Spectral Signatures of Prime Factorization

We present a protocol for integer factorization for all integers <i>N</i> below a certain cut-off Λ=2d, grounded in the theory of quantum measurement. In this framework, the factorization of an integer N≤Λ is achieved in a number of steps equal to the total number <i>I</i> of primes present in its factorization; explicitly, the procedure consists of a sequence of <i>I</i> quantum measurements. The method requires a single-purpose quantum device designed to perform measurements of an observable with a prescribed spectrum. Crucially, the construction of this device involves solving, once and for all, a set of approximately 2d differential equations, independently of the specific integer to be factorized. We argue that the initialization task of this device can be efficiently implemented on a quantum computer in <i>d</i> steps, thereby decoupling the computational cost of device preparation from the factorization process itself.