Grover’s Model with Single Oracle Call and Log(N) Different Diffusion Calls

The author has recently presented the Single-Iteration Quantum Search Algorithm, which achieves amplitude-amplification with exactly one oracle call and one π/4 rotation [1]. For N = 2ⁿ search spaces, the algorithm achieves success probability exactly 1/2 with one iteration of oracle operator and one π/4 rotation with log(N) different diffusion calls, compared to O($\sqrt{N}$ ) iterations (O($\sqrt{N}$ ) oracle calls and O($\sqrt{N}$ ) same diffusion calls) for Grover’s algorithm. This work presents a simpler and cleaner derivation for one oracle call and log(N) diffusion calls, while the original proof was based on Cartan-Dieudonné theorem [1]. The standard lower bounds for unstructured quantum search still apply even though the number of oracle calls is reduced from O($\sqrt{N}$ ) to O(1) and the number of diffusion calls is reduced from O($\sqrt{N}$ ) to O(logN).