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Quantum Tanner Good-Family Presentation-Invariant 2D Barrier

Claim/Theorem

For any infinite Quantum Tanner family whose existence is certified by Leverrier and Zemor's Theorem 17 (p. 23) together with Theorem 18 (p. 24), the minimal static-2D lower bound is already presentation-invariant.

Indeed, [[quantum-tanner-expander-anchor.md]] records that these families have

\[ k=\Theta(n), \qquad d=\Theta(n). \]

Applying [[2d-syndrome-depth-from-code-parameters.md]] gives, for any 2D-local syndrome-extraction circuit on \(m\) physical qubits,

\[ \Delta\;\in\;\Omega\!\left(\frac{k\sqrt{d}}{m}\right) \;=\; \Omega\!\left(\frac{n^{3/2}}{m}\right). \]

Therefore, in the linear-space regime \(m=\Theta(n)\),

\[ \Delta\;=\;\Omega(\sqrt{n}). \]

This statement is basis-independent: it applies to any syndrome-extraction routine for the code family, not just the chosen local-generator presentation. Since SWAP-only compilation is a special case of the stronger 2D-local syndrome-extraction model, the same lower bound holds a fortiori for SWAP-only compilation on a static near-square grid.

So, for theorem-level Quantum Tanner families, the minimal static-2D target of Conjecture 3 is already solved without any need to analyze generator presentation or intrinsic cut rank.

Dependencies

  • [[quantum-tanner-expander-anchor.md]]
  • [[2d-syndrome-depth-from-code-parameters.md]]

Conflicts/Gaps

  • This node is purely 2D-specific. It does not address more general hardware families, nor does it identify the conjectured congestion-dilation functional.
  • The argument uses only the good code parameters \(k,d\), so it does not exploit the Tanner-expansion structure. For the full Conjecture-3 program, the chosen-presentation and cut-rank routes remain important because they are closer to the desired CD(T_n,\mathfrak G) statement.
  • The theorem applies to theorem-level Quantum Tanner families already guaranteed by Theorem 17 or Theorem 18. If one wants an explicit deterministic local component-code construction, that remains a separate frontier.

Sources

  • 10.48550/arXiv.2202.13641 (Theorem 17, p. 23; Theorem 18, p. 24; via the local PDF bundle built from the Zotero attachment)
  • 10.48550/arXiv.2302.04317 (Theorem 24, p. 12)