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

TL;DR

This work analyzes the quantum query complexity of local search on graphs, linking lower bounds to graph geometry. Using the strong weighted adversary method on a construction from Branzei, Choo, and Recker (2024), the authors prove $Q(G) = \Omega\left(\dfrac{n^{3/4}}{\sqrt{g}}\right)$ where $g$ is the vertex congestion, and derive a corollary for $\beta$-expanders with maximum degree $\Delta$ giving $Q(G) = \Omega\left(\dfrac{\sqrt{\beta}\, n^{1/4}}{\sqrt{\Delta}\, \log n}\right)$. On constant-degree expanders, this yields $Q(G) = \Omega\left(\dfrac{n^{1/4}}{\sqrt{\log n}}\right)$, improving prior quantum lower bounds but still leaving a gap to the best known upper bound $O(n^{1/3})$ for such graphs. The results illuminate how congestion and expansion govern quantum query complexity for local search and suggest a rich gap between lower and upper bounds in the quantum regime. The methodology hinges on translating local search into a decision problem and applying a sophisticated weight-based adversary framework to a carefully constructed input family.

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 $Ω\bigl( \frac{n^{\frac{3}{4}}}{\sqrt{g}} \bigr)$, where $g$ is the vertex congestion of the graph. For a $β$-expander with maximum degree $Δ$, this implies a lower bound of $ Ω\bigl(\frac{\sqrtβ \; n^{\frac{1}{4}}}{\sqrtΔ \; \log{n}} \bigr)$. We obtain these bounds by applying the strong weighted adversary method to a construction by Brânzei, Choo, and Recker (2024). As a corollary, on constant degree expanders, we derive a lower bound of $Ω\bigl(\frac{n^{\frac{1}{4}}}{ \sqrt{\log{n}}} \bigr)$. This improves upon the best prior quantum lower bound of $Ω\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.

Theorems & Definitions (15)

• Theorem 0
• Corollary 1
• Corollary 2
• proof
• Definition 1: Staircase
• Definition 2: Tail of a staircase
• Definition 3: The functions $f_{\mathbf{x}}$ and $g_{\mathbf{x},b}$; the set $\mathcal{X}$
• Definition 4: The map $\mathcal{H}$
• Theorem 0
• proof