  We introduce an inductive $n$-qubit pure-state estimation method. This is
based on projective measurements on states of $2n+1$ separable bases or $2$
entangled bases plus the computational basis. Thus, the total number of
measurement bases scales as $O(n)$ and $O(1)$, respectively. Thereby, the
proposed method exhibits a very favorable scaling in the number of qubits when
compared to other estimation methods. Monte Carlo numerical experiments show
that the method can achieve a high estimation fidelity. For instance, an
average fidelity of $0.88$ on the Hilbert space of $10$ qubits is achieved with
$21$ separable bases. The use of separable bases makes our estimation method
particularly well suited for applications in noisy intermediate-scale quantum
computers, where entangling gates are much less accurate than local gates. We
experimentally demonstrate the proposed method in one of IBM's quantum
processors by estimating 4-qubit Greenberger-Horne-Zeilinger states with a
fidelity close to $0.875$ via separable bases. Other $10$-qubit separable and
entangled states achieve an estimation fidelity in the order of $0.85$ and
$0.7$, respectively.
