Deterministic Quantum Search for Arbitrary Initial Success Probabilities

Abstract: Unstructured search remains as one of the significant challenges in computer science, as classical search algorithms become increasingly impractical for large-scale systems due to their linear time complexity. Quantum algorithms, notably Grover's algorithm, leverages quantum parallelism to achieve quadratic speedup over classical approaches. Its generalization, the amplitude amplification algorithm, extends this advantage to a broader class of search problems. However, both algorithms are inherently probabilistic and fail to guarantee success with certainty, also they offer no quantum advantage when the initial success probability exceeds 0.5. This work presents a deterministic quantum search algorithm that operates effectively for arbitrary initial success probabilities, providing guaranteed success in searching target states. The proposed approach introduces at most one additional iteration beyond the optimal count required by probabilistic quantum search algorithms. The algorithm preserves the quadratic speedup characteristic of quantum search methods. Additionally, an approach is proposed for the exact search of multiple target states within a bounded number of steps. A complete circuit-level implementation of the proposed algorithm is also presented.