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Cross-Cut Stabilizer Rank

Claim/Theorem

Let \(\mathcal S\) be the binary vector space of commuting Pauli operators measured by a syndrome-extraction routine, and fix a cut \(L\) of the physical qubits. Define the local stabilizer subspaces

\[ \mathcal S_L=\{s\in\mathcal S:\operatorname{supp}(s)\subseteq L\}, \qquad \mathcal S_R=\{s\in\mathcal S:\operatorname{supp}(s)\subseteq L^c\}. \]

Define the cross-cut stabilizer rank by

\[ \chi_L(\mathcal S) \;:=\; \dim \mathcal S-\dim(\mathcal S_L+\mathcal S_R) \;=\; \dim\!\left(\mathcal S/(\mathcal S_L+\mathcal S_R)\right). \]

Then \(\chi_L(\mathcal S)\) depends only on the stabilizer space \(\mathcal S\) and the cut \(L\), not on any chosen generating set. Moreover, if \(G\subseteq \mathcal S\) is any generating family of \(\mathcal S\), then the number \(n_{\mathrm{cut}}(G,L)\) of independent generators in \(G\) whose support intersects both \(L\) and \(L^c\) satisfies

\[ n_{\mathrm{cut}}(G,L)\;\ge\;\chi_L(\mathcal S). \]

Indeed, generators supported entirely on one side map to zero in the quotient \(\mathcal S/(\mathcal S_L+\mathcal S_R)\), so the quotient must be spanned by the images of the cross-cut generators.

Thus \(\chi_L(\mathcal S)\) is the intrinsic presentation-invariant version of “how many independent stabilizers must cross the cut.”

Dependencies

  • [[stabilizer-measurement-cut-lower-bound.md]]

Conflicts/Gaps

  • This node is purely algebraic. It does not yet lower-bound \(\chi_L(\mathcal S)\) from Tanner expansion, code parameters, or geometry.
  • The statement is still about commuting-Pauli stabilizer spaces, not arbitrary compiled interaction graphs.
  • A large cross-cut generator count in one chosen presentation does not automatically imply the same lower bound on \(\chi_L(\mathcal S)\); that extra step is exactly one of the current frontiers.

Sources

  • 10.48550/arXiv.2109.14599