Asymptotic optimality of Grover-Radhakrishnan-Korepin algorithm

TL;DR

This paper proves the asymptotic optimality of the Grover-Radhakrishnan-Korepin algorithm for partial quantum search in the large-block limit using control theory.

Contribution

It provides the first rigorous proof of the GRK algorithm's optimality by formulating the problem as a time-optimal control task and analyzing extremal structures.

Findings

Proves the asymptotic optimality of the GRK algorithm.

Uses Pontryagin maximum principle to analyze control structures.

Shows the optimal extremal has a global-local-global form.

Abstract

Grover's algorithm is a cornerstone of quantum algorithms and is strictly optimal in oracle-query complexity. While the full search problem admits no further improvement, one may trade accuracy for speed in the partial search problem, where the task is to identify only the block containing the target item. The best known quantum algorithm for the partial search problem is the Grover-Radhakrishnan-Korepin (GRK) algorithm, whose optimality has long been conjectured but not proved. In this work, we prove the optimality of GRK in the large-block limit. We formulate partial search as a time-optimal control problem and apply the Pontryagin maximum principle to derive the switching-function dynamics, establish the bang-bang structure of regular extremals, and exclude non-optimal switching patterns. As a result, we show that the optimal regular extremal has the global-local-global form, which yields a control-theoretic proof of the asymptotic optimality of the GRK algorithm in oracle-query complexity.