A Bi-directional Quantum Search Algorithm

TL;DR

BDGS tackles the scaling inefficiencies of Grover's search by fusing Partial Grover Search with a bi-directional strategy. The method executes forward and backward passes concurrently, fixing next bits via Partial Grover Search and using smaller oracles to reduce circuit depth, while preserving $O(\sqrt{N})$ scaling. The authors derive a closed-form iteration bound $\frac{\pi}{4\sqrt{2}}\sqrt{N}\left(1-\sqrt{\frac{1}{b^{r/2k}}}\right)$ for the average oracle calls and demonstrate performance gains in Qiskit simulations against Depth-First Grover Search and standard Grover. The results indicate BDGS can substantially reduce runtime for structured search problems and point to future work on multi-solution and hybrid quantum-classical approaches for broader applicability.

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 $\fracπ{4\sqrt{2}}\sqrt{N}(1-\sqrt{\frac{1}{b^{r/2k}}})$ iterations over regular Grover Search and Partial Grover Search (PGS), which takes $\fracπ{4}\sqrt{N}\sqrt{1-\frac{1}{b}}$ (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).

Figures (5)

• Figure 1: Quantum Circuit schematic for Bi-directional Grover's Quantum Search ($U_\omega$ and $U_s$ are oracle and diffuser, respectively.)
• Figure 2: Partial Grover Search with Bi-directional Search (BDGS) with $b = 4$. The BDGS steps with $b = 4$ denote dividing the database into four parts for every layer in a bi-directional search with a partial Grover search to get the next two bits of the solution address.
• Figure 3: Design of Oracle for the proposed Bi-directional Grover Search for $b=4$
• Figure 4: Plot for Runtime (s) vs. $\#$Qubit search space for the proposed BDGS, DFGS guo2022, and standard GS.
• Figure 5: Plot for $\#$Iteration vs. $\#$Qubit search space for BDGS, DFGS guo2022, and standard GS.