An interactive Three.js explorer for multivariate multicycle (MM) quantum error-correcting codes and their bicycle, tricycle, toric, and cubic-code subfamilies.
Open the app: https://doc.quantumsavory.org/mm-code-explorer/
- Edit cyclic ideals and 2–6 binary generating polynomials independently of the lattice dimension.
- Explore periodic 2D and 3D lattices, or a 4D lattice as consecutive, labeled 3D slices along the fourth axis.
- See exactly one check representative per X/Z translation family, with polynomial colors, distinct qubit blocks, hover identification, and exact support tables.
- Adjust lattice spacing, block separation, and fourth-axis slice spacing; rotate, pan, and zoom the Three.js views.
- Start with seven presets: BB144, TT72, MM486, MM648, 4D toric96, Haah's cubic code at L=4, and the 2D toric code at L=5.
The app is static and runs entirely in the browser. GitHub Actions tests and builds it, then deploys it to GitHub Pages.
The app implements the Koszul construction described in Mian, Gwilliam, and Krastanov, Multivariate Multicycle Codes for Complete Single-Shot Decoding. See also the authors' code database.
Let
S = F₂[x₁,…,x_D] / ⟨x₁^L₁ − 1,…,x_D^L_D − 1⟩,
q = floor(t/2), F₁,…,F_t ∈ S.
A physical qubit is identified by a q-element subset B of {1,…,t} and a lattice coordinate g. There are binomial(t,q) × product(L_i) physical qubits. The number of variables D and the number of polynomials t are independent.
For each monomial with exponent vector e in F_j:
- The X family indexed by a (q−1)-subset R contains
(B = R ∪ {j}, g = +e mod L)for each j outside R. - The Z family indexed by a (q+1)-subset T contains
(B = T \ {j}, g = −e mod L)for each j inside T.
Every displayed representative is anchored at the origin. Adding a common lattice coordinate to its support gives the other members of that family. No translated checks are drawn. Exponents reduce modulo the periods, and coincident monomials cancel over F₂ before supports are constructed. Distinct blocks at the same coordinate remain distinct physical qubits.
This fixes a consistent row-support/transpose convention. Publications may reorder or complement block labels, reverse coordinate orientations, or exchange X and Z names. Here a positive polynomial exponent appears in an X support, and its negative appears in the paired Z construction. Binary-matrix tests verify CSS commutation for all translations and recover known logical dimensions.
The UI uses axes (x,y,z,w); four-variable examples written as (w,x,y,z) in the manuscript have been consistently renamed. Small within-site block offsets are display aids, not additional lattice coordinates. Lines identify check membership, not local hardware couplings. In 4D, all slices in a check panel share a camera, preserving their left-to-right order as they rotate; support links are drawn only in the actual anchor slice w=0.
The ideal field accepts, for example, x^12 - 1, y^6 - 1 or I = ⟨x^{12}-1, y^{6}-1⟩. Generator order is immaterial. Only independent cyclic relations are supported. Period 1 collapses an axis.
Enter one polynomial per line. Examples include x^3 + y + y^2, (1+x)*(1+y), implicit multiplication xy, integer coefficients reduced modulo 2, and inverse monomial shifts such as x^-1. No expression is evaluated as JavaScript.
Interactive limits: 2–4 variables, 2–6 polynomials, periods 1–32, at most 12 fourth-axis slices, 4,096 sites, and 24,576 physical qubits. Code distances in preset names are attributed to the cited literature; the app does not calculate distances or make decoding-performance claims for custom inputs.
Requires Node.js 22 or later.
npm ci
npm run dev
npm test
npm run buildnpm test verifies polynomial arithmetic, input validation, qubit-block identity, full translated X/Z commutation, and independently calculated binary ranks for published examples, plus five- and six-generator constructions.
For the browser smoke test, start the dev server and install Chromium or use a local installation:
CHROMIUM_PATH=/usr/bin/chromium npm run test:browserAPP_URL can point the same browser checks at a deployed site. The tests cover all presets, dimension views, filters, custom input, validation, camera interaction, spacing, support tables, and mobile overflow. Screenshots are written to ignored .artifacts/.
Drag to rotate 3D views; drag to pan 2D views. Scroll to zoom and right-drag to pan. Focus a view and use arrow keys to rotate/pan and + / − to zoom. Exact supports remain accessible in HTML tables if WebGL is unavailable.
The repository's Pages source is GitHub Actions. Pushes to main run .github/workflows/pages.yml, which tests, builds, and publishes dist/. Relative asset paths allow hosting under the repository subpath. No backend or environment secrets are required.
Application source is MIT licensed. The cited research belongs to its respective authors.