← Four-qubit error-detecting 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.

    • global-parity canonical

      The exact definition is two commuting global parity questions, XXXX and ZZZZ. Their common +1 eigenspace is the whole code; the square shown by the viewer is an incidence layout, not a locality claim.

      References: vaidman1996error, rains1997distance

  2. R2

    Structure

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

    • binary-css derived by four_two_two.build plus CheckMatrix.to_css 1

      The X and Z parity-check matrices are each the one-row all-ones matrix. Their even overlap vanishes over GF(2), and their two independent rows leave two logical qubits.

      References: rains1997distance

  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.

    • syndrome-quotient derived by single-qubit symplectic syndrome enumeration 1

      All four X faults have one syndrome, all four Z faults another, and all four Y faults a third. A nonzero result certifies a bad block, while the position has been quotiented away.

      References: linke2017faulttolerant

    • erasure-side-information curated from grassl1997erasure

      If an external process supplies the damaged position, the same distance-two block corrects one erasure. This is a different information model from an unknown-position Pauli channel, not a contradiction.

      References: grassl1997erasure, gingrich2003optical

    • logical-clifford-action derived by Pauli conjugation modulo the stabilizer row space 1

      A weight-two symplectic logical basis makes the block's dense Clifford action explicit: uniform physical H swaps the two logical labels while applying H to each, and uniform physical S induces logical CZ.

      References: meier2012magic, harper2019faulttolerant