← Toric 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 ChainComplex.to_checkmatrix plus toric logical-support validation toric-square-v2

      Expanding every vertex star and face boundary gives sparse binary X and Z matrices. Their stacked symplectic rows are the executable baseline for commutation, rank, exact small-instance distance, logical-loop, and stacked-block CNOT checks.

      References: kitaev1997imperfect, kitaev2003fault

  2. R2

    Structure

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

    • periodic-chain-complex canonical

      Qubits are the one-cells of a square cellulation of the torus. The face boundary d2 and vertex incidence d1 satisfy d1 d2=0; HX=d1 and HZ=d2^T. Cycles modulo local boundaries leave the two noncontractible homology directions and hence two logical qubits.

      References: kitaev1997quantum, kitaev2003fault, dennis2002topological

    • cyclic-hypergraph-product derived by pedagogy/reproduce.py cyclic_hgp_checkmatrix cyclic-hgp-exact-v1

      Let C_L be the square cyclic repetition parity map with one check per periodic neighbor pair. HGP(C_L, C_L^T) reproduces the constructor's ordered star and plaquette matrices exactly. The square cyclic factors have both kernel and transpose-kernel sectors; this is why the output has two logical sectors, unlike HGP of the open repetition factors used by the planar Surface entry.

      References: tillich2009quantum, kovalev2012improved

  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.

    • anyon-strings curated from kitaev2003fault

      Open primal and dual strings create endpoint excitations of the two charge types. Closed contractible strings act locally, while crossing noncontractible strings have odd symplectic intersection and therefore realize conjugate logical Paulis.

      References: kitaev2003fault

    • statistical-mechanics curated from dennis2002topological

      Corrections with one syndrome split into homology classes whose total likelihoods become disorder-sector partition functions. Matching keeps a light representative; maximum-likelihood decoding compares class free energies. The phase boundary depends on the noise and syndrome model.

      References: dennis2002topological, wang2003confinement, chubb2021statistical

    • tensor-network curated from schuch2010peps

      A PEPS virtual Z2 symmetry encodes the same topological degeneracy and local indistinguishability. In decoding, tensor contractions instead sum probabilities inside logical classes; the state representation and the decoder use related mathematics but answer different questions.

      References: schuch2010peps, bravyi2014efficient

    • fermionic — majorana-pairs curated from bravyi2018coherent

      Under restricted coherent-noise channels the correction calculation can be mapped to free Majorana fermions and simulated efficiently. This is a channel-specific algorithmic dictionary, not a claim that the physical qubits are tetrons or that arbitrary coherent noise becomes Pauli noise.

      References: bravyi2018coherent, iverson2020coherence

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