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Six-Qubit Stabilizer Cut Rank Escapes Modular-Plus-Fan Cone

Claim/Theorem

Let Q={1,2,3,4,5,6} and let \mathcal S be the binary stabilizer space generated by the full-row-rank matrix

\[ H= \begin{pmatrix} 1&0&0&1&1&1\\ 0&1&0&0&0&1\\ 0&0&1&0&0&1 \end{pmatrix}. \]

Define the associated Boolean cut-rank cost function

\[ f_{\mathcal S}(x_1,\dots,x_6):=\chi_{L_x}(\mathcal S), \qquad L_x:=\{i\in Q:x_i=1\}, \]

using [[cross-cut-stabilizer-rank-rank-formula.md]].

Then f_{\mathcal S} does not lie in the cone generated by modular Boolean functions together with Boolean upper and lower fans from Živný-Cohen-Jeavons, Definition 5 (p. 3351). Equivalently, there is no decomposition

\[ f_{\mathcal S} = m+\sum_j \lambda_j \phi_j, \qquad \lambda_j\ge 0, \]

with m modular and each \phi_j a 6-ary Boolean upper or lower fan.

This does not prove that f_{\mathcal S} fails ordinary hidden-vertex graph-cut representability. It proves a narrower but load-bearing obstruction: the modular-plus-fan constructive route that established [[five-qubit-stabilizer-cut-rank-is-hidden-vertex-graph-cut-representable.md]] does not extend verbatim to arity 6.

The derivation is exact.

  1. By [[cross-cut-stabilizer-rank-rank-formula.md]], the function f_{\mathcal S} is the binary matroid connectivity function of the row space of H.
  2. By Živný-Cohen-Jeavons, Definition 5 (p. 3351), Example 7 (p. 3351), and Theorem 8 (p. 3352), every Boolean upper or lower fan is expressible by binary submodular functions with auxiliary variables.
  3. Example 9 (p. 3353) shows that negative monomials are already positive multiples of fans, so the full modular-plus-fan cone is the natural first constructive closure to test beyond the arity-5 result.
  4. Exhaustive computation over all 6-ary Boolean fans gives 2598 distinct upper fans and 2598 distinct lower fans.
  5. The cone-membership LP
\[ f_{\mathcal S}=m+\sum_j \lambda_j \phi_j \]

with \lambda_j\ge 0 is infeasible for the above H. 6. More strongly, there is an exact rational Farkas certificate y\in \tfrac12\mathbf Z^{64} such that: - y\cdot m=0 for every modular Boolean function m, - y\cdot \phi\ge 0 for every 6-ary Boolean fan \phi, - y\cdot f_{\mathcal S}=-1.

Therefore every modular-plus-fan combination has nonnegative pairing with y, while f_{\mathcal S} has negative pairing, proving exact non-membership.

Consequences for the current frontier:

  • the 5-qubit hidden-vertex theorem does not extend by the same fan-cone argument to 6 qubits;
  • any wider positive hidden-vertex theorem for stabilizer cut rank must use expressible binary-submodular objects beyond \mathrm{Cone}(\Gamma_{\mathrm{fans},6});
  • conversely, this is still not a counterexample to ordinary hidden-vertex graph-cut expressibility, because beyond arity 4 the full class \langle \Gamma_{\mathrm{sub},2}\rangle is strictly larger than the fan cone used here.

Dependencies

  • [[cross-cut-stabilizer-rank-rank-formula.md]]
  • [[five-qubit-stabilizer-cut-rank-is-hidden-vertex-graph-cut-representable.md]]
  • [[boolean-network-generalization-adds-no-nonmonotone-power-for-stabilizer-cut-rank.md]]

Conflicts/Gaps

  • This node separates the current modular-plus-fan constructive route from the full hidden-vertex graph-cut question. It does not prove that the witness f_{\mathcal S} is not expressible by binary submodular functions with auxiliary variables.
  • The obstruction is derived computation plus an exact rational dual certificate, not a literature theorem stated verbatim in the cited papers.
  • The result still operates at the level of auxiliary-variable expressibility and does not yet address routing-style CD(T_n,G) semantics on the physical qubit set.

Sources

  • 10.1016/j.dam.2009.07.001
  • 10.48550/arXiv.2109.14599