Quantum Hamiltonian simulation (QHS) is one of the fundamental quantum sub-routines in quantum physics and quantum computing. QHS prepares a unitary approximately within some error bound for a given Hamiltonian. It has immense applications in science and engineering, from solving linear systems of equations to finding the ground state energy of an atomic configuration. We focus on the algorithmic side of the QHS problem for different practical Hamiltonian systems (such as signal processing, communication, and chemistry) and analyse their performance. Further, we look for interesting applications where these algorithms may bring some quantum advantage. One of the exciting research problems is to simulate the potential energy Hamiltonian on a practical quantum computer for some atomic configuration. Given a 1-dimensional potential energy function (of an arbitrary nature), we propose a Quantum polynomial approximate encoding algorithm that can offer a complexity improvement over the existing Hadamard basis encoding with complexity for some qubit size.
