Lightly-Crossed Direct-Sum Local Blocks Force Balanced Cut Rank

Claim/Theorem

Keep the notation of [[quantum-tanner-needs-balanced-local-block-rank-accumulation.md]], [[local-quotient-image-span-controls-rank-accumulation.md]], [[small-side-local-cut-gives-full-local-cross-rank.md]], [[cross-cut-rank-not-monotone-under-stabilizer-extension.md]], [[stabilizer-cut-rank-functional.md]], and [[balanced-cut-rank-to-syndrome-depth.md]].

There is a concrete conditional local-block criterion that yields direct linear balanced cut rank without passing through a cut-uniform dense \(k\)-connected set.

Let \(H\) be the binary qubit-support matrix of the chosen local-generator presentation of a Quantum Tanner code, with local row blocks \(H(v)\) supported on \(Q(v)\) and local quotient images

\[ W_v(L) := \frac{S(v)+B(L)}{B(L)} \le \frac{S}{B(L)} \]

as in [[local-quotient-image-span-controls-rank-accumulation.md]].

Fix a balance parameter \(0<\beta\le 1/2\). Assume there is a constant \(c_{\mathrm{pack}}>0\) such that for every \(\beta\)-balanced cut \(L \subseteq Q\) one can choose a vertex family \(F(L)\) satisfying all of the following.

1. Light local crossing:

   for every \(v \in F(L)\),

   \[ m_v(L) := \min\{|L \cap Q(v)|,\ |R \cap Q(v)|\} < \delta \Delta, \]

   so by [[small-side-local-cut-gives-full-local-cross-rank.md]],

   \[ \chi_L(S(v)) = m_v(L). \]

2. No survival loss on the selected local blocks:

   \[ \dim W_v(L)=m_v(L) \qquad \forall\,v\in F(L). \]

3. No overlap loss on the selected local blocks:

   \[ \sum_{v\in F(L)} W_v(L) \text{ is a direct sum in } S/B(L). \]

4. Linear total light-side mass:

   \[ \sum_{v\in F(L)} m_v(L)\ge c_{\mathrm{pack}}\,|Q|. \]

Then every \(\beta\)-balanced cut \(L\) satisfies

\[ \lambda_{M(H)}(L) = \chi_L(S) \ge c_{\mathrm{pack}}\,|Q|. \]

Consequently, by [[balanced-cut-rank-to-syndrome-depth.md]], any bounded-degree hardware family whose separator cuts are \(\beta\)-balanced obeys the corresponding linear cut-rank syndrome-depth lower bound. In particular, on a static near-square 2D grid with N=\Theta(|Q|),

\[ \operatorname{depth}(C)=\Omega(\sqrt{|Q|}). \]

Proof.

1. By [[small-side-local-cut-gives-full-local-cross-rank.md]], each selected local block has full local intrinsic cut rank

   \[ \chi_L(S(v))=m_v(L). \]

2. Hypothesis 2 says that on the selected family there is no survival loss when passing from local cross rank to the global quotient:

   \[ \dim W_v(L)=\chi_L(S(v))=m_v(L). \]

3. Hypothesis 3 says there is also no overlap loss on the selected family, so

   \[ \dim\Big(\sum_{v\in F(L)}W_v(L)\Big) = \sum_{v\in F(L)}\dim W_v(L). \]

4. Since [[local-quotient-image-span-controls-rank-accumulation.md]] gives

   \[ \lambda_{M(H)}(L) = \dim\Big(\sum_v W_v(L)\Big), \]

   one obtains

   \[ \lambda_{M(H)}(L) \ge \dim\Big(\sum_{v\in F(L)}W_v(L)\Big) = \sum_{v\in F(L)} m_v(L) \ge c_{\mathrm{pack}}\,|Q|. \]

This isolates the exact local-block hypothesis that would close the direct balanced-cut-rank route. The remaining task is not to build a global cut-uniform dense connected set, but to prove that every balanced cut contains a linear-mass family of lightly crossed local blocks whose full local cross rank both survives the quotient and packs independently there.

Dependencies

• [[quantum-tanner-needs-balanced-local-block-rank-accumulation.md]]
• [[local-quotient-image-span-controls-rank-accumulation.md]]
• [[small-side-local-cut-gives-full-local-cross-rank.md]]
• [[cross-cut-rank-not-monotone-under-stabilizer-extension.md]]
• [[stabilizer-cut-rank-functional.md]]
• [[balanced-cut-rank-to-syndrome-depth.md]]

Conflicts/Gaps

• This is a conditional theorem, not a sourced family theorem for Quantum Tanner codes.
• The hypotheses are deliberately strong. In particular, [[cross-cut-rank-not-monotone-under-stabilizer-extension.md]] shows why no-survival-loss and no-overlap-loss cannot be inferred by naive monotonicity from a favorable local subfamily.
• The node therefore does not close the route from the currently sourced family package. It isolates the exact local-block statement still missing if one wants a direct balanced-cut-rank theorem without passing through dense connected sets or tangle breadth.
• The canonical frontier remains [[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]]. This node only sharpens one subordinate direct local-block route.

Sources

• 10.48550/arXiv.2202.13641
• 10.48550/arXiv.0805.2199
• 10.48550/arXiv.2206.07571
• 10.48550/arXiv.2508.05095
• 10.48550/arXiv.2109.14599