← Surface 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 derived by surface.build HGP output -> CheckMatrix.from_css v1

      Expanding the two repetition-factor boundary maps gives sparse binary X and Z check matrices. Their stacked symplectic rows are the universal executable baseline used for commutation, rank, logical, and distance calculations.

      References: tillich2009quantum

  2. R2

    Structure

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

    • hypergraph-product canonical

      Two open length-L repetition chains form the two axes. The two HGP qubit blocks are bit×bit and check×check, while X and Z checks let one parity map act at a time; open-chain corank is exactly what leaves one logical qubit rather than the toric code's two.

      References: tillich2009quantum

    • homological curated from bravyi1998quantum

      The checks are the boundary and coboundary maps of a relative chain complex. Logical strings are relative cycles modulo boundaries, and distance is the shortest nontrivial primal or dual relative cycle.

      References: bravyi1998quantum, bombin2007homological

  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.

    • statistical-mechanics curated from dennis2002topological

      Error chains with a fixed syndrome become disorder sectors of a random-bond Ising model. Maximum-likelihood decoding compares their free energies; minimum-weight matching is the zero-temperature approximation that keeps energy but discards path entropy.

      References: dennis2002topological, wang2003confinement

    • zx curated from kissinger2022phasefree

      A phase-free ZX diagram and a CSS code encode the same binary linear subspace. In this language rough/smooth merge and split operations become spider maps, so lattice surgery is diagram rewriting rather than an opaque stabilizer schedule.

      References: kissinger2022phasefree, debeaudrap2020zx, bombin2023unifying

    • fermionic — quon curated from liu2019quantized

      The toric bulk underlying the planar patch becomes a quantized graph: star and plaquette checks are unified as cycle operators, e/m duality is graph self-duality, and charged string diagrams expose the fermionic composite and twist-defect bookkeeping.

      References: liu2017quon, liu2019quantized

      Open machine-derivation problem →

    • tensor-network curated from schuch2010peps

      The ground space is a PEPS whose virtual Z2 symmetry generates topological degeneracy, local indistinguishability, and anyonic excitations. The code's topological order is therefore a constraint on the local tensor, not merely a property asserted from its Hamiltonian.

      References: schuch2010peps