any horizontal ZZ gauge edge meets that strip on zero or two sites, so it commutes. The perpendicular construction gives a Z strip over two adjacent columns. Thus all r-1+c-1 strips commute with the entire gauge group.
3. Count the quotient. There are c(r-1) independent X gauge rows and r(c-1) independent Z gauge rows. Their symplectic supports occupy different halves of the binary matrix, so
rank(G) = c(r-1) + r(c-1) = 2rc-r-c.
The center basis has rank rank(S)=r+c-2. The subsystem identities
k = n - (rank(G)+rank(S))/2
and
g = (rank(G)-rank(S))/2
give one protected qubit and (r-1)(c-1) gauge qubits. The rank ledger derives these formulas from open-chain incidence matrices rather than assuming the generators are independent.
4. Find bare logicals. X on one complete row commutes with every gauge generator: each horizontal Z edge overlaps it twice. Z on one complete column commutes for the perpendicular reason. The two strings cross once and anticommute. Moving to a parallel row or column multiplies by a center strip, which is itself in the gauge group, so the protected action is unchanged.
5. Dress and reduce. Multiplying an X logical by vertical X gauge edges can move its one X within each column independently. What survives gauge motion is the parity in every column. A nontrivial X logical has odd parity in all c columns, so it needs at least one occupied site per column and has weight at least c. A row realizes the bound. Likewise, a Z logical has odd parity in all r rows and has weight exactly r. The dressed-distance proof makes the quotient and the lower bounds explicit.
6. Read two repetition syndromes. Modulo horizontal Z gauge edges, a Z error is completely specified by one parity bit per row. An X center strip between rows i and i+1 reports the XOR of the two adjacent row parities. That is the open repetition-check matrix of length r. X errors reduce to column parities and the length-c repetition check in the other direction. Perfect-syndrome decoding is therefore two classical decisions; noisy gauge extraction is a spacetime inference problem.
7. Gauge-fix without confusing logical names. On the 3 × 3 member, promote all six horizontal ZZ gauge pairs to stabilizers and retain the two X center strips. Their row space equals the repository's nine-qubit Shor stabilizer row space. The neighboring Shor entry chooses encoded-basis names in which its logical X is the Z column and its logical Z is the X row, the reverse of this card's Pauli-type labels. The operation and gauge-fixing page records both the exact row-space equality and this logical-basis dictionary.
G is nonabelian, S=Z(G) fixes the codespace, and protected Paulis live in the centralizer of S modulo G. This is the executable picture used by constructor.py.The representations and realizations ledger states the equivalences and non-equivalences. The guided gauge-to-center tour lets the reader inspect the exact pair edges, center strips, bare logicals, gauge fix, and rectangular counterfactual from constructor data.
Why n=rc. The rectangle has one physical qubit per site and no boundary identifications or deleted sites.
Why only one protected qubit. The local gauge span has rank 2rc-r-c; its center has rank r+c-2. Substituting those two independently recomputed ranks into the subsystem dimension formula leaves k=1 for every r,c >= 2. The remaining (r-1)(c-1) encoded degrees of freedom are gauge qubits, not extra protected logicals.
Why the distances follow the opposite dimension. An X-type dressed logical must carry odd parity in each column and therefore occupies at least c sites; a complete row attains that weight. A Z-type dressed logical must carry odd parity in each row and therefore has weight r; a complete column attains it. The scalar distance is the smaller sector.
Why the square members look symmetric. For r=c=L, X and Z distances both equal L, while the gauge count becomes (L-1)^2. Symmetry exchanges X and Z only after transposing the grid; physical Hadamards without that permutation do not restore the declared orientation.
Why the 3 × 5 member is useful. Its X logical crosses five columns while its Z logical crosses three rows. Aspect ratio can therefore match a noise bias, but the optimal rectangle also depends on physical error rates, extraction circuits, objective, and operation costs. It is not a code-only constant.
instances.yaml marks n, k, and CSS status verified. Distances and gauge counts remain structured literature claims because the shared verifier has no subsystem quotient implementation; pedagogy/reproduce.py supplies an entry-specific independent audit without silently upgrading their provenance.
XX and an incident horizontal ZZ anticommute. Requiring simultaneous +1 eigenvalues would define no common stabilizer subspace. Only center products are stabilizers.n-rank(S) protected qubits. That expression also counts the disposable gauge subsystem. The protected count uses both rank(G) and rank(S); the 3 × 3 block has one protected qubit and four gauge qubits, not five protected qubits.Pairwise CNOT. Apply one physical CNOT between corresponding sites of two equally shaped blocks. An X gauge on the control maps to the product of the matching X gauges on control and target; a Z gauge on the target maps to the matching pair on both blocks. The other two gauge types stay in their own blocks. The gauge group and its center are preserved, and the row-X/column-Z logical pair undergoes logical CNOT. Routing, extraction, and recovery are additional resources.
Hadamard on a square. Physical H swaps X and Z, turning vertical X edges into vertical Z edges and horizontal Z edges into horizontal X edges. A row/ column transposition then restores the declared gauge orientation and swaps the logical row and column. The permutation is part of the gate.
Gauge fixing. Measuring and fixing the horizontal Z gauge edges on the 3 × 3 block yields the Shor stabilizer presentation. More general choices produce compass-code descendants. Which gauges are promoted, how their outcomes set the Pauli frame, and how faults are handled are operation data, not consequences of the word "gauge fix."
Geometry-assisted non-Clifford routes. Dimension-matched rectangular blocks support cited transversal multi-controlled-Z constructions, while biased-noise architectures combine strong aspect ratios with teleportation and verified ancillas. Subsystem lattice surgery consumes compatible boundaries and repeated joint measurements. Measurement-free deformation replaces mid-circuit measurement with coherent correction and extra qubits. The operations ledger separates these mechanisms and their resource boundaries.
Dynamical descendant. Floquet–Bacon–Shor logical operations require a period-four schedule, time-dependent logical tracking, detector decoding, and feed-forward. They are evidence for the descendant named in the card, not verified operations of the static check matrix exported here.
Q1: Why may two Bacon–Shor gauge generators anticommute? A: Gauge outcomes describe an unprotected subsystem and need not be simultaneous stabilizer eigenvalues. The protected subspace is fixed by center elements, which commute with the whole gauge group.
Q2: How does a product of anticommuting local checks become a commuting stabilizer? A: Multiply every parallel pair check across a complete gap. Any perpendicular gauge edge then intersects the product on zero or two qubits, so the total symplectic overlap is even.
Q3: Why does a 3 × 5 block have d_X=5 and d_Z=3 in this convention? A: X gauge motion preserves one parity per column, so a nontrivial X logical needs one occupied site in all five columns. Z gauge motion preserves one parity per row, so a nontrivial Z logical needs one site in all three rows.
Q4: If the 3 × 3 center has four stabilizers, why is k not five? A: Four additional encoded degrees of freedom are gauge qubits. Using rank(G)=12 and rank(S)=4 gives k=9-(12+4)/2=1 and g=(12-4)/2=4.
Q5: What information does perfect Z-error decoding actually use? A: Horizontal Z gauge motion erases all detail except each row's error parity. The two X center syndromes on a three-row block report adjacent differences of those three bits, exactly the classical repetition syndrome.
Q6: Does gauge-fixing Bacon–Shor to Shor preserve the names logical X and logical Z used by both entry cards? A: It preserves the protected Pauli algebra and exactly reproduces the Shor stabilizer row space, but the Shor entry's encoded-basis convention calls the Z column logical X and the X row logical Z. The labels are swapped relative to this card's Pauli-type names.