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Bridging Classical and Quantum String Matching: A Computational Reformulation of Bit-Parallelism

Simone Faro, Arianna Pavone, Caterina Viola

TL;DR

This work addresses accelerating string matching by bridging classical bit-parallel algorithms with quantum computation. It introduces a general framework to translate Shift-And and Shift-Add into quantum circuits using QRAM and reversible bitwise operations, then leverages Grover’s search to achieve sublinear, quadratic-speedups for the unstructured search problem. The key contributions include (i) a systematic method to convert bit-parallel automata into quantum form, (ii) explicit quantum versions of Shift-And and Shift-Add with near-linear circuit depth, and (iii) two Grover-based strategies that reach $\tilde{\mathcal{O}}(m\sqrt{n})$ and $\tilde{\mathcal{O}}(\sqrt{n})$ time, respectively. These results offer a practical blueprint for quantum-accelerated text search and provide a platform for extending quantum speedups to non-standard string matching tasks.

Abstract

String matching is a fundamental problem in computer science, with critical applications in text retrieval, bioinformatics, and data analysis. Among the numerous solutions that have emerged for this problem in recent decades, bit-parallelism has significantly enhanced their practical efficiency, leading to the development of several optimized approaches for both exact and approximate string matching. However, their potential in quantum computing remains largely unexplored. This paper presents a novel pathway that not only translates bit-parallel string matching algorithms into the quantum framework but also enhances their performance to achieve a quadratic speedup through Grover's search. By embedding quantum search within a bit-parallel model, we reduce the time complexity of string matching, establishing a structured pathway for transforming classical algorithms into quantum solutions with provable computational advantages. Beyond exact matching, this technique offers a foundation for tackling a wide range of non-standard string matching problems, opening new avenues for efficient text searching in the quantum era. To demonstrate the simplicity and adaptability of the technique presented in this paper, we apply this translation and adaptation process to two landmark bit-parallel algorithms: Shift-And for exact pattern matching and Shift-Add for approximate string matching with up to k errors.

Bridging Classical and Quantum String Matching: A Computational Reformulation of Bit-Parallelism

TL;DR

This work addresses accelerating string matching by bridging classical bit-parallel algorithms with quantum computation. It introduces a general framework to translate Shift-And and Shift-Add into quantum circuits using QRAM and reversible bitwise operations, then leverages Grover’s search to achieve sublinear, quadratic-speedups for the unstructured search problem. The key contributions include (i) a systematic method to convert bit-parallel automata into quantum form, (ii) explicit quantum versions of Shift-And and Shift-Add with near-linear circuit depth, and (iii) two Grover-based strategies that reach and time, respectively. These results offer a practical blueprint for quantum-accelerated text search and provide a platform for extending quantum speedups to non-standard string matching tasks.

Abstract

String matching is a fundamental problem in computer science, with critical applications in text retrieval, bioinformatics, and data analysis. Among the numerous solutions that have emerged for this problem in recent decades, bit-parallelism has significantly enhanced their practical efficiency, leading to the development of several optimized approaches for both exact and approximate string matching. However, their potential in quantum computing remains largely unexplored. This paper presents a novel pathway that not only translates bit-parallel string matching algorithms into the quantum framework but also enhances their performance to achieve a quadratic speedup through Grover's search. By embedding quantum search within a bit-parallel model, we reduce the time complexity of string matching, establishing a structured pathway for transforming classical algorithms into quantum solutions with provable computational advantages. Beyond exact matching, this technique offers a foundation for tackling a wide range of non-standard string matching problems, opening new avenues for efficient text searching in the quantum era. To demonstrate the simplicity and adaptability of the technique presented in this paper, we apply this translation and adaptation process to two landmark bit-parallel algorithms: Shift-And for exact pattern matching and Shift-Add for approximate string matching with up to k errors.

Paper Structure

This paper contains 12 sections, 1 equation, 4 figures.

Table of Contents

  1. Introduction
  2. String Matching and Bit-Parallelism
  3. Bitwise Operations: A Bridge from Classical to Quantum Computing
  4. Basic Quantum Operators
  5. Adapting Classical Bitwise Operations for Quantum Computation
  6. Quantum String Matching Algorithms in $\tilde{\mathcal{O}}(n)$
  7. Quantum Shift-And
  8. Quantum Shift-Add
  9. Enhancing Quantum String Matching via Grover’s Search
  10. A First Jump From $\tilde{\mathcal{O}}(n)$ to $\tilde{\mathcal{O}}(m\sqrt{n})$
  11. A Final Shift From $\tilde{\mathcal{O}}(m\sqrt{n})$ to $\tilde{\mathcal{O}}(\sqrt{n})$
  12. Final Discussion

Figures (4)

  • Figure 1: The Quantum Shift-And algorithm, which translates the classical bit-parallel Shift-And algorithm into a quantum framework. (Top) High-level pseudocode describing the procedure executed by the circuit. (Bottom) The quantum circuit implementing the algorithm.
  • Figure 2: The Quantum Shift-Add algorithm, which translates the classical bit-parallel Shift-Add algorithm into a quantum framework. (Top) High-level pseudocode describing the procedure executed by the circuit. (Bottom) The quantum circuit implementing the algorithm.
  • Figure 3: The Quantum String Matching algorithm which makes use of the Grover unstructured search, working in $\tilde{\mathcal{O}}(m\sqrt{n})$ time.
  • Figure 4: The Quantum String Matching algorithm which makes use of the Grover unstructured search, working in $\tilde{\mathcal{O}}(\sqrt{n})$ time.