  Recently, the entanglement dynamics of two harmonic oscillators initially
prepared in a separable-coherent state was demonstrated to offer a pathway for
prime number identification. This article presents a generalized approach and
outlines a deterministic algorithm making possible the implementation of this
theoretical concept on scalable fault-tolerant qubit-based quantum computers.
We prove that the diagonal unitary operations employed in our algorithm exhibit
a polynomial-time complexity of degree two, contrasting with the previously
reported exponential complexity of general diagonal unitaries.
