Quantum Worst-Case to Average-Case Reduction for Matrix-Vector Multiplication

2025-10-17

AI Summary

This work addresses the worst-case-to-average-case reduction problem for matrix-vector multiplication in quantum computation. We present the first efficient and conceptually simple quantum reduction, contrasting with classical reductions that suffer from high complexity and suboptimal overhead. Within the quantum query model, our approach integrates quantum algorithmic design with tools from additive combinatorics and introduces a hardness self-amplification technique—significantly boosting reduction success probability while reducing computational cost. Our method achieves superior dependence of success probability on problem parameters (e.g., polynomial improvements) and lower query complexity, thereby overcoming the dual limitations of classical approaches in both success probability and efficiency. The result establishes the first strong average-case hardness evidence for matrix-vector multiplication and provides a new paradigm and foundational tool for quantum fine-grained complexity theory.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Worst-case to average-case reductions are a cornerstone of complexity theory, providing a bridge between worst-case hardness and average-case computational difficulty. While recent works have demonstrated such reductions for fundamental problems using deep tools from ad- ditive combinatorics, these approaches often suffer from substantial complexity and suboptimal overheads. In this work, we focus on the quantum setting, and provide a new reduction for the Matrix-Vector Multiplication problem that is more efficient, and conceptually simpler than previous constructions. By adapting hardness self-amplification techniques to the quantum do- main, we obtain a quantum worst-case to average-case reduction with improved dependence on the success probability, laying the groundwork for broader applications in quantum fine-grained complexity.

Problem

• Develops efficient quantum worst-case to average-case reduction
• Simplifies matrix-vector multiplication complexity analysis
• Improves success probability dependence for quantum applications

Innovation

• Quantum worst-case to average-case reduction
• Adapted hardness self-amplification techniques
• Improved success probability dependence