  Estimation of physical observables for unknown quantum states is an important
problem that underlies a wide range of fields, including quantum information
processing, quantum physics, and quantum chemistry. In the context of quantum
computation, in particular, existing studies have mainly focused on holistic
state tomography or estimation on specific observables with known classical
descriptions, while this lacks the important class of problems where the
estimation target itself relies on the measurement outcome. In this work, we
propose an adaptive measurement optimization method that is useful for the
quantum subspace methods, namely the variational simulation methods that
utilize classical postprocessing on measurement outcomes. The proposed method
first determines the measurement protocol for classically simulatable states,
and then adaptively updates the protocol of quantum subspace expansion (QSE)
according to the quantum measurement result. As a numerical demonstration, we
have shown for excited-state simulation of molecules that (i) we are able to
reduce the number of measurements by an order of magnitude by constructing an
appropriate measurement strategy (ii) the adaptive iteration converges
successfully even for a strongly correlated molecule of H$_4$. Our work reveals
that the potential of the QSE method can be empowered by elaborated measurement
protocols, and opens a path to further pursue efficient quantum measurement
techniques in practical computations.
