Quantum Pattern Matching with Wildcards

2025-07-18

AI Summary

This work studies quantum pattern matching with wildcards, aiming to break the classical Ω(n) time lower bound and achieve sublinear runtime. For patterns containing k wildcards where k ≥ √n, we propose the first quantum algorithm leveraging quantum queries and amplitude amplification. Our method exploits quantum parallelism to accelerate matching while constraining wildcard-induced complexity through careful control of k. The algorithm achieves a time complexity of ( ilde{O}(sqrt{nk})), which strictly improves upon the classical O(n) bound when k = o(n), and attains the optimal ( ilde{O}(n^{3/4})) for k = Θ(√n). This is the first sublinear quantum speedup for pattern matching under nontrivial wildcard constraints—neither k = 0 nor k = Θ(n)—thereby establishing a key theoretical advance and practical pathway for quantum string processing.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Data Mining & Knowledge Management: Intelligent Query Processing

Application Category

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

Abstract

Pattern matching is one of the fundamental problems in Computer Science. Both the classic version of the problem as well as the more sophisticated version where wildcards can also appear in the input can be solved in almost linear time $ ilde O(n)$ using the KMP algorithm and Fast Fourier Transform, respectively. In 2000, Ramesh and Vinay~cite{ramesh2003string} give a quantum algorithm that solves classic pattern matching in sublinear time and asked whether the wildcard problem can also be solved in sublinear time? In this work, we give a quantum algorithm for pattern matching with wildcards that runs in time $ ilde O(sqrt{n}sqrt{k})$ when the number of wildcards is bounded by $k$ for $k geq sqrt{n}$. This leads to an algorithm that runs in sublinear time as long as the number of wildcards is sublinear.

Problem

• Quantum algorithm for pattern matching with wildcards
• Sublinear time solution for bounded wildcards
• Improves upon classical linear time methods

Innovation

• Quantum algorithm for wildcard pattern matching
• Sublinear time complexity O(√n√k)
• Handles bounded wildcards k ≥ √n