  Quantum algorithms designed for realistic quantum many-body systems, such as
chemistry and materials, usually require a large number of measurements of the
Hamiltonian. Exploiting different ideas, such as {importance sampling,}
observable compatibility, or classical shadows of quantum states, different
advanced measurement schemes have been proposed to greatly reduce the large
measurement cost. Yet, the underline cost reduction mechanisms seem distinct
from each other, and how to systematically find the optimal scheme remains a
critical challenge. Here, we address this challenge by proposing a unified
framework of quantum measurements, incorporating advanced measurement methods
as special cases. Our framework allows us to introduce a general scheme~ --
~overlapped grouping measurement, which simultaneously exploits the advantages
of most existing methods. An intuitive understanding of the scheme is to
partition the measurements into overlapped groups with each one consisting of
compatible measurements. We provide explicit grouping strategies and
numerically verify its performance for different molecular Hamiltonians with up
to 16 qubits. Our numerical result shows significant improvements over existing
schemes. Our work paves the way for efficient quantum measurement and fast
quantum processing with current and near-term quantum devices.
