Shor's algorithm operates on the principle that integer factorization can be efficiently solved by reducing it to a period-finding problem within a modular exponentiation group, leveraging quantum superposition and entanglement to evaluate periodicity in parallel. The core mechanism involves utilizing the Quantum Fourier Transform (QFT) to extract frequency information from the interference patterns generated by unitary operations that map input states $|x\rangle$ to function values modulo a composite integer $N$, thereby revealing the period $r$. This procedure exploits quantum mechanical phenomena, such as state collapse upon measurement of remainders, to transition computational complexity from exponential in classical computing to polynomial time for factorization.
Shor's algorithm operates on the principle that integer factorization can be efficiently solved by reducing it to a period-finding problem within a modular exponentiation group, leveraging quantum su…