Limitation of Quantum Walk Approach to the Maximum Matching Problem
Alcides Gomes Andrade Júnior, Akira Matsubayashi
TL;DR
The paper investigates whether the quantum walk framework can yield a faster quantum algorithm for Maximum Matching beyond the known $O(n^{7/4})$ upper bound. By formalizing the query model and Markov-chain-based search, it proves that a quantum walk algorithm with $O(n^{2-ε})$ queries, using a Markov chain independent of the input graph, would require super-polynomial time on some graphs, including bipartite ones. This establishes a fundamental limitation of the naive quantum-walk approach for this problem, suggesting that improvements require more sophisticated, input-adaptive techniques or hybrids with other quantum methods. The result informs the design of future quantum algorithms for combinatorial problems by delineating the boundary of the standard quantum-walk toolkit.
Abstract
The Maximum Matching problem has a quantum query complexity lower bound of $Ω(n^{3/2})$ 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 $O(n^{2-ε})$ queries with any constant $ε>0$, and if the underlying classical random walk is independent of an input graph, then the guaranteed time complexity is larger than any polynomial of $n$.