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The time evolution operator plays a crucial role in the precise computationof chemical experiments on quantum computers and holds immense promise foradvancing the fields of physical and computer sciences, with applicationsspanning quantum simulation and machine learning. However, the construction oflarge-scale quantum computers poses significant challenges, prompting the needfor innovative and resource-efficient strategies. Traditional methods likephase estimation or variational algorithms come with certain limitations suchas the use of classical optimization or complex quantum circuitry. Onesuccessful method is the Trotterization technique used for quantum simulation,specifically in atomic structure problems with a gate complexity ofapproximately O(n^2) for an n-qubit realization. In this work, we have proposeda new encoding method, namely quantum approximate time evolution (QATE) for thequantum implementation of the kinetic energy operator as a diagonal unitaryoperator considering the first quantization level. The theoretical foundationsof our approach are discussed, and experimental results are obtained on an IBMquantum machine. Our proposed method offers gate complexity in sub-quadraticpolynomial with qubit size $n$ which is an improvement over previous work.Further, the fidelity improvement for the time evolution of the Gaussian wavepacket has also been demonstrated.

Hamiltonian Encoding for Quantum Approximate Time Evolution of Kinetic Energy Operator