The object. Start from a honeycomb tiling, put data qubits on its vertices, and color the faces red, green, and blue so that neighboring faces have different colors. Every retained face f carries two checks on the same vertex set,
X_f = product of X on the vertices of f,
Z_f = product of Z on the vertices of f.
The entry fixes one finite family rather than the bulk tiling alone: an odd-distance triangular patch with one boundary of each color. Bulk faces have six vertices and clipped boundary faces have four. The binary face incidence matrix H is used twice, H_X=H_Z=H, so the code is a self-dual CSS code. The face-incidence derivation turns the constructor's integer-grid rule into the exact family ledger
n=(3d^2+1)/4, number of faces=3(d^2-1)/8, and k=1.
For the shipped patches the Atlas verifies [[7,1,3]] and [[19,1,5]] exactly. It recomputes n=37 and k=1 for the distance-seven patch, while its distance remains literature-claimed because the shared verifier does not yet carry a general lower-bound certificate at that size. The distinction is recorded on the logical and distance page and routed to the distance-certificate gap.
The trick. A face color and a Pauli type are independent labels. A green hexagon supports both one X check and one Z check. The three face colors tell you how string excitations branch, deform, and terminate at boundaries; X versus Z tells you which Pauli sector is being checked. Keeping those axes separate is the shortest route through most color-code notation.
One sentence. The triangular 6.6.6 color code uses one three-colored vertex-face incidence pattern for both CSS sectors, leaving one conjugate pair of boundary logical strings and a transversal Clifford structure.
A two-dimensional topological memory wants local bounded-weight checks while its logical operators grow with the patch. A surface-code face has one Pauli role: primal and dual cellulations carry different check sectors. The color code asks whether one cellulation can support both sectors and thereby make some logical symmetries literal physical symmetries.
Three geometric conditions make that possible:
The last condition gives H H^T=0 over F_2. Thus the same matrix may be used for X and Z checks without violating commutation. Because the two check families have identical support, physical Hadamard exchanges them. Because the vertex graph is bipartite, alternating S and S-dagger preserves every face pair and implements a phase gate in the declared logical convention. Pairwise CNOT between aligned blocks follows from the ordinary CSS conjugation rules. The Clifford operations page derives those ideal Pauli maps and keeps routing, syndrome extraction, and recovery outside the word "transversal."
The design buys more symmetry, not a free universal gate set. A bounded-depth, bounded-range logical gate on a growing-distance 2D local stabilizer family is Clifford by the Bravyi–König theorem. A logical non-Clifford operation therefore needs injection, distillation, switching, surgery, or another mechanism that supplies an extra resource or leaves the theorem's assumptions.
The finite boundary is also part of the code, not packaging. The label 6.6.6 specifies that three hexagons meet at a bulk vertex. It does not specify which faces are clipped, which excitations condense, how many logical qubits remain, or the qubit-count formula. This entry's literal boundary parameter is three-color-triangle; a torus, a Pauli-boundary patch, and a different triangular staircase are different formal objects.
1. Build a triangular index set. Write d=2t+1 and take integer points (r,c) satisfying 0<=c<=r<=3t. In row r, mark a face center when c mod 3 = 2-(r mod 3) and treat every other point as a data vertex. The six fixed offsets around a face center list its bulk vertices; clipping those offsets against the triangle produces weight-four boundary faces.
2. Count faces and qubits. Rows 3a, 3a+1, and 3a+2 contain respectively a, a+1, and a+1 face centers. Summing the complete row triples gives 3t(t+1)/2 faces. Subtracting them from all integer points in the triangle leaves 3t^2+3t+1 data vertices. Replacing t=(d-1)/2 gives the formulas in Essence.
3. Use one incidence matrix twice. Let H[f,q]=1 exactly when data qubit q lies on face f. Every row has weight four or six. Distinct rows overlap on either no vertex or the two endpoints of one shared edge. Therefore every entry of H H^T is even, so H_X H_Z^T=0 when H_X=H_Z=H.
