← Shor 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 canonical

      Six Z rows are three disjoint copies of the length-three open-chain parity matrix. Two X rows are the outer length-three parity matrix with each bit expanded across one complete inner block. This executable row order fixes every syndrome and normalizer statement on the entry.

      References: shor1995scheme, gottesman1997stabilizer

    • logical-normalizer-pair derived by constructor.logical_paulis + qeclib.gf2 row-space checks v1

      In the encoded-basis convention, logical X is Z on one site in each inner block and logical Z is X across any complete block. Both have weight three, commute with every stabilizer, lie outside its row space, and anticommute with each other.

      References: shor1995scheme, gottesman1997stabilizer

  2. R2

    Structure

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

    • nested-repetition curated from shor1995scheme

      The inner bit-flip code compares neighboring computational-basis bits. The outer phase-flip code compares the phase of neighboring cat-state blocks. A single Y fault has both components, so its syndrome is the XOR of the two repetition decisions rather than a third decoding problem.

      References: shor1995scheme, gottesman1997stabilizer

    • stabilizer-incidence derived by constructor.build -> checkmatrix_scene v1

      The constructor-derived bipartite incidence view places data in three rows. Six short Z-check spokes stay within rows, while two long X-check spokes join adjacent rows. The geometry is a block-structure diagram, not a claim of hardware locality.

      References: gottesman1997stabilizer

    • bacon-shor-gauge-fix curated from bacon2006operator

      The square three-by-three Bacon–Shor subsystem block recovers this stabilizer presentation by promoting its horizontal ZZ gauge pairs to stabilizers while retaining its two X center generators. The directed typed edge therefore runs from Bacon–Shor to Shor; this stabilizer code does not carry the discarded gauge subsystem.

      References: shor1995scheme, bacon2006operator

  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.

    • projective-plane-homology curated from freedman1998projective

      A suitable projective-plane cellulation realizes the same stabilizer subspace as a homological code. This supplies a topological dictionary for the fixed block, not an identity edge to every projective-plane code or a new scalable family claim.

      References: freedman1998projective, bombin2007homological

    • tableau-to-graph-boundary curated from khesin2025universalgraph

      General stabilizer-to-graph conversion can encode an equivalent tableau as graph data after local Clifford bookkeeping. A graph state is not by itself the Shor code: the logical subspace and the equivalence map must remain explicit.

      References: khesin2025universalgraph

    • parameter-identity-boundary curated from cross2025small

      The parameter label [[9,1,3]] does not identify this code uniquely. Other nine-qubit stabilizer constructions can be inequivalent, so an identity claim needs a stabilizer or encoded-basis equivalence witness rather than a matching parameter tuple.

      References: huang2021ninequbit, cross2025small