Exact bounds on quantum partial search algorithm and improving the parallel search

Grover's algorithm provides a quadratic speedup over classical algorithms for searching unstructured databases and is known to be strictly optimal in oracle query complexity, with tight bounds on its success probability. Although the standard Grover search cannot be further accelerated in the full-search setting, a trade-off between accuracy and query complexity gives rise to the partial search problem. The Grover-Radhakrishnan-Korepin (GRK) algorithm is widely regarded as the optimal protocol for this task. In this work, we provide strong evidence for the strict optimality of the GRK operator sequence among all admissible compositions of global and local Grover operators. By exhaustively examining all operator sequences with a fixed number of oracle queries, we show that the GRK structure universally maximizes the success probability. Building on this result, we derive an asymptotically tight upper bound on the maximal success probability for partial search and establish a matching lower bound on the minimal expected number of oracle queries. Furthermore, we investigate parallel quantum search within the partial-search framework. While a direct GRK-based parallelization does not outperform established parallel Grover schemes, we demonstrate that a hybrid strategy combining partial and full search protocols achieves a strictly improved parallel efficiency. Our results clarify the fundamental limits of quantum partial search and its role in optimizing parallel quantum search algorithms.