Limitation of Quantum Walk Approach to the Maximum Matching Problem

TL;DR

This paper demonstrates that the quantum walk approach cannot significantly improve the query complexity for the Maximum Matching problem, establishing fundamental limitations of this technique.

Contribution

It proves the inherent limitations of quantum walk methods in achieving faster algorithms for maximum matching, even under ideal conditions.

Findings

Quantum walk technique fails to improve the upper bound on query complexity.

Any quantum walk algorithm with certain conditions requires super-polynomial time.

Limits are established for quantum algorithms based on known quantum walk frameworks.

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 $n$.