4. Count the logical dimension. Boundary peeling makes the face rows independent: any nonempty mod-two sum has an outermost retained face whose exposed boundary vertex is not canceled; remove that layer and repeat. If m=3(d^2-1)/8, the X and Z halves therefore contribute 2m independent stabilizers. Since n-2m=1, the patch encodes one qubit. The reproduction transcript independently runs GF(2) elimination on all three shipped instances.
5. Exhibit the logical pair. Put X on the d data vertices of the left side. Every face meets that side evenly, so the operator commutes with all Z checks. It is not an X stabilizer because the same support filled with Z overlaps it on odd d sites. The X and Z boundary strings therefore anticommute and form one logical pair. This proves an upper bound d_code<=d.
6. Separate the upper witness from the lower bound. A drawn weight-d string never proves that no lighter logical exists. Exact CSS enumeration closes both sides for d=3 and d=5. For d=7, the Atlas recomputes the weight-seven witness but retains the cited family lower bound as a claimed origin. The distance ledger shows exactly where computation stops and literature begins.
7. Decode the correlations you actually measured. One data-qubit Pauli fault touches the three colored faces meeting that vertex in the bulk. A projection decoder turns color-pair sectors into surface-code matching problems and then lifts a compatible correction. A Möbius decoder instead folds all three triangular boundary sectors into one nonorientable matching graph. With repeated extraction, detector events and correlated circuit faults replace the static syndrome. The decoding page keeps these inputs separate.
8. Read operations from the same algebra. Physical H swaps every paired X_f,Z_f. Alternating S on even row-major qubits with S-dagger on odd ones sends X_f to X_f Z_f with no residual face phase and sends the declared logical X to i X_L Z_L. Pairwise physical CNOT gives the CSS logical CNOT map. Surgery and magic-state protocols are then additional spacetime constructions, not more consequences of H_X=H_Z.
H_X=H_Z=H is the universal executable picture. It decides commutation, rank, the finite logical quotient, exact small distances, and the incidence scene without relying on a drawing.The representation and realization ledger states what each dictionary preserves. The guided face-to-boundary tour lets the reader switch the constructor-derived distance-three and distance-five scenes among face color, Pauli incidence, and logical-string modes.
Why odd d. The chosen triangular staircase has side support of odd length. That makes the X and Z representatives on the same side overlap oddly and therefore anticommute. The constructor rejects even inputs rather than silently switching to another boundary family.
Why n=(3d^2+1)/4. For d=2t+1, the integer triangle contains (3t+1)(3t+2)/2 positions and the modular rule marks 3t(t+1)/2 of them as face centers. The remainder is 3t^2+3t+1 data qubits. This is a boundary-specific count, not a consequence of the bulk Schläfli symbol by itself.
Why k=1. There are m=3(d^2-1)/8 independent face supports. Each appears once in the X sector and once in the Z sector, so the stabilizer rank is 2m. Substitution gives k=n-2m=1. The red, green, and blue string descriptions are related representatives of that one quotient.
Why the declared logical has weight d. The left side contains one data site at each step of the boundary staircase after face centers are removed. Every adjacent face meets it on zero or two sites. The support therefore has zero syndrome, and its X/Z copies anticommute because d is odd.
Why the code distance is not automatically verified at every d. The boundary string supplies only an upper witness. A lower bound must exclude every lighter normalizer element outside the stabilizer row space. The shared exact enumerator does that for the seven- and nineteen-qubit instances, but the thirty-seven-qubit member uses the fetch-verified triangular-family result of Chamberland et al. rather than pretending an unavailable machine certificate exists.
Why the family is qLDPC but low-rate. Face-check weight is at most six and each bulk qubit touches three faces in each Pauli sector, independent of d. The family therefore has bounded local degrees, while k=1 and n=Theta(d^2) make its rate vanish. Bounded check weight does not imply positive rate.
