A Proposed Quantum Hamiltonian Encoding Framework for Time Evolution Operator Design of Potential Energy Function

The exploration of potential energy operators in quantum systems holds paramount significance, offering profound insights into atomic behaviour, defining interactions, and enabling precise prediction of molecular dynamics. By embracing the Born-Oppenheimer picture, we delve into the intricate quantum evolution due to potential energy, facilitating accurate modelling and simulation of atomic phenomena with improved quantum fidelity. This research delves into time evolution operation due to potential energy functions for applications spanning quantum chemistry and condensed matter physics. Challenges in practical implementation, encompassing the formidable curse of dimensionality and intricate entangled interactions, are thoughtfully examined. Drawing upon seminal works, we lay a robust foundation for comprehensive investigations into potential energy landscapes with two proposed algorithms. In one methodology, we have shown a systematic decomposition of the potential energy function into Hadamard bases with composite construction of Pauli-Z, identity and RZ gates which can construct the unitary time evolution operator corresponding to the potential energy with a very high fidelity. The other method is a trade-off between complexity and fidelity, where we propose a novel quantum framework that can reduce the gate complexity from $Θ(2^n)$ to $Θ(^nC_r )$ (for some $r < n$). The proposed quantum algorithms are capable of efficiently simulating potential energy operators. The algorithms were implemented in simulators and IBM quantum hardwares to prove their efficacy