Quantum Search on Bipartite Multigraphs

2025-04-17

AI Summary

This work addresses the marked vertex search problem on 2-tessellable graphs—including bipartite multigraphs—and presents the first quantum search algorithm tailored to this graph class. Methodologically, it extends both the Szegedy and staggered quantum walk models to 2-tessellable graphs for the first time, integrating the AGJK subroutine with a quantum oracle query mechanism. Theoretically, the algorithm achieves a query complexity of $O(1/sqrt{varepsilon})$, where $varepsilon$ is a lower bound on the stationary distribution probability of the marked vertex—yielding a provable quadratic speedup over classical Markov chain search. Crucially, this result lifts prior structural constraints requiring bipartiteness or regularity, thereby significantly broadening the scope of graph families amenable to quantum walk–based search. The framework provides a scalable quantum acceleration paradigm for a wider range of graph-theoretic problems.

Technology Category

• Search and Optimization: Distributed Search
• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Search

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Quantum walks provide a powerful framework for achieving algorithmic speedup in quantum computing. This paper presents a quantum search algorithm for 2-tessellable graphs, a generalization of bipartite graphs, achieving a quadratic speedup over classical Markov chain-based search methods. Our approach employs an adapted version of the Szegedy quantum walk model (adapted SzQW), which takes place on bipartite graphs, and an adapted version of Staggered Quantum Walks (Adapted StQW), which takes place on 2-tessellable graphs, with the goal of efficiently finding a marked vertex by querying an oracle. The Ambainis, Gily'en, Jeffery, and Kokainis' algorithm (AGJK), which provides a quadratic speedup on balanced bipartite graphs, is used as a subroutine in our algorithm. Our approach generalizes existing quantum walk techniques and offers a quadratic speedup in the number of queries needed, demonstrating the utility of our adapted quantum walk models in a broader class of graphs.

Problem

• Develop quantum search algorithm for 2-tessellable graphs
• Achieve quadratic speedup over classical search methods
• Generalize quantum walk techniques for broader graph classes

Innovation

• Adapted Szegedy quantum walk model
• Adapted Staggered Quantum Walks
• AGJK algorithm as subroutine