Quantum circuit for the NP-Complete problem of Quadratic Congruences

This work proposes quantum circuits for the NP-complete problem of Quadratic Congruences and its optimization version. Given integers A, B, and C, the problem consists of finding x < C such that x² ≡ A (mod B). The optimization version aims to minimize the difference x² − A (mod B). The proposed circuits achieve a quadratic speedup compared to brute-force testing of all possible solutions. Simulation results for these circuits validate their feasibility for finding solutions in both versions.