  The time evolution operator plays a crucial role in the precise computation
of chemical experiments on quantum computers and holds immense promise for
advancing the fields of physical and computer sciences, with applications
spanning quantum simulation and machine learning. However, the construction of
large-scale quantum computers poses significant challenges, prompting the need
for innovative and resource-efficient strategies. Traditional methods like
phase estimation or variational algorithms come with certain limitations such
as the use of classical optimization or complex quantum circuitry. One
successful method is the Trotterization technique used for quantum simulation,
specifically in atomic structure problems with a gate complexity of
approximately O(n^2) for an n-qubit realization. In this work, we have proposed
a new encoding method, namely quantum approximate time evolution (QATE) for the
quantum implementation of the kinetic energy operator as a diagonal unitary
operator considering the first quantization level. The theoretical foundations
of our approach are discussed, and experimental results are obtained on an IBM
quantum machine. Our proposed method offers gate complexity in sub-quadratic
polynomial with qubit size $n$ which is an improvement over previous work.
Further, the fidelity improvement for the time evolution of the Gaussian wave
packet has also been demonstrated.
