The object. Put data qubits on the even-even and odd-odd sites of a (2L−1) × (2L−1) checkerboard. The two blocks contain L² and (L−1)² qubits. Put X-checks on the even-odd sites and Z-checks on the odd-even sites; a check acts on the data qubits one lattice step away. Interior checks have weight four and boundary checks have weight three. The resulting family has
[[2L²−2L+1, 1, L]].
The shipped instances are [[13,1,3]], [[25,1,4]], and [[41,1,5]]. The verifier recomputes n and k for all three, enumerates d=3,4 exactly, and reruns a seeded weight-five upper-bound witness for L=5; the matching lower bound is literature-backed. The complete calculation is in the HGP construction transcript and the distance proof.
The idea. A repetition code stores one observable nonlocally and leaves its conjugate local: with checks Z_i Z_{i+1}, one can choose X_L=X⊗...⊗X but Z_L=Z_1. In the Ising picture, a one-site order parameter distinguishes the two ground states. That is fatal for a quantum memory: the Knill–Laflamme condition requires every operator supported in a correctable region to act as a scalar inside the code space.
Gauging the Ising Z₂ symmetry replaces the local order parameter by a local constraint. Qubits live on square-lattice edges; a star A_v=∏_{e∋v}X_e imposes the Gauss law and a plaquette B_p=∏_{e∈∂p}Z_e imposes zero flux. At this commuting-projector fixed point, every sufficiently local operator has only scalar matrix elements within the ground space, while nontrivial encoded action requires a noncontractible string. On a torus there are two independent string qubits. Cutting to a disk with two rough and two smooth boundary segments leaves one. This is the planar surface code introduced through lattice boundaries by Bravyi and Kitaev, built from Kitaev's Z₂ topological code.
One sentence. The surface code is the repetition code with its local order parameter gauged away, then cut so that one conjugate pair of boundary-to-boundary strings survives.
A distance-d quantum code detects every Pauli error of weight below d and corrects arbitrary errors on at most ⌊(d−1)/2⌋ unknown qubits. It must therefore protect both X-like and Z-like directions. Concatenated codes do this recursively, and the threshold theorem proves that sufficiently weak local noise can be suppressed to arbitrary accuracy (Aharonov–Ben-Or).
The engineering problem is not whether concatenation works, but what interaction contract its recursive gadgets impose. Higher levels require protected inner gadgets, verified ancillas, and either long-range connectivity or routing. The surface-code alternative is to hold the microscopic contract fixed:
This locality makes repeated measurement natural and turns decoding into a bulk phase-transition problem. It also imposes two unavoidable prices. First, two-dimensional commuting local checks obey the Bravyi–Poulin–Terhal tradeoff, k d² = O(n). Second, bounded-spread local logical gates in two dimensions are Clifford by the Bravyi–König theorem, so a native logical T gate is unavailable. The surface code is the answer to a specific design choice: pay geometric area and magic-state overhead in exchange for a local syndrome circuit with a reachable threshold.
Physical route: gauge, close, then cut.
Z₂ symmetry gives edge qubits with star and plaquette constraints.L × L torus there are 2L² edge qubits. The L² stars have one product relation and so do the L² plaquettes, giving k=2L²−(2L²−2)=2. The logicals are the two homologically nontrivial loop classes and their duals.Algebraic route: hypergraph product of two open chains.
Let H be the (L−1) × L repetition parity matrix with adjacent 11 rows. For two copies, the constructor forms
H_X = [H ⊗ I_L | I_(L−1) ⊗ Hᵀ],
H_Z = [I_L ⊗ H | Hᵀ ⊗ I_(L−1)].
The two qubit blocks contain L² bit×bit coordinates and (L−1)² check×check coordinates. Commutation is designed in:
H_X H_Zᵀ = H ⊗ Hᵀ + H ⊗ Hᵀ = 0 over F₂.
The open-chain matrix has rank L−1, hence classical dimensions k=1 and kᵀ=0. The HGP dimension formula k_Q=k₁k₂+k₁ᵀk₂ᵀ therefore gives one logical qubit. Closing each chain adds a dependent parity row: kᵀ becomes one and the same product gives the toric code's two logical qubits. Thus “boundary versus handle” is the corank of the classical factor matrix, not a metaphor.
