← Quantum repetition 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.

    • parity-chain-stabilizer canonical

      The open-chain incidence matrix has one row for each neighboring pair. Reading it as Z checks diagnoses bit flips; reading it as X checks diagnoses phase flips. The constructor uses the same binary matrix in both cases, so the basis label changes the protected sector without changing the chain geometry.

      References: gottesman1997stabilizer, chen2021repetitive

    • normalizer-asymmetry derived by constructor.logical_supports + symplectic normalizer checks v1

      In the bit-flip convention, X on every site commutes with all parities and is not a stabilizer, while Z on any one site is its conjugate logical. Their roles exchange in the phase-flip convention. This normalizer calculation exposes both the growing protected-sector distance and the weight-one logical that fixes the full quantum distance.

      References: gottesman1997stabilizer

  2. R2

    Structure

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

    • domain-wall-boundary derived by open-chain parity incidence over F_2 v1

      A connected string of protected-sector flips changes only the parities at its endpoints. Interior flips cancel from the syndrome because every interior check sees two changed qubits. Decoding therefore reconstructs a chain with the observed boundary, not the unknowable microscopic error itself.

      References: chen2021repetitive

  3. R3

    Dynamics

    Ordered measurement schedules and their instantaneous stabilizer groups.

    • spacetime-detection-graph curated from kelly2015preservation

      Repeating the parity measurements adds a time direction. Differences between neighboring syndrome rounds become detection events, and a data fault and a measurement fault occupy differently oriented edges of a planar spacetime graph. Majority vote on the final data is therefore not a decoder for the circuit-level history.

      References: kelly2015preservation, chen2021repetitive, cao2025exactrepetition

  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.

    • ising-inference curated from cao2025exactrepetition

      Under a specified stochastic noise model, candidate error histories can be written as spin configurations and logical decoding compares their partition functions. This statistical-mechanical dictionary belongs to the decoder and noise model; it is not a second stabilizer family.

      References: cao2025exactrepetition, chen2025nishimori