← Haah cubic code

Entry facet

Representation ladder

R1 checks through R5 cross-model views, with canonicality and provenance.

  1. R1

    Checks

    Sparse Pauli generators in symplectic form, including stabilizer and gauge rows.

    • symplectic-css-checks derived by haah_cubic.build periodic offsets -> CheckMatrix.from_css v1

      Periodic translation expands the two local X patterns and their inverted Z partners into sparse binary CSS matrices. The stacked symplectic rows are the finite executable baseline used for commutation, rank, logical, and small-distance calculations.

      References: haah2011local

  2. R2

    Structure

    CSS pairs, chain complexes, lattices, and other structure behind the checks.

    • cube-corner-css canonical

      At each cubic vertex, orbital A and orbital B receive different Pauli factors. For the X generator their supports are the monomials of 1+x+y+z and 1+xy+yz+zx. Spatial inversion followed by exchanging the orbitals gives the Z generator. Keeping the complete corner table is essential: the phrase “two checks per cube” also describes inequivalent models in the founding classification.

      References: haah2011local

  3. R3

    Dynamics

    Ordered measurement schedules and their instantaneous stabilizer groups.

    No representation at this rung is recorded for this entry.

  4. R4

    Circuit & detectors

    Named syndrome-extraction circuits and detector error models.

    No representation at this rung is recorded for this entry.

  5. R5

    Cross-model views

    Executable or illuminating algebraic, ZX, fermionic, tensor, and statistical dictionaries.

    • laurent-polynomial-module derived by constructor._X_OFFSETS/_Z_OFFSETS -> periodic binary CSS rows v1

      Translation becomes multiplication by Laurent monomials, so the infinite stabilizer family is stored in two compact polynomial columns. A finite periodic instance imposes x^L=y^L=z^L=1 and expands every translate into binary CSS rows. Matrix rank then computes k for that particular L; it must not be replaced by a guessed smooth scaling law.

      References: haah2011local, williamson2016fractal

    • no-string-logical-segments curated from haah2011local

      “No string logical operator” is a statement about bounded-width logical segments whose length can grow without bound. The proof deforms long candidate segments until their nontrivial part disconnects. Periodic plane and fractal logical operators remain, so replacing “no string” by “no logical” would reverse the content of the theorem.

      References: haah2011local

    • fractal-syndrome-geometry curated from bravyi2011selfcorrection

      Local Pauli faults create bounded clusters of cube defects, but isolated fracton charge cannot be transported by an ordinary Pauli string. Error patterns and neutral clusters therefore organize recursively rather than as pairs joined by shortest paths, motivating the family-specific renormalization decoder.

      References: haah2011local, bravyi2011selfcorrection

    • bifurcating-renormalization curated from haah2014bifurcation

      Coarse-graining does not return one copy of the same phase as an ordinary fixed-point topological model would. The cubic model bifurcates into multiple sectors, which is a structural phase diagnostic rather than a numerical decoder claim or a finite-size degeneracy formula.

      References: haah2014bifurcation, dua2019bifurcating

    • ungauged-fractal-symmetry curated from williamson2016fractal

      Ungauging turns the stabilizer constraints into a model with fractal subsystem symmetries. This dictionary explains why the natural conserved structures are neither ordinary global symmetries nor freely deformable strings; it changes representation and should not be reported as a logical gate on a fixed periodic block.

      References: williamson2016fractal, vijay2016fracton