A Prime Sieve Scheme Based on Quantum Statevector Verification and Classical Codebook Deletion

I propose a quantum-classical collaborative prime sieve scheme. The quan- tum part uses a single-ancilla interference circuit to precisely encode the co- sine function of each prime wave into the quantum state amplitudes, and rigorously verifies the correctness of the peak positions through statevector simulation. The classical part uses the phase codebook parameters derived from orbit regularization to directly generate the set of composite positions on the classical side and delete them from the orbit-1 candidate set. Within the range of 31^2 = 961, this scheme sieves out all 17 primes with 100% precision, zero false positives, and zero missed primes. The circuit has been rigorously verified on three independent simulation platforms—Qiskit, MindQuantum, and pyQPanda—with a precision of 10−17. This scheme provides a logically rigorous and hardware-feasible verification tool for quantum number theory.