  Estimation of the energy of quantum many-body systems is a paradigmatic task
in various research fields. In particular, efficient energy estimation may be
crucial in achieving a quantum advantage for a practically relevant problem.
For instance, the measurement effort poses a critical bottleneck for
variational quantum algorithms.
  We aim to find the optimal strategy with single-qubit measurements that
yields the highest provable accuracy given a total measurement budget. As a
central tool, we establish new tail bounds for empirical estimators of the
energy. They are helpful for identifying measurement settings that improve the
energy estimate the most. This task constitutes an NP-hard problem. However, we
are able to circumvent this bottleneck and use the tail bounds to develop a
practical, efficient estimation strategy, which we call ShadowGrouping. As the
name suggests, it combines shadow estimation methods with grouping strategies
for Pauli strings. In numerical experiments, we demonstrate that ShadowGrouping
outperforms state-of-the-art methods in estimating the electronic ground-state
energies of various small molecules, both in provable and practical accuracy
benchmarks. Hence, this work provides a promising way, e.g., to tackle the
measurement bottleneck associated with quantum many-body Hamiltonians.
