The non-stabilizerness cost of quantum state estimation

We study the non-stabilizer resources required to achieve informational completeness in single-setting quantum state estimation scenarios. We consider fixed-basis projective measurements preceded by quantum circuits acting on $n$-qubit input states, allowing ancillary qubits to increase retrievable information. We prove that when only stabilizer resources are allowed, these strategies are always informationally equivalent to projective measurements in a stabilizer basis, and therefore never informationally complete, regardless of the number of ancillas. We then show that incorporating $T$ gates enlarges the accessible information. Specifically, we prove that at least ${2n}/{\log_2 3}$ such gates are necessary for informational completeness, and that $2n$ suffice. We conjecture that $2n$ gates are indeed both necessary and sufficient. Finally, we unveil a tight connection between entanglement structure and informational power of measurements implemented with $t$-doped Clifford circuits. Our results recast notions of ``magic'' and stabilizerness - typically framed in computational terms - into the setting of quantum metrology.