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Stabilizer Cut-Rank Functional

Claim/Theorem

Let \(\mathcal S\) be the stabilizer space measured by a Clifford syndrome-extraction circuit on a hardware graph \(G_{\mathrm{hw}}\). Define

\[ \Xi(\mathcal S,G_{\mathrm{hw}}) \;:=\; \max_{L\subseteq V(G_{\mathrm{hw}}),\ |\partial L|>0} \frac{\chi_L(\mathcal S)}{|\partial L|}, \]

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

Then any Clifford circuit measuring a generating family of \(\mathcal S\) obeys

\[ \operatorname{depth}(C)\;\ge\;\frac{1}{64}\,\Xi(\mathcal S,G_{\mathrm{hw}}). \]

This follows by combining [[cross-cut-stabilizer-rank.md]] with [[stabilizer-measurement-cut-lower-bound.md]] cut by cut.

Therefore the current graph already contains a generator-choice-invariant lower-bound functional that is closer to the conjectured CD(T_n,\mathfrak G) than any specific Tanner-presentation theorem:

  • the hardware side enters only through cut capacities \(|\partial L|\),
  • the code side enters only through the intrinsic quotient ranks \(\chi_L(\mathcal S)\).

So one clean route to Conjecture 3 is now: lower-bound \(\Xi(\mathcal S,G_{\mathrm{hw}})\) for expander-style QLDPC stabilizer spaces on constrained hardware.

Dependencies

  • [[cross-cut-stabilizer-rank.md]]
  • [[cross-cut-stabilizer-rank-rank-formula.md]]
  • [[stabilizer-measurement-cut-lower-bound.md]]

Conflicts/Gaps

  • This functional is rigorous only for Clifford circuits measuring commuting Pauli operators. It is not yet a statement about arbitrary compilers or non-measurement-based emulation strategies.
  • The hard remaining step is structural: current expansion nodes lower-bound cross-cut generators for a chosen presentation, but they do not yet lower-bound the intrinsic quantity \(\chi_L(\mathcal S)\) or, equivalently, the matroid-connectivity quantity from [[cross-cut-stabilizer-rank-rank-formula.md]].
  • The functional is cut-based rather than path-based, so additional work is still needed to identify it directly with a congestion-dilation invariant of a guest graph.

Sources

  • 10.48550/arXiv.2109.14599