A Bi-directional Quantum Search Algorithm

TL;DR

This paper introduces a novel bi-directional quantum search algorithm that combines partial Grover's search with bi-directional tactics, significantly improving efficiency over traditional Grover's algorithms for larger qubit systems.

Contribution

It proposes the Bi-Directional Grover Search (BDGS) algorithm, integrating bi-directional search with partial Grover's search, and demonstrates its superior performance through benchmarking.

Findings

Requires fewer iterations than traditional Grover's algorithms.

Outperforms Depth-First Grover's Search and standard Grover's Search in tests.

Implementation available on Github for further use and validation.

Abstract

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).