Abstract
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.