6.6.6 as a finite specification. The bulk symbol does not determine boundary clipping, n, k, or the logical basis. Numerical claims from another triangular staircase, a torus, or a 4.8.8 patch do not transfer automatically.d string a distance proof. It proves d_code<=d. The reverse inequality is an exact enumeration or a cited family theorem, and its provenance must remain visible.construction:.Hadamard. Physical H on every data qubit maps X_f to Z_f and Z_f to X_f. Since the two face-support matrices are identical, the stabilizer group is preserved. The same conjugation swaps the declared left-boundary logical X and Z, so it is logical H in this basis.
Phase. The physical vertex graph is bipartite. In the constructor's row-major order the two parts are the even- and odd-numbered qubits. Apply S on the even part and S-dagger on the odd part. Every retained face has equal numbers from the two parts, so the conjugated X check is exactly X_f Z_f with its phase canceled. The odd left boundary contains one more even site, so X_L maps to i X_L Z_L while Z_L stays fixed: logical S in the declared convention. Reversing the bipartition implements the dagger.
Pairwise CNOT. Align two identical patches and apply a physical CNOT to each corresponding pair. Control X checks acquire the matching target X check, target Z checks acquire the matching control Z check, and the other two check types remain fixed. The boundary logical pair undergoes the same CNOT conjugation. Physical alignment, transport, extraction, and recovery remain additional costs.
Lattice surgery. Compatible colored or Pauli boundary sectors can be joined by a temporary strip of checks whose product measures a joint logical Pauli. Fault tolerance requires repeated seam extraction, spacetime decoding, and a Pauli-frame update. The low-overhead cited layout deliberately mixes 6.6.6 and 4.8.8 regions, so its resource result is not a property of an isolated 6.6.6 memory.
Non-Clifford completion. A prepared or distilled magic state, logical Pauli measurement, and feed-forward supply the non-Clifford resource. The fixed-block trapped-ion and multi-distance superconducting demonstrations combine these resources with color-code Clifford operations; they do not demonstrate a universal transversal gate set on one 2D patch. The operations ledger separates the ideal code automorphisms from these protocol and hardware layers.
Q1: Why may the same face support define both an X check and a Z check? A: Every face has even weight and distinct faces overlap on zero or two vertices, so H H^T=0 over F_2. The X/Z symplectic overlap is therefore even for every pair.
Q2: Why does a red face not mean an X-type stabilizer? A: Red is one of three geometric face colors and controls the colored-string dictionary. That red support appears once as an X stabilizer and again as a Z stabilizer; Pauli type is a separate label.
Q3: How do the formulas n=(3d^2+1)/4 and k=1 know about the boundary? A: They count the retained vertices and independent clipped faces of this specific modular triangular staircase. A different cut changes those counts even if its bulk faces are also hexagons.
Q4: What does the left-boundary string prove about distance? A: Its weight d, zero syndrome, and anticommutation with the conjugate string prove an upper bound. Exact enumeration supplies the lower bound for d=3,5; the distance-seven lower bound remains cited and claimed.
Q5: Why is a surface-code projection not already a color-code decoder? A: One physical error contributes to several color-pair projections. The projected matches are correlated and must be lifted to one correction whose full color-code syndrome is correct, including the triangular boundaries.
Q6: Why does alternating S and S-dagger preserve an X face check? A: Each four- or six-vertex face contains equal numbers from the two bipartition classes. The local conjugation phases cancel and the result is X_f Z_f, the product of two existing stabilizers on that face.
Q7: Do transversal H, S, and CNOT make the family universal? A: No. They generate Clifford operations. A non-Clifford gate in the cited protocols comes from a magic state, injection, distillation, switching, surgery, or another resource outside the isolated constant-depth 2D stabilizer gate.
Q8: What exactly was demonstrated at distance three? A: The seven-qubit patch is also the Steane block, so fixed-block experiments verify encoded operations on the smallest family member. They do not by themselves demonstrate increasing-distance suppression across the 6.6.6 family.