Skip to content

Quantum Tanner LTC Package Still Misses Dense Intrinsic Connectivity

Claim/Theorem

The current source-grounded Quantum Tanner / left-right-Cayley package still does not imply balanced intrinsic cut rank.

More precisely, the graph already establishes all of the following:

  1. [[quantum-tanner-theorem17-parity-expander.md]] gives a deterministic parity-Tanner local-expander theorem for the chosen local-generator presentation of the explicit Quantum Tanner family.
  2. [[left-right-cayley-ltc-from-local-agreement-plus-expansion.md]] gives local testability of the adjacent left-right-Cayley square code from local agreement plus two-direction expansion.
  3. [[good-code-parameters-do-not-imply-cut-rank.md]] shows that good global parameters [n,k,d] alone do not force balanced intrinsic connectivity.
  4. [[good-ltc-does-not-imply-balanced-cut-rank.md]] shows that local testability alone does not force balanced intrinsic connectivity either.
  5. [[ltc-sparse-cut-product-decomposition.md]] and [[strong-ltc-constraint-graph-small-set-expander.md]] remain tester-graph statements: they constrain sparse cuts of a chosen constraint graph and imply approximate product structure there, but [[cut-rank-is-interface-state-dimension.md]] shows that the Conjecture-3 intrinsic target is exact interface dimension, not approximate tester decomposition.

Therefore the strongest currently supported intrinsic route is matroidal:

  • first prove a dense intrinsically connected object in the original parity-check matroid, such as the dense-breadth hypothesis in [[dense-tangle-breadth-forces-balanced-cut-rank.md]], or an equally strong dense \(k\)-connected set / dense large-rank lean bag;
  • then convert that intrinsic connectivity to stabilizer cut rank, interface-state dimension, and SWAP-only depth.

So the present intrinsic toolkit does not fail because the compiler-side bridge is missing. It fails because the explicit-family package has not yet been upgraded from tester expansion and local agreement to dense intrinsic connectivity in the parity-check matroid itself.

This also clarifies the right language ordering on the live frontier:

  1. matroid connectivity is the strongest exact structural formulation;
  2. interface-state dimension is the operationally equivalent reformulation;
  3. LTC / sparse-cut language is currently only adjacent evidence, not the final bridge.

Dependencies

  • [[quantum-tanner-theorem17-parity-expander.md]]
  • [[left-right-cayley-ltc-from-local-agreement-plus-expansion.md]]
  • [[good-code-parameters-do-not-imply-cut-rank.md]]
  • [[good-ltc-does-not-imply-balanced-cut-rank.md]]
  • [[ltc-sparse-cut-product-decomposition.md]]
  • [[strong-ltc-constraint-graph-small-set-expander.md]]
  • [[cut-rank-is-interface-state-dimension.md]]
  • [[dense-tangle-breadth-forces-balanced-cut-rank.md]]

Conflicts/Gaps

  • This is a synthesis obstruction, not a negative theorem about Quantum Tanner codes themselves.
  • It does not rule out that the left-right-Cayley geometry secretly enforces dense tangle breadth or some equivalent dense connected-set hypothesis; it isolates that as the missing theorem.
  • The node also does not prove that a tester-side sparse-cut theorem can never be upgraded to intrinsic cut rank. It only records that no such exact bridge is currently on the graph.

Sources

  • 10.1109/FOCS54457.2022.00117
  • 10.1145/3519935.3520024
  • dinurLocallyTestableCodes
  • 10.37236/12467
  • 10.48550/arXiv.2109.14599
  • 10.48550/arXiv.0711.1383