Deterministic Grover search with a restricted oracle

TL;DR

This paper introduces a deterministic version of Grover's quantum search algorithm, ensuring certainty in finding the correct result with minimal additional steps, using generalized phase rotations.

Contribution

The authors develop a two-parameter D2p protocol that guarantees success without oracle control, maintaining Grover's quadratic speedup.

Findings

Achieves 100% success rate with at most one extra iteration

Uses generalized phase rotations instead of phase inversions

Provides Bloch sphere visualization for intuition

Abstract

Grover's quantum search algorithm provides a quadratic quantum advantage over classical algorithms across a broad class of unstructured search problems. The original protocol is probabilistic, returning the desired result with significant probability on each query, but in general, requiring several iterations of the algorithm. We present a modified version to return the correct result with certainty without having user control over the quantum search oracle. Our deterministic, two-parameter "D2p" protocol utilizes generalized phase rotations replacing the phase inversions after a standard oracle query. The D2p protocol achieves a 100% success rate in no more than one additional iteration compared to the optimal number of steps in the original Grover's search enabling the same quadratic speed up. We also provide a visualization using the Bloch sphere for enhanced geometric intuition.