A note on quantum lower bounds for local search via congestion and expansion

TL;DR

This paper establishes new quantum lower bounds for local search problems on graphs, relating complexity to graph properties like congestion and expansion, and improves upon previous bounds.

Contribution

It introduces graph-geometry-based quantum lower bounds for local search, using the adversary method, and enhances previous results for expanders.

Findings

Quantum lower bound of (n^{3/4}/\u221a{g}) for local search

Improved lower bound of (n^{1/4}/7{n}) for constant degree expanders

Gap remains between quantum lower and upper bounds for certain graphs

Abstract

We consider the quantum query complexity of local search as a function of graph geometry. Given a graph $G = (V,E)$ with $n$ vertices and black box access to a function $f : V \to \mathbb{R}$, the goal is find a vertex $v$ that is a local minimum, i.e. with $f(v) \leq f(u)$ for all $(u,v) \in E$, using as few oracle queries as possible. We show that the quantum query complexity of local search on $G$ is $\Omega\bigl( \frac{n^{\frac{3}{4}}}{\sqrt{g}} \bigr)$, where $g$ is the vertex congestion of the graph. For a $\beta$-expander with maximum degree $\Delta$, this implies a lower bound of $ \Omega\bigl(\frac{\sqrt{\beta} \; n^{\frac{1}{4}}}{\sqrt{\Delta} \; \log{n}} \bigr)$. We obtain these bounds by applying the strong weighted adversary method to a construction by Br\^anzei, Choo, and Recker (2024). As a corollary, on constant degree expanders, we derive a lower bound of $\Omega\bigl(\frac{n^{\frac{1}{4}}}{ \sqrt{\log{n}}} \bigr)$. This improves upon the best prior quantum lower bound of $\Omega\bigl( \frac{n^{\frac{1}{8}}}{\log{n}}\bigr) $ by Santha and Szegedy (2004). In contrast to the classical setting, a gap remains in the quantum case between our lower bound and the best-known upper bound of $O\bigl( n^{\frac{1}{3}} \bigr)$ for such graphs.