Span-program-based quantum algorithm for tree detection

TL;DR

This paper introduces a quantum algorithm based on span programs that efficiently detects whether a given graph contains a specific tree as a subgraph or not as a minor, with optimal query complexity and near-optimal running time.

Contribution

It presents the first time-efficient quantum algorithm for the subgraph/not-a-minor problem for trees using span programs, achieving optimal query complexity.

Findings

Query complexity is O(n), which is proven to be optimal.

Time complexity is near-linear, up to polylogarithmic factors.

The algorithm effectively distinguishes subgraph presence or minor absence in graphs.

Abstract

Span program is a linear-algebraic model of computation originally proposed for studying the complexity theory. Recently, it has become a useful tool for designing quantum algorithms. In this paper, we present a time-efficient span-program-based quantum algorithm for the following problem. Let $T$ be an arbitrary tree. Given query access to the adjacency matrix of a graph $G$ with $n$ vertices, we need to determine whether $G$ contains $T$ as a subgraph, or $G$ does not contain $T$ as a minor, under the promise that one of these cases holds. We call this problem the subgraph/not-a-minor problem for $T$. We show that this problem can be solved by a bounded-error quantum algorithm with $O(n)$ query complexity and $\tilde{O}(n)$ time complexity. The query complexity is optimal, and the time complexity is tight up to polylog factors.