  Many hybrid quantum-classical algorithms for the application of ground state
energy estimation in quantum chemistry involve estimating the expectation value
of a molecular Hamiltonian with respect to a quantum state through measurements
on a quantum device. To guide the selection of measurement methods designed for
this observable estimation problem, we propose a benchmark called CSHOREBench
(Common States and Hamiltonians for ObseRvable Estimation Benchmark) that
assesses the performance of these methods against a set of common molecular
Hamiltonians and common states encountered during the runtime of hybrid
quantum-classical algorithms. In CSHOREBench, we account for resource
utilization of a quantum computer through measurements of a prepared state, and
a classical computer through computational runtime spent in proposing
measurements and classical post-processing of acquired measurement outcomes. We
apply CSHOREBench considering a variety of measurement methods on Hamiltonians
of size up to 16 qubits. Our discussion is aided by using the framework of
decision diagrams which provides an efficient data structure for various
randomized methods and illustrate how to derandomize distributions on decision
diagrams. In numerical simulations, we find that the methods of decision
diagrams and derandomization are the most preferable. In experiments on IBM
quantum devices against small molecules, we observe that decision diagrams
reduces the number of measurements made by classical shadows by more than 80%,
that made by locally biased classical shadows by around 57%, and consistently
require fewer quantum measurements along with lower classical computational
runtime than derandomization. Furthermore, CSHOREBench is empirically efficient
to run when considering states of random quantum ansatz with fixed depth.
