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Quantum Tanner Needs Balanced Local-Block Rank Accumulation

Claim/Theorem

Let \(H\) be the binary qubit-support matrix of the chosen local-generator presentation of a Quantum Tanner code. For each vertex \(v\) of the left-right Cayley complex, let \(H(v)\) denote the local row block supported on the \(\Delta^2\) qubits in the face neighborhood \(Q(v)\) and zero elsewhere. For a qubit cut \(L \sqcup R = Q\), define the local crossing mass

\[ \mu_H(L) \;:=\; \sum_v \Big( \operatorname{rank} H(v)_L + \operatorname{rank} H(v)_R - \operatorname{rank} H(v) \Big). \]

Then the abstract dense-breadth hypothesis from [[dense-tangle-breadth-forces-balanced-cut-rank.md]] can be replaced, at the actual Quantum-Tanner construction level, by the concrete qubit-side surrogate

$$ H'_{\mathrm{loc\text{-}acc}}(\beta): \quad \forall\,L\subseteq Q, |L|\in[\beta |Q|,(1-\beta)|Q|], $$ such that

\[ \mu_H(L)=\Omega(|Q|) \qquad\text{and}\qquad \lambda_{M(H)}(L)\ge c\,\mu_H(L) \]

for some constant \(c>0\) independent of \(|Q|\).

Under \(H'_{\mathrm{loc\text{-}acc}}(\beta)\), every \(\beta\)-balanced cut has

\[ \chi_L(\mathcal S)=\lambda_{M(H)}(L)=\Omega(|Q|), \]

so the interface-state dimension is exponential in \(|Q|\) across every balanced decomposition edge by [[cut-rank-is-interface-state-dimension.md]], and the SWAP-only lower-bound route from [[stabilizer-cut-rank-functional.md]] and [[swap-only-compiler-extraction-reduces-cd-to-stabilizer-cut-rank.md]] closes without any appeal to tangles.

This is the sharpest construction-level replacement for the previous abstract missing hypothesis \(H_{\mathrm{dense}}\): the Quantum Tanner source package already exposes the local blocks \(H(v)\) exactly, while the only genuinely new qubit-side statement still missing is the accumulation inequality

\[ \lambda_{M(H)}(L)\ge c\,\mu_H(L). \]

That inequality says that locally crossed tensor-code neighborhoods contribute linearly many independent cross-cut rank constraints in the global parity-check matrix. It is strictly more concrete than dense tangle breadth because it is stated directly on the original qubit matrix.

Construction-level evidence from the actual source package:

  1. The Quantum-Tanner construction places qubits on LRCC faces and, at each vertex \(v\), inserts a translated copy of a fixed local tensor-code basis on \(Q(v)\); this is exactly the block decomposition \(H=\sum_v H(v)\) in qubit coordinates. See the construction on p. 3 of 10.48550/arXiv.2508.05095 and [[quantum-tanner-local-generator-blowup.md]].
  2. For the explicit [36,8,3] instance in Appendix C / Table V of 10.48550/arXiv.2508.05095, with
\[ G=D_4,\quad A=\{s,r,r^3\},\quad B=\{sr,sr^3,r^2\},\quad C_A=[1\ 1\ 1],\quad C_B= \begin{bmatrix} 1&0&0\\ 1&1&1 \end{bmatrix}, \]

the reconstructed qubit-support matrix has \(36\) columns and \(16\) local neighborhoods of size \(9\). 3. In an exact reconstruction of that instance, pairwise neighborhood overlaps occur only in sizes 0, 2, and 3, with multiplicities 40, 24, and 56 among the 120 unordered vertex pairs. 4. Across 5000 random balanced cuts with \(|L|=18\), the global cut rank

\[ \lambda_{M(H)}(L) = \operatorname{rank} H_L + \operatorname{rank} H_R - \operatorname{rank} H \]

ranged from 5 to 11, while the local crossing mass \(\mu_H(L)\) ranged from 19 to 32, and every sampled cut split between 14 and 16 of the 16 local neighborhoods.

So the smallest explicit family member already shows that balanced cuts produce abundant local crossing structure. What is not yet proved is that these local contributions accumulate with bounded cancellation in the global row space. This identifies the exact construction-level invariant still missing.

Dependencies

  • [[quantum-tanner-left-right-cayley-source-package-stops-at-tester-side-structure.md]]
  • [[quantum-tanner-diagonal-expansion-structure.md]]
  • [[quantum-tanner-local-generator-blowup.md]]
  • [[quantum-tanner-incidence-spectral-gap.md]]
  • [[stabilizer-cut-rank-functional.md]]
  • [[cut-rank-is-interface-state-dimension.md]]

Conflicts/Gaps

  • No current source on disk proves the lower bound \(\mu_H(L)=\Omega(|Q|)\) for every balanced qubit cut from the cited Quantum-Tanner family theorems alone, although the incidence/diagonal expansion package makes this the most plausible part of \(H'_{\mathrm{loc\text{-}acc}}\).
  • More importantly, no current source proves the accumulation inequality \(\lambda_{M(H)}(L)\ge c\,\mu_H(L)\). This is the exact place where the argument still leaves sourced construction facts and would need new qubit-side matroid information.
  • The [36,8,3] computation is evidence, not a theorem. It shows that local crossing mass and split-neighborhood counts by themselves do not automatically become global balanced cut rank.
  • The matrix used here is the binary qubit-support matrix attached to the chosen local-generator presentation, as in [[stabilizer-cut-rank-functional.md]]. The explicit-instance paper's [n,k,d] parameters concern the underlying CSS code, not the rank of this support matrix.

Sources

  • 10.48550/arXiv.2206.07571
  • 10.48550/arXiv.2508.05095