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arXiv · 2510.26246

Limitation of Quantum Walk Approach to the Maximum Matching Problem

Alcides Gomes Andrade Júnior·Akira Matsubayashi
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Abstract

The Maximum Matching problem has a quantum query complexity lower bound of $\Omega(n^{3/2})$ for graphs on $n$ vertices represented by an adjacency matrix. The current best quantum algorithm has the query complexity $O(n^{7/4})$, which is an improvement over the trivial bound $O(n^2)$. Constructing a quantum algorithm for this problem with a query complexity improving the upper bound $O(n^{7/4})$ is an open problem. The quantum walk technique is a general framework for constructing quantum algorithms by transforming a classical random walk search into a quantum search, and has been successfully applied to constructing an algorithm with a tight query complexity for another problem. In this work we show that the quantum walk technique fails to produce a fast algorithm improving the known (or even the trivial) upper bound on the query complexity. Specifically, if a quantum walk algorithm designed with the known technique solves the Maximum Matching problem using $O(n^{2-\epsilon})$ queries with any constant $\epsilon>0$, and if the underlying classical random walk is independent of an input graph, then the guaranteed time complexity is larger than any polynomial of $n$.

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Alcides Gomes Andrade Júnior, Akira Matsubayashi. 2025-10-30. Limitation of Quantum Walk Approach to the Maximum Matching Problem. https://arxiv.org/abs/2510.26246

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