Contracted Expansion To Cross-Cut Stabilizers

Claim/Theorem

Let a stabilizer code have maximum stabilizer weight at most \(w\), and let \(\bar T\) be the contracted Tanner graph of its chosen stabilizer generators. If \(\bar T\) has local Cheeger constant \(h_\varepsilon(\bar T)>0\), then for any data-qubit subset \(L\) with \(|D\cap L|\le \varepsilon n/2\), the number \(n_{\mathrm{cut}}\) of stabilizer generators whose support intersects both \(L\) and its complement satisfies

\[ n_{\mathrm{cut}}\;\ge\;\frac{2\,h_\varepsilon(\bar T)}{w(w-1)}\,|D\cap L|. \]

Hence any balanced data-qubit cut in a bounded-weight local-expander code cuts linearly many stabilizers.

Dependencies

• None.

Conflicts/Gaps

• The claim depends on the contracted Tanner graph of a fixed stabilizer-generator set. Changing generators can change the connectivity seen by this theorem.
• The theorem counts stabilizers cut by the partition, not routed Tanner edges or SWAP paths. It is therefore naturally matched to stabilizer-measurement circuit lower bounds rather than directly to compilation maps.
• To invoke this theorem from Conjecture 3's Tanner-expansion hypothesis, one still needs [[tanner-to-contracted-expansion-transfer.md]].

Sources

• 10.48550/arXiv.2109.14599