Sample Complexity for Embedded Multipartite Entanglement Witness via Pauli and Clifford Classical Shadows

Detecting multipartite entanglement in many qubit systems is measurement-intensive, motivating protocols that estimate only selected observables with provable efficiency. In this work we use the classical shadow protocol to study the sample complexity required to estimate a family of subsystem $n$-partite entanglement witness embedded in an larger $N$-qubit system. We derive ensemble dependent variance bounds that lead to qualitatively distinct scaling for the snapshots cost at fixed additive error $\epsilon$ with numerical simulations confirm these trends, exhibiting a clear crossover from Pauli favorable performance for local witness to Clifford favorable performance as the witness becomes more global.