← Steane 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-stabilizer derived by qeclib.logicals.logical_basis + qeclib.distance.exact_distance v1

      Six commuting binary symplectic rows are the executable object. Their centralizer modulo their row span exposes one anticommuting logical Pauli pair, and exact enumeration finds the smallest representative at weight three.

      References: gottesman1997stabilizer

  2. R2

    Structure

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

    • hamming-css canonical

      The three rows of one 3-by-7 Hamming parity matrix generate both the X and Z checks. Its seven nonzero columns make a perfect address book for single-qubit syndromes, while Hamming modulo simplex is the one-bit logical quotient.

      References: steane1996error, steane1996multiple, calderbank1996good

    • triangular-color-code curated from bombin2006topological

      Seven qubits occupy the vertices of a three-face triangular patch. Each face carries paired X and Z checks; the color-code geometry turns the equality of the two check spaces into a visual explanation of the transversal Clifford gates.

      References: bombin2006topological

  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.

    • zx curated from duncan2014quantomatic

      The encoder and recovery become a ZX diagram whose stabilizer identities are proved by graph rewriting. This view is useful for checking whole circuits, not just the commutation of the final six generators.

      References: duncan2014quantomatic

    • tensor-network curated from cao2022quantumlego

      Two [[4,2,2]] code tensors compose into the Steane encoding tensor. Operator pushing explains how local tensor constraints become the six stabilizers and makes the block reusable inside larger networks.

      References: cao2022quantumlego

    • fermionic — majorana-pairs curated from bravyi2010majorana

      The CSS-to-Majorana dictionary maps a single odd-length Steane block to a formal half-integer Majorana encoding; two copies are needed for an ordinary Majorana code with paired odd logical operators. The caveat is part of the dictionary, not a new seven-Majorana realization.

      References: bravyi2010majorana

      Open machine-derivation problem →