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Stabilizer Measurement Cut Lower Bound

Claim/Theorem

Let \(C\) be a Clifford circuit measuring commuting Pauli operators \(S_1,\dots,S_r\). For any subset \(L\) of circuit qubits,

\[ \operatorname{depth}(C)\;\ge\;\frac{n_{\mathrm{cut}}}{64\,|\partial L|}, \]

where \(n_{\mathrm{cut}}\) is the number of independent measured operators whose support intersects both \(L\) and its complement, and \(\partial L\) is the set of connectivity-graph edges crossing the partition. This gives a direct guest-cut versus hardware-cut lower bound for stabilizer-measurement circuits and is the cleanest rigorous object on the current graph that resembles the desired congestion-dilation functional.

Dependencies

  • None.

Conflicts/Gaps

  • The theorem applies to Clifford circuits measuring commuting Pauli operators, not to arbitrary compilation maps or non-Clifford emulation strategies.
  • The quantity \(n_{\mathrm{cut}}\) counts independent cross-cut measured operators, not routed Tanner edges directly. Turning it into a statement about CD(T_n,\mathfrak G) still requires a compilation model that identifies the relevant guest interactions.
  • The theorem is architecture-agnostic, but to get concrete lower bounds one still needs a useful partition \(L\) and a lower bound on \(n_{\mathrm{cut}}\) from code structure.

Sources

  • 10.48550/arXiv.2109.14599