The geometric checkerboard is exactly these Kronecker blocks laid out in the plane. The worked HGP page prints the matrices, coordinate dictionary, locality checks, ranks, and commutation cancellation from the shipped constructor.
H_Zᵀ and H_X are consecutive boundary maps. Z-logicals are relative cycles modulo face boundaries; X-logicals are the dual relative cycles. Multiplying by a stabilizer deforms a string within its relative homology class. Distance is the shortest nontrivial primal or dual relative cycle. See the CSS chain-complex dictionary.2d²−2d+1 data qubits. Recutting the same bulk by 45 degrees puts the data on a d × d array while keeping both crossing lengths equal to d, giving [[d²,1,d]]. With one dedicated check ancilla per independent stabilizer, the totals are respectively 4d²−4d+1 and 2d²−1. The saving is boundary geometry, not a new phase; the rotated construction is described explicitly in Horsman et al..e and m anyons. Moving one around the other produces a minus sign, so their composite has fermionic self-statistics. An e↔m duality wall can end at a twist defect; the patch corners are boundary-changing versions of the same object. This is why twist encodings inherit Majorana-like fusion bookkeeping without containing microscopic Majorana particles.0.143% ± 0.003% logical error per cycle and an empirical suppression factor Λ=2.14 ± 0.02 for a distance increase of two. Λ is a device-and-decoder fit, not a direct measurement of p_th/p.Length. The HGP has one L² qubit block and one (L−1)² block, so n=2L²−2L+1. The constructor and verifier recompute this for every shipped instance.
Dimension. Each open repetition matrix has k=1 and transpose dimension zero. Künneth bookkeeping gives 1·1+0·0=1. Direct rank computation agrees: both quantum check matrices have rank L(L−1), hence k=n−2L(L−1)=1.
Distance. A row of the L × L block is an X-logical and a column is a Z-logical, so d≤L. For the reverse inequality, the Z-logical class has L pairwise-disjoint column representatives, all equivalent modulo Z-stabilizers. Any logical anticommuting with that class must touch every one of those disjoint supports and therefore has weight at least L; the same argument with the X rows handles the conjugate class. Thus d=L. The distance page proves the general statement and then checks the representatives computationally.
Verification status. L=3,4 use exhaustive CSS logical enumeration. For L=5, the lower bound d≥5 is carried by the family theorem and its literature provenance, while the verifier reruns a seeded low-weight search that finds a weight-five logical. The instance is therefore marked bounded, not verified.
Rate and locality. k/n→0 while d=Θ(√n). This saturates the two-dimensional k d²=O(n) tradeoff up to a constant. The complete argument, including exactly where Euclidean bounded range enters, lives on the BPT derivation page. Likewise, the absence of a constant-depth local T gate follows from the full Bravyi–König derivation, not from a slogan about transversality.
4d²−4d+1 physical qubits per logical memory. The rotated patch reduces this to 2d²−1. These totals exclude routing lanes, spare qubits, and factory space, which must be stated separately rather than hidden in “physical qubits.”Transversal CNOT. For two identical CSS blocks, pairwise physical CNOT maps a control X-stabilizer to the product of corresponding control and target X-stabilizers, and maps a target Z-stabilizer to the corresponding product in the opposite direction. The same conjugation sends X_L⊗I → X_L⊗X_L and I⊗Z_L → Z_L⊗Z_L. It is therefore a depth-one logical CNOT when corresponding qubits can interact. Movable neutral-atom arrays have demonstrated improvement of this two-qubit logical operation while scaling surface-code distance from three to seven (Bluvstein et al.); correlated errors still require joint decoding. The transversal operation was the fault-tolerant ingredient: preparation of the distance-five and distance-seven logical input states was not itself fault-tolerant, so this result must not be read as an end-to-end below-threshold memory demonstration.
