Quantum Submodular Maximization

2026-10-03

AI Summary

This study investigates the quantum query complexity of maximizing non-negative submodular functions. Focusing on both unconstrained and cardinality-constrained settings, we design novel quantum query algorithms based on a reversible numerical oracle model and analyze them through the lens of complexity theory. Our primary contributions are threefold. First, we overcome the query bottleneck of classical randomized algorithms by achieving an approximation ratio of $(1/2-\varepsilon)$ using only $O(\log n)$ queries. Second, under cardinality constraints, our algorithms attain quadratic to exponential speedups over their classical counterparts. Finally, we establish new quantum lower bounds that rigorously characterize the limits of achievable speedup for specific approximation thresholds. Collectively, these results demonstrate the fundamental advantages of quantum computing for this class of combinatorial optimization problems.

Abstract

We study the quantum query complexity of maximizing a non-negative submodular function, considering both the unconstrained setting and, for monotone functions, a cardinality constraint $k$ on an $n$-element ground set. In the exact reversible digital value-oracle model, our unconstrained algorithm achieves an expected $(1/2-\varepsilon)$-approximation using only $O_\varepsilon(\log n)$ queries. In contrast, any classical randomized algorithm that attains a fixed expected ratio above $1/4$ requires $\Omega(n/\log n)$ queries (Li, Feldman, Kazemi, and Karbasi, 2022), establishing an exponential separation in query complexity. For cardinality-constrained maximization, we give a bounded-error quantum algorithm that achieves a $(1-1/e-\varepsilon)$-approximation using $\widetilde O_\varepsilon(\min\{\sqrt n,n/k\})$ queries. When $k=o(n)$, our algorithm achieves at least a quadratic speedup up to logarithmic factors over classical randomized algorithms (Mirzasoleiman, Badanidiyuru, Karbasi, Vondr\'ak, and Krause, 2015; Peng and Rubinstein, 2025). Moreover, when $k=cn$ for any fixed rational $c<1-1/e-\varepsilon$, the query complexity reduces to $O_{\varepsilon,c}(\log n)$, yielding an exponential separation from the classical $\Omega(n/\log n)$ lower bound (Li, Feldman, Kazemi, and Karbasi, 2022). We further prove quantum lower bounds of $\exp(\Omega(\varepsilon^2n))$ queries for achieving a ratio beyond $1/2+\varepsilon$ without constraints, and $\exp(\Omega(\varepsilon^2k))$ queries for exceeding $1-1/e+\varepsilon$ when $k/n\le\varepsilon$. These barriers demonstrate that quantum computation offers no exponential speedup at these approximation thresholds.

Problem

• submodular maximization
• quantum query complexity
• cardinality constraint
• approximation ratio
• quantum lower bounds

Innovation

• Quantum query complexity
• Submodular maximization
• Exponential separation
• Approximation algorithms
• Quantum lower bounds