  Obtaining the expectation value of an observable on a quantum computer is a
crucial step in the variational quantum algorithms. For complicated observables
such as molecular electronic Hamiltonians, a common strategy is to present the
observable as a linear combination of measurable fragments. The main problem of
this approach is a large number of measurements required for accurate sampling
of the observable's expectation value. We consider several partitioning schemes
based on grouping of commuting multi-qubit Pauli products with the goal of
minimizing the number of measurements. Three main directions are explored: 1)
grouping commuting operators using the greedy approach, 2) involving non-local
unitary transformations for measuring, and 3) taking advantage of compatibility
of some Pauli products with several measurable groups. The last direction gives
rise to a general framework that not only provides improvements over previous
methods but also connects measurement grouping approaches with recent advances
in techniques of shadow tomography. Following this direction, we develop two
new measurement schemes that achieve a severalfold reduction in the number of
measurements for a set of model molecules compared to previous state-of-the-art
methods.
