Representation ladder

• symplectic-css-checks derived by quantum_expander.build -> CheckMatrix.from_css v1

  Applying the symmetric hypergraph product expands the finite factor into sparse binary X- and Z-check matrices. Their stacked symplectic rows are the executable baseline for commutation, rank, logical witnesses, and finite-distance checks.

  References: tillich2009quantum, leverrier2015quantumexpander

• two-sided-tanner-expansion curated from leverrier2015quantumexpander

  Write the classical parity matrix H as a biregular bipartite graph with variable side A and check side B. For every nonempty S within the chosen cutoff, compare its distinct neighborhood with Delta times |S|. The decoder theorem needs this inequality independently on A and B with both deficit parameters strictly below one sixth; a spectral-gap label or expansion of the later quantum Tanner graph is not a substitute.

  References: leverrier2015quantumexpander, hoory2006expander

• symmetric-hypergraph-product canonical

  Apply HGP(H,H). Physical qubits split into A-by-A and B-by-B sectors, while X checks occupy B-by-A and Z checks occupy A-by-B. Each CSS row combines one factor incidence with its transpose partner. The two mixed Kronecker terms therefore appear twice in HX HZ^T and cancel over GF(2).

  References: tillich2009quantum, leverrier2015quantumexpander

• factor-distance-lift derived by constructor._minimum_kernel_word + factor-derived X logical v1

  A kernel word of H can be lifted into one HGP product sector to give a quantum logical witness. The HGP distance theorem combines the classical distance of H with the transpose distance of H^T. The finite constructor exhausts both small kernels and exports one lifted logical, but scalable lower-bound certification remains a separate problem.

  References: tillich2009quantum

No representation at this rung is recorded for this entry.

• nongeometric-syndrome-circuits curated from delfosse2021measurement

  Bounded check weight does not imply bounded-depth nearest-neighbor extraction in a low-dimensional layout. Expansion distributes Tanner neighbors non-geometrically, so a device needs long-range couplers, routing, or growing local depth. Theoretical constant-overhead recovery accounts for a full architecture; it is not an experimental realization of the finite blocks exported here.

  References: delfosse2021measurement, fawzi2018constant

• small-set-flip curated from leverrier2015quantumexpander

  Given a perfect stabilizer syndrome, inspect subsets contained inside one bounded stabilizer support and apply a subset whose flip decreases syndrome weight enough. Expansion ensures that a sufficiently small adversarial error exposes a useful move and that zero syndrome below the protected scale cannot conceal a logical class. This is a generator- local greedy decoder, not bitwise majority voting.

  References: leverrier2015quantumexpander

• decoder-regime-split curated from fawzi2017efficient

  The founding small-set-flip theorem treats bounded-weight adversarial Pauli errors with perfect syndrome. The later random-error analysis uses probability to exclude bad local structures, and peeling follows resolvable erasure structure. These are three statements with different inputs and failure events; none licenses one model-free threshold.

  References: leverrier2015quantumexpander, fawzi2017efficient, prabhu2025peeling