  Quantum algorithms on the noisy intermediate-scale quantum (NISQ) devices are
expected to simulate quantum systems that are classically intractable to
demonstrate quantum advantages. However, the non-negligible gate error on the
NISQ devices impedes the conventional quantum algorithms to be implemented.
Practical strategies usually exploit hybrid quantum classical algorithms to
demonstrate potentially useful applications of quantum computing in the NISQ
era. Among the numerous hybrid algorithms, recent efforts highlight the
development of quantum algorithms based upon quantum computed Hamiltonian
moments, $\langle \phi | \hat{\mathcal{H}}^n | \phi \rangle$ ($n=1,2,\cdots$),
with respect to quantum state $|\phi\rangle$. In this tutorial, we will give a
brief review of these quantum algorithms with focuses on the typical ways of
computing Hamiltonian moments using quantum hardware and improving the accuracy
of the estimated state energies based on the quantum computed moments.
Furthermore, we will present a tutorial to show how we can measure and compute
the Hamiltonian moments of a four-site Heisenberg model, and compute the energy
and magnetization of the model utilizing the imaginary time evolution in the
real IBM-Q NISQ hardware environment. Along this line, we will further discuss
some practical issues associated with these algorithms. We will conclude this
tutorial review by overviewing some possible developments and applications in
this direction in the near future.
