Grover's search algorithms, including various partial Grover searches, experience scaling problems as the number of iterations rises with increased qubits, making implementation more computationally expensive. This paper combines Partial Grover's search algorithm and Bi-directional Search to create a fast Grover's quantum search algorithm, referred to as Bi-Directional Grover Search (BDGS). We incorporated a bi-directional search tactic with a partial Grover search, starting from an initial state and a single marked state in parallel. We have shown in this article that our novel approach requires $\frac{\pi}{4\sqrt{2}}\sqrt{N}(1-\sqrt{\frac{1}{b^{r/2k}}})$ iterations over regular Grover Search and Partial Grover Search (PGS), which takes $\frac{\pi}{4}\sqrt{N}\sqrt{1-\frac{1}{b}}$ (here, $N=2^r$ elements, $b$ is the branching factor of partial search, and $k= \lceil\log_2b \rceil$). The proposed BDGS algorithm is benchmarked against the state-of-the-art Depth-First Grover's Search (DFGS) and generic Grover's Search (GS) implementations for $2$ to $20$ qubits and provides promising results. The Qiskit Python implementation of the proposed BDGS algorithm is available on Github (https://github.com/hafeezzwiz21/DFGS-BDGS).

翻译：Grover搜索算法（包括各种部分Grover搜索）随着量子比特数增加导致迭代次数上升，实现的计算复杂度随之升高。本文将部分Grover搜索算法与双向搜索相结合，提出了一种快速Grover量子搜索算法——双向Grover搜索（BDGS）。我们采用部分Grover搜索的双向搜索策略，并行地从初始态和单一标记态开始搜索。本文证明，该新方法需要$\frac{\pi}{4\sqrt{2}}\sqrt{N}(1-\sqrt{\frac{1}{b^{r/2k}}})$次迭代，而常规Grover搜索和部分Grover搜索（PGS）需要$\frac{\pi}{4}\sqrt{N}\sqrt{1-\frac{1}{b}}$次迭代（其中$N=2^r$个元素，$b$为部分搜索的分支因子，$k= \lceil\log_2b \rceil$）。针对$2$至$20$量子比特，将所提出的BDGS算法与当前最先进的深度优先Grover搜索（DFGS）及通用Grover搜索（GS）实现进行基准测试，取得了良好结果。所提出的BDGS算法的Qiskit Python实现已发布在GitHub上（https://github.com/hafeezzwiz21/DFGS-BDGS）。