Deterministic quantum search with adjustable parameters: implementations and applications

TL;DR

This paper introduces a flexible quantum search framework that allows adjustable parameters, enabling deterministic solutions to problems where previous schemes with fixed parameters were ineffective.

Contribution

It presents a novel, adjustable parameter framework for deterministic quantum search, expanding the mathematical and practical applicability of Grover's algorithm.

Findings

Framework allows arbitrary phase parameters for deterministic search.

Successfully applied to two problems previously unsolvable by fixed-parameter schemes.

Enhances the versatility and scope of quantum search algorithms.

Abstract

Grover's algorithm provides a quadratic speedup over classical algorithms to search for marked elements in an unstructured database. The original algorithm is probabilistic, returning a marked element with bounded error. There are several schemes to achieve the deterministic version, by using the generalized Grover's iteration $G(\alpha,\beta):=S_r(\beta)\, S_o(\alpha)$ composed of phase oracle $S_o(\alpha)$ and phase rotation $S_r(\beta)$. However, in all the existing schemes the value range of $\alpha$ and $\beta$ is limited; for instance, in the three early schemes $\alpha$ and $\beta$ are determined by the proportion of marked states $M/N$. In this paper, we break through this limitation by presenting a search framework with adjustable parameters, which allows $\alpha$ or $\beta$ to be arbitrarily given. The significance of the framework lies not only in the expansion of mathematical form, but also in its application value, as we present two disparate problems which we are able to solve deterministically using the proposed framework, whereas previous schemes are ineffective.