Efficient Implementation of a Quantum Search Algorithm for Arbitrary N

This paper presents an enhancement to Grover's search algorithm for instances where the number of items (or the size of the search problem) N is not a power of 2. By employing an efficient algorithm for the preparation of uniform quantum superposition states over a subset of the computational basis states, we demonstrate that a considerable reduction in the number of oracle calls (and Grover's iterations) can be achieved in many cases. For special cases (i.e., when N is of the form such that it is slightly greater than an integer power of 2), the reduction in the number of oracle calls (and Grover's iterations) asymptotically approaches 29.33%. This improvement is significant compared to the traditional Grover's algorithm, which handles such cases by rounding N up to the nearest power of 2. The key to this improvement is our algorithm for the preparation of uniform quantum superposition states over a subset of the computational basis states, which requires gate complexity and circuit depth of only O \(log₂ (N\)), without using any ancilla qubits.