HGP(rep_L, rep_L) is the unrotated planar surface code

Claim. The shipped constructor produces local CSS checks on n=L²+(L−1)² qubits and encodes exactly one logical qubit.

Distance is intentionally deferred to its own proof. This page establishes the object and its dimension.

1. The classical factor

Let H=rep_parity(L) be the (L−1) × L binary matrix

1 1 0 0 ...
0 1 1 0 ...
      ...
... 0 1 1

Its rows are independent: beginning at the left edge, column zero occurs only in row zero, so a zero linear combination forces the first coefficient to vanish; repeating column by column forces every coefficient to vanish. Therefore

rank(H)=L−1, dim ker(H)=1, and dim ker(Hᵀ)=0.

The one vector in ker(H) is the all-ones repetition word. The zero transpose dimension is the algebraic signature of an open chain.

2. The quantum product

For two copies of H, the hypergraph product puts qubits in two blocks:

bit×bit: L² qubits;
check×check: (L−1)² qubits.

It defines

H_X = [ H ⊗ I_L       | I_(L−1) ⊗ Hᵀ ]
H_Z = [ I_L ⊗ H       | Hᵀ ⊗ I_(L−1) ]

The CSS commutator has one contribution from each block:

(H ⊗ I)(I ⊗ H)ᵀ       = H ⊗ Hᵀ
(I ⊗ Hᵀ)(Hᵀ ⊗ I)ᵀ     = H ⊗ Hᵀ

Their sum is zero over F₂. Block two is not decorative: it supplies the second copy of the would-be commutation defect.

At L=3, the matrices asserted by the shipped computation are

H_X =
100100000 | 1000
010010000 | 1100
001001000 | 0100
000100100 | 0010
000010010 | 0011
000001001 | 0001

H_Z =
110000000 | 1000
011000000 | 0100
000110000 | 1010
000011000 | 0101
000000110 | 0010
000000011 | 0001

Before reduction modulo two, every entry of H_X H_Zᵀ is either zero or two. After reduction, all entries vanish.

3. Why the checks are geometrically local

Use the constructor's coordinates:

bit×bit qubit (a,b) is at (2b,2a);
check×check qubit (a,b) is at (2b+1,2a+1);
X-check (a,b) is at (2b,2a+1);
Z-check (a,b) is at (2b+1,2a).

Every 1 in a check row then joins sites at Manhattan distance exactly one. Interior checks see four neighbors. At the left/right or top/bottom edge one neighbor is absent, leaving weight three. reproduce.py checks this adjacency for every shipped size, not only for the drawn L=3 patch.

The L=3 interleaved lattice is

y=4  q6   Z4   q7   Z5   q8
y=3  X3   q11  X4   q12  X5
y=2  q3   Z2   q4   Z3   q5
y=1  X0   q9   X1   q10  X2
y=0  q0   Z0   q1   Z1   q2

Weight-three Z-checks lie on the top and bottom rough boundaries. Weight-three X-checks lie on the left and right smooth boundaries.

4. Why `k=1`

There are L(L−1) rows in each quantum check matrix. The HGP rank identity gives

rank(H_X)
  = rank(H)L + (L−1)rank(H) − rank(H)²
  = L(L−1),

and the same for H_Z. Hence

k = [L²+(L−1)²] − 2L(L−1) = 1.

Equivalently, the HGP dimension formula gives

k_Q = k₁k₂ + k₁ᵀk₂ᵀ = 1·1 + 0·0 = 1.

If each open chain is closed into an L-edge cycle, the added parity row is dependent. The ordinary kernel remains one-dimensional and the transpose kernel becomes one-dimensional, giving 1·1+1·1=2: the toric code.

5. Recompute it

source .venv/bin/activate
python atlas/surface/pedagogy/reproduce.py