Problem-Solution Symmetry in Grover’s Quantum Search Algorithm

Grant Eberle, Gonzalo E. Ordonez

Haoxiang Guo

We demonstrated the detailed construction of the hybrid quantum-classical computer. Based on this architecture, the useful concept of amplitude interception is illustrated. It is then embedded into a combination of Depth-First Search and Grover's algorithm to generate a novel approach, the Depth-First Grover Search(DFGS), to handle multi-solution searching problems on unstructured databases with an unknown number of solutions. Our new algorithm attains an average complexity of $\mathcal{O}(m\sqrt{N})$ which performs as efficient as a normal Grover Search, and a $\mathcal{O}(\sqrt{p}N)$ complexity with a manually determined constant $p$ for the case with all elements are solutions, where a normal Grover Search will degenerate to $\mathcal{O}(N\sqrt{N})$. The DFGS algorithm is more robust and stable in comparison.

Hao-Peng 昊鹏 Wang 王, Ji-Ze 纪泽 Cai 蔡, Xiong 雄 Fang 方 +3 more

Abstract The objective of Grover’s quantum algorithm is to efficiently identify a single solution within a search space. However, there are search problems that require finding all possible solutions. A direct, repeated application of Grover’s algorithm is unsuitable for these problems, as it lacks an internal mechanism to avoid rediscovering previous solutions. Here we introduce a quantum algorithm that efficiently and deterministically discovers all solutions by preparing special initial states of the index register, a step accomplished by leveraging knowledge from solutions obtained during execution. We experimentally demonstrate our algorithm on a superconducting quantum processor. This work provides quantum examples for search scenarios with some knowledge about solutions.

Kyoichi Okamoto, Osamu Watanabe

治 渡辺, OSAMU WATANABE

Debanjan Konar, Zain Hafeez, Vaneet Aggarwal

Abstract Grover’s search algorithms, including various partial Grover searches (PGS), experience scaling problems when multiple solutions are sought, as the number of iterations increases with the number of solutions or marked states, making implementation more computationally expensive. Inspired by recent PGS algorithms for multi-solution searchers, this article proposes a scalable Grover quantum search algorithm, referred to as Bi-directional Multi-solution scalable Grover Search (BMGS), to efficiently search for an arbitrary number of solutions from an unstructured database. We introduced a novel multi-segment bidirectional search tactic with PGS across multiple equal segments of each state, starting from an initial state and multiple marked states in parallel, obviating the need for merge operations. We have shown in this work that for each solution our novel approach requires at most $$\sqrt{\mathcal {N}}\left( 1- \sqrt{\frac{1}{b^{\lfloor \frac{r}{dk}\rfloor }}}\right) $$ N 1 - 1 b ⌊ r dk ⌋ iterations (here, $$\mathcal {N}=2^r$$ N = 2 r elements, $$k=\log _2 b$$ k = log 2 b , d is the number of equal segments on r qubits, and b is the branching factor). Our proposed BMGS algorithm is benchmarked against state-of-the-art Depth-First Grover Search (DFGS) and PGS implementations for an arbitrary number of solutions, ranging from 2 to 20 qubits, as a proof of concept. We also show that our BMGS requires fewer iterations for shallow quantum circuits and achieves an optimal $$\mathcal {O}$$ O ( $$\sqrt{s\mathcal {N}}$$ s N ) average complexity for s solutions, when $$dk < r$$ d k < r . The Qiskit Python implementation of the proposed BMGS algorithm is available on GitHub ( https://github.com/hafeezzwiz21/Multi-Solution-DFGS-BMGS ).