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Five-Qubit Stabilizer Cut Rank Satisfies Fsep

Claim/Theorem

Let Q={1,2,3,4,5} and let \mathcal S be any stabilizer space on Q. Define the associated 5-ary Boolean cost function

\[ f_{\mathcal S}(x_1,\dots,x_5):=\chi_{L_x}(\mathcal S), \qquad L_x:=\{i\in Q:x_i=1\}, \]

where \chi_L(\mathcal S) is the intrinsic cut-rank functional from [[cross-cut-stabilizer-rank-rank-formula.md]].

Let F_{\mathrm{sep}}:{0,1}^5\to{0,1}^5 be the conservative Hamming-distance-nonincreasing operation defined in Živný-Cohen-Jeavons, Fig. 2 (p. 9). By Proposition 15 (p. 8) and Theorem 14 (p. 8), F_{\mathrm{sep}} is a multimorphism of every binary submodular language, hence of every function expressible by binary submodular functions with auxiliary variables.

Then every 5-qubit stabilizer cut-rank function satisfies this universal necessary condition:

\[ F_{\mathrm{sep}}\in \operatorname{Mul}(\{f_{\mathcal S}\}). \]

Equivalently, no 5-qubit stabilizer space furnishes an F_{\mathrm{sep}}-based obstruction to hidden-vertex graph-cut expressibility.

This is a derived finite theorem from exhaustive computation:

  1. Enumerate all 374 subspaces of GF(2)^5.
  2. For each subspace, form the cut-rank connectivity function L\mapsto \chi_L(\mathcal S) using [[cross-cut-stabilizer-rank-rank-formula.md]].
  3. Deduplicate these to 92 distinct 5-ary functions on 2^Q.
  4. For each of the 92 functions, check the multimorphism inequality for F_{\mathrm{sep}} on all 32^5=33,554,432 input 5-tuples.

The outcome is exact:

  • no F_{\mathrm{sep}} violation occurs for any of the 92 distinct 5-qubit stabilizer cut-rank functions;
  • the specific 5-qubit binary example from [[simple-binary-connectivity-violates-nonnegative-hypergraph-cut-condition.md]] is among these functions and also satisfies F_{\mathrm{sep}}.

Consequences for the current frontier:

  • the arity-4 positive result of [[four-qubit-stabilizer-cut-rank-is-hidden-vertex-graph-cut-representable.md]] does not break immediately at arity 5;
  • the known exact nonnegative-hypergraph obstruction at 5 qubits does not automatically upgrade to an auxiliary-vertex graph-cut obstruction;
  • the present literature-based hidden-vertex obstruction method on disk, namely the universal F_{\mathrm{sep}} multimorphism test, does not separate the stabilizer subclass even at 5 terminals.

So the live gap is now sharper: either one needs a deeper multimorphism or expressibility obstruction than F_{\mathrm{sep}}, or the binary-matroid/stabilizer subclass may admit a genuinely stronger hidden-vertex graph-cut realization than generic submodular functions do.

Dependencies

  • [[cross-cut-stabilizer-rank-rank-formula.md]]
  • [[four-qubit-stabilizer-cut-rank-is-hidden-vertex-graph-cut-representable.md]]
  • [[simple-binary-connectivity-violates-nonnegative-hypergraph-cut-condition.md]]

Conflicts/Gaps

  • Passing F_{\mathrm{sep}} is only a necessary condition for expressibility over binary submodular functions in arity 5; unlike arity 4, it is not known here to be sufficient.
  • Therefore this node does not prove hidden-vertex graph-cut realizability for all 5-qubit stabilizer cut-rank functions. It proves only that the current universal necessary witness from Živný-Cohen-Jeavons does not obstruct them.
  • The exhaustive check is finite and exact, but it is a derived computation rather than a literature theorem.
  • Even a positive auxiliary-variable expressibility theorem would still not by itself produce 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