← Bacon–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.

    • gauge-center-quotient canonical

      Vertical XX pairs and horizontal ZZ pairs generate a nonabelian gauge group. Multiplying every vertical pair across two neighboring rows gives an X stabilizer; multiplying every horizontal pair across two neighboring columns gives a Z stabilizer. The constructor verifies that these center generators commute with the full gauge group and lie in its span.

      References: bacon2003decoherence, bacon2006operator, aliferis2007subsystem

  2. R2

    Structure

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

    • repetition-quotient derived by qeclib.gf2.rank + Bacon–Shor gauge-span and logical-pair checks v1

      Perfect gauge readout reduces the two Pauli channels to classical repetition inference in perpendicular directions. A bare logical X may be placed on any complete row and a bare logical Z on any complete column; parallel representatives differ only by gauge action. The engine constructs the protected quotient C(S)/G, exactly enumerates the three shipped finite instances, and independently validates any seeded upper-bound witness. Scalable lower bounds and presentation-equivalence certificates remain open under wanted/subsystem-dressed-distance.

      References: bacon2006operator, aliferis2007subsystem, napp2013optimal

  3. R3

    Dynamics

    Ordered measurement schedules and their instantaneous stabilizer groups.

    • period-four-descendant curated from alam2025board

      Floquet–Bacon–Shor replaces the static gauge-center object with a period-four measurement schedule whose detectors and logical representatives evolve in time. Its superconducting logical-operation demonstration is therefore evidence for a dynamical descendant, not a hardware realization of this card's static check matrix.

      References: alam2025board, sun2025floquetbacon

  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.

    • compass-hamiltonian curated from dorier2005compass

      Reading the same XX/ZZ links as Hamiltonian terms produces the square quantum-compass pattern and its rigid line symmetries. This dictionary does not turn the ordinary static Bacon–Shor model into a topological or self-correcting memory; locality belongs to the gauge generators, while the stabilizer products grow with a linear dimension.

      References: dorier2005compass, bacon2006operator, sarma2013gauge

    • tensor-leg-reassignment curated from cao2022quantumlego

      A surface-code tensor network can be reorganized by declaring selected dangling legs logical rather than physical. The reassignment exposes the Bacon–Shor weight-two gauge structure and explains the relationship as a change of subsystem bookkeeping, not an equality of stabilizer tables.

      References: cao2022quantumlego

    • frustration-graph-boundary curated from chapman2022freefermion

      The gauge-term anticommutation graph records which perpendicular XX and ZZ links meet once. It is a useful contrast with free-fermion subsystem constructions, but the ordinary Bacon–Shor compass Hamiltonian itself is not labeled free-fermion solvable: the fetched source explicitly treats it as the obstructed relative as of 2026-08-02.

      References: chapman2022freefermion