Factoring some integers using Shor’s algorithm on real quantum hardware

Experimental realizations of Shor’s integer factorization algorithm on NISQ devices often rely on heavy circuit compilation that inadvertently encodes prior knowledge of the solution, bypassing the core challenge of quantum arithmetic. In this work, we present a hardware-execution baseline for the quantum order-finding routine where modular exponentiation is kept explicit as a sequence of controlled modular multiplications. We explicitly acknowledge that our approach relies on strong prior structural knowledge: the moduli are deliberately chosen as products of Fermat primes, and that ensures the multiplication constants \(c_k = A^{2^k} \bmod N\) will reduce to identities or powers of two. However, unlike compiled demonstrations that hardcode the final state, this structural advantage is used strictly as an experimental proxy to make explicit arithmetic feasible on current hardware. Every non-trivial controlled modular multiplication is executed and verified by intermediate checkpoint measurements. We ran the resulting circuits on IBM superconducting devices for \(N \in \{51, 85, 255, 771\}\) . For moduli up to \(N = 255\) , we observed Hellinger fidelities up to 0.946, while for \(N = 771\) we quantitatively characterize the signal degradation caused by increased circuit depth and accumulated hardware noise. By reporting logical and device-compiled resource counts, this methodology provides a reproducible reference point for studying depth-induced degradation in structured instances of order-finding and evaluating future hardware-aware optimizations.