State preparation and evolution in quantum computing: A perspective from Hamiltonian moments

Quantum algorithms on noisy intermediate-scale quantum ( NISQ) devices are expected to soon simulate quantum systems that are classically intractable. However, the non- negligible gate error present on NISQ devices impedes the implementation of many purely quantum algorithms, necessitating the use of hybrid quantum-classical algorithms. One such hybrid quantum- classical algorithm, is based upon quantum computed Hamiltonian moments.j <^> H n j.D E on 1/4 1,2, THORN, with respect to quantum state j.i. In this tutorial review, 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. We also present a computation of the Hamiltonian moments of a four-site Heisenberg model and compute the energy and magnetization of the model utilizing the imaginary time evolution on current IBM-Q hardware. Finally, we discuss some possible developments and applications of Hamiltonian moment methods.