Lattice surgery. Joining two smooth boundaries introduces local seam checks whose product is Z_L⊗Z_L; the dual rough merge measures X_L⊗X_L. A merge reduces k by one because the measured logical parity is projected, and a split restores two output patches with a recorded Pauli frame. Repeating the merged syndrome for d rounds supplies timelike distance, but the fault-tolerant outcome comes from spacetime decoding, not from a bare majority vote once data and circuit faults are allowed. The worked surgery page derives the seam product, checks the L=3 rectangular merge as n=28,k=1,d=3, and separates the toy measurement-only limit from the full decoder.
Hadamard. Applying H to every data qubit exchanges X and Z checks. For this patch it maps the code to its dual, which is the same geometry rotated by 90 degrees. Transversal H followed by that relabeling is logical H. On fixed hardware, restoring the original orientation is a distance-preserving code deformation taking order d rounds (Horsman et al.).
Folded Clifford gates. If the surface code is maintained in a folded, mirror-dual layout, paired qubits support a constant-depth logical S using phase gates on the fold and CZ-type gates between mirror partners. The logical gate is constant depth on an already folded layout; physically folding or unfolding a planar patch is itself an order-d routing or deformation operation. This distinction is explicit in Moussa's construction.
Braiding and twists. Turning off checks creates holes. A loop around one hole and a string between like holes form a logical pair; moving a primal hole around a dual hole implements CNOT by code deformation. Boundary-type changes create twists, whose braiding supplies single-qubit Clifford operations. These are alternatives within the Clifford menu, not ways around Bravyi–König.
Universal completion. Logical T is supplied by a non-stabilizer resource: inject a noisy magic state, distill it using Clifford operations, and consume it by gate teleportation. The operations listed in card.yaml distinguish surgery, transversal, braiding, twist, and fold-transversal mechanisms so that hardware geometry is never mistaken for an intrinsic property of the code.
Q1: What single rank fact distinguishes this planar code from the toric code? A: The open-chain repetition matrix is (L−1)×L with full row rank, so its transpose code has dimension zero. Closing the chain creates one row relation, making the transpose dimension one. In the HGP formula this changes 1·1+0·0=1 into 1·1+1·1=2.
Q2: Why is d=L a proof rather than a picture of one length-L string? A: A length-L row and column prove only the upper bound. The lower bound uses L disjoint representatives of each logical class: anything anticommuting with the class must touch every representative, hence at least L qubits. The representative equivalence and disjointness are checked in the distance transcript.
Q3: Why does XZZX reach 50% under pure dephasing while the CSS surface code does not? A: Pure Z errors activate only one syndrome type in either description. XZZX adds diagonal parity conservation, splitting that two-dimensional problem into independent repetition codes. CSS lacks this one-dimensional decomposition and retains its roughly 10.9% optimal code-capacity threshold.
Q4: May one compare 10.9%, 2.9%, and 0.57% as three estimates of the same threshold? A: No. They refer respectively to perfect-syndrome code-capacity noise, repeated phenomenological data/measurement noise, and a particular scheduled circuit-level model. Their elementary fault locations and decoders differ. See the model-by-model ledger.
Q5: Why are d surgery rounds not simply “majority vote in time”? A: If only independent seam-outcome flips occur, repeated parity samples do reduce to a repetition code. Real syndrome circuits also contain data and propagated faults, which create correlated spacetime detection events. The logical parity is inferred jointly with those events by a spacetime decoder.
Q6: How many qubits does the unrotated patch consume when ancillas are counted? A: It has 2d²−2d+1 data qubits and one fewer independent checks, so one dedicated ancilla per check gives 4d²−4d+1 total. The rotated totals are d² data plus d²−1 ancillas, or 2d²−1.
Q7: Is physical H on every qubit already a logical H? A: It exchanges the code with its dual. The logical gate is transversal H plus the 90-degree geometry relabeling; restoring a fixed planar footprint requires code deformation. An S gate is constant depth only once the mirror-dual folded layout is available.
Q8: What precisely forbids logical T? A: A bounded-depth, bounded-range circuit on a growing-distance two-dimensional topological Pauli stabilizer code induces a level-two Clifford-hierarchy operation. T is level three. Injection and code switching work because they add a resource or leave an assumption of that theorem.