  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 {\Theta}(2n) to
{\Theta}(nCr ) (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 hardware to prove their efficacy
