01 / DEFINE THE CODE

From polynomials
to parity checks.

Explore a code

02 / ADJUST THE VIEW

Lattice spacing1.0×

Qubit block separation1.0×

Lattice guides Support links

Drag to rotate in 3D. Scroll to zoom. Right-drag to pan. In 2D, drag to pan. Focus a view to use the arrow keys and + / −.

How to read the lattice

Every colored point is a physical qubit in the displayed parity check. Its color identifies the polynomial that contributes it. Faint points show the other qubits.

At each lattice site, small offsets distinguish qubit blocks. A qubit is specified by both its site and its block. The offsets are only for readability.

We display one representative per family at the origin. Translating its support modulo the lattice periods generates that family. Lines indicate support membership, not nearest-neighbor hardware couplings.

With t polynomials, q = ⌊t/2⌋. Blocks are q-element subsets B. X families are (q−1)-subsets R: Fⱼ contributes to B = R ∪ {j} at +e. Z families are (q+1)-subsets T: Fⱼ contributes to B = T ∖ {j} at −e. All coordinates are reduced modulo the periods.

The negative exponents in Z checks implement the circulant transpose. Identical terms cancel over 𝔽₂, including terms that coincide after wrapping. Family labels use this fixed Koszul convention; block order and X/Z naming can differ between publications.

Limits: 2–6 polynomials, 1–32 sites per axis, up to 12 fourth-axis slices, 4,096 lattice sites and 24,576 qubits. Code distances in presets are reported by the cited sources, not computed here.

THE LATTICE / ONE CHECK PER FAMILY

Bivariate bicycle

2D periodic lattice

◇ origin