ADAPT-VQE stands out as a robust algorithm for constructing compact ans\"atze
for molecular simulation. It enables to significantly reduce the circuit depth
with respect to other methods, such as UCCSD, while achieving higher accuracy
and not suffering from so-called barren plateaus that hinder the variational
optimisation of many hardware-efficient ans\"atze. In its standard
implementation, however, it introduces a considerable measurement overhead in
the form of gradient evaluations trough estimations of many commutator
operators. In this work, we mitigate this measurement overhead by exploiting a
recently introduced method for energy evaluation relying on Adaptive
Informationally complete generalised Measurements (AIM). Besides offering an
efficient way to measure the energy itself, Informationally Complete (IC)
measurement data can be reused to estimate all the commutators of the operators
in the operator pool of ADAPT-VQE, using only classically efficient
post-processing. We present the AIM-ADAPT-VQE scheme in detail, and investigate
its performance with several H4 Hamiltonians and operator pools. Our numerical
simulations indicate that the measurement data obtained to evaluate the energy
can be reused to implement ADAPT-VQE with no additional measurement overhead
for the systems considered here. In addition, we show that, if the energy is
measured within chemical precision, the CNOT count in the resulting circuits is
close to the ideal one. With scarce measurement data, AIM-ADAPT-VQE still
converges to the ground state with high probability, albeit with an increased
circuit depth in some cases