Grover's Algorithm with Diffusion and Amplitude Steering

We review the basic theoretical underpinnings of the Grover algorithm, providing a rigorous and well motivated derivation. We then present a generalization of Grover's algorithm that searches an arbitrary subspace of the multi-dimensional Hilbert space using a diffusion operation and an amplitude amplification procedure that has been biased by unitary {\em steering operators}. We also outline a generalized Grover's algorithm that takes into account higher level correlations that could exist between database elements. In the traditional Grover algorithm, the Hadamard gate selects a uniform sample of computational basis elements when performing the phase selection and diffusion. In contrast, steered operators bias the selection process, thereby providing more flexibility in selecting the target state. Our method is a generalization of the recently proposal pattern matching algorithm of Hiroyuki et al..