Efficient quantum tomography of a polynomial subspace

We present a generalized scheme of direct fidelity estimation and introduce a method for performing tomography of the neighborhood space from pure states. Such a tomography method is self-verifying and can be extended to multiple base states with constant overhead, and the condition of the base state is not bounded by limited entanglement. It only requires direct fidelity estimation of the states within the neighborhood space, where the efficiency is determined by the set of possible measurements. We identified a generalized sufficient condition to perform neighborhood tomography with a constant factor overhead irrelevant to the total number of parties of the whole system given the set of possible measurements, and presented some common sets of states satisfying such conditions with common measurement sets