A Concept of Controlling Grover Diffusion Operator: A New Approach to Solve Arbitrary Boolean-Based Problems

Abstract A controlled diffuser is designed as a new approach for Grover’s algorithms to search for solutions for arbitrary Boolean oracles, since the conventional diffuser is not capable of searching for solutions for arbitrary Boolean and Phase oracles. This controlled diffuser relies on the states of functional (output) qubit as the reflection of Boolean decisions from a Boolean oracle, without relying on the phase kickback. This article discusses the problems that are designed as Boolean and Phase oracles using the structures of POS, SOP, ESOP, digital logic circuits, and CSP-SAT. Our work concludes that the conventional diffuser only finds the solutions for these problems when their collector gates are in the form of a Boolean AND gate (Toffoli) for Boolean oracles or a multiple-controlled Z gate for Phase oracles, while the controlled diffuser successfully finds the solutions for all Boolean oracles regardless of different Boolean gates of their collector gates, in Grover iterations of O\(\left(\sqrt{\text{N}}\right)\) times. In this article, we proposed new terms of Collector Gate, Calibration Test, and calculated Quantum Cost for this controlled diffuser.