Quantum Fine-Grained Lower Bounds for SetDisjointness via Sub-Linear Reductions from 3SUM

2026-09-30

AI Summary

This study addresses the absence of 3SUM-based fine-grained conditional lower bounds for Set Disjointness in quantum settings and the failure of classical reductions in quantum algorithms. We propose a general fine-grained reduction framework for Abelian 3-orthogonal arrays that combines sublinear-time reductions with near-linear hashing constructions, establishing the first sublinear-time quantum reduction from 3SUM to online Set Disjointness. This approach derives a preprocessing-query time trade-off lower bound of p+2q≥1, which is further extended to the 3XOR problem. By providing the first conditional lower bounds for quantum Set Disjointness and demonstrating that the quantum 3SUM conjecture implies specific time trade-off constraints, this work bridges a critical gap in conditional complexity theory for quantum computing and strengthens the theoretical foundations of quantum data structure complexity.

Abstract

In classical fine-grained complexity, the 3SUM Conjecture is used to prove a variety of conditional lower bounds on data structure and graph problems via an initial reduction to the SetDisjointness problem. However, there is an $\tilde{O}(n)$-time quantum algorithm for 3SUM and a direct application of Grover's algorithm to SetDisjointness queries beats the state-of-the-art classical conditional bound by Kopelowitz, Pettie, and Porat (SODA 2016); this shows that these classical bounds do not apply in the quantum setting. Thus establishing analogous conditional lower bounds in the quantum setting requires applying the quantum 3SUM Conjecture to a \emph{quantum} fine-grained reduction from 3SUM to SetDisjointness. We give the first sub-linear time quantum reductions from 3SUM to online SetDisjointness. Via our reduction, the quantum 3SUM conjecture implies a $p + 2q \geqslant 1$ tradeoff bound for quantum SetDisjointness algorithms with $O(N^p)$ preprocessing time and $O(N^q)$ query time. We also give an analogous reduction from 3XOR. These results are derived from a general framework for fine-grained reductions to SetDisjointness which applies to any Abelian 3-Orthogonal Array (3OA) problem with suitable almost-linear hash functions.

Problem

• Quantum fine-grained complexity
• 3SUM conjecture
• SetDisjointness
• Sub-linear reductions
• Conditional lower bounds

Innovation

• Quantum fine-grained reductions
• Sub-linear time
• SetDisjointness
• 3SUM Conjecture
• Abelian 3-Orthogonal Array