Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Parameterized quantum algorithms for closest string problems

Josh Cudby, Sergii Strelchuk

TL;DR

Three quantum algorithms for the Closest String Problem (CSP) and the related Closest Substring Problem (CSSP) are presented, demonstrating improved performance over classical counterparts in specific parameter regimes, highlighting the promise of quantum approaches in structured combinatorial settings.

Abstract

Parameterized complexity enables the practical solution of generally intractable NP-hard problems when certain parameters are small, making it particularly useful in real-world applications. The study of string problems in this framework has been particularly fruitful, yielding many state-of-the-art classical algorithms that run efficiently in certain parameter regimes contrary to their worst- or average-case performance. Motivated by the dramatic increase in genomic data and its growing computational demands, we initiate the study of the quantum parameterized complexity of the Closest String Problem (CSP) and the related Closest Substring Problem (CSSP). We present three quantum algorithms for the CSP and one for the CSSP. Each algorithm demonstrates improved performance over classical counterparts in specific parameter regimes, highlighting the promise of quantum approaches in structured combinatorial settings. We also derive a conditional lower bound for the CSP with binary alphabets, showing that our first algorithm is tight in its dominant scaling factor.

Parameterized quantum algorithms for closest string problems

TL;DR

Three quantum algorithms for the Closest String Problem (CSP) and the related Closest Substring Problem (CSSP) are presented, demonstrating improved performance over classical counterparts in specific parameter regimes, highlighting the promise of quantum approaches in structured combinatorial settings.

Abstract

Parameterized complexity enables the practical solution of generally intractable NP-hard problems when certain parameters are small, making it particularly useful in real-world applications. The study of string problems in this framework has been particularly fruitful, yielding many state-of-the-art classical algorithms that run efficiently in certain parameter regimes contrary to their worst- or average-case performance. Motivated by the dramatic increase in genomic data and its growing computational demands, we initiate the study of the quantum parameterized complexity of the Closest String Problem (CSP) and the related Closest Substring Problem (CSSP). We present three quantum algorithms for the CSP and one for the CSSP. Each algorithm demonstrates improved performance over classical counterparts in specific parameter regimes, highlighting the promise of quantum approaches in structured combinatorial settings. We also derive a conditional lower bound for the CSP with binary alphabets, showing that our first algorithm is tight in its dominant scaling factor.
Paper Structure (17 sections, 23 theorems, 29 equations, 3 tables, 2 algorithms)

This paper contains 17 sections, 23 theorems, 29 equations, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Outlook
  3. Notation and Preliminaries
  4. Strings and distances
  5. Parameterized Complexity
  6. String Selection Problems
  7. Quantum Algorithms
  8. Quantum Algorithms for the Closest String Problem
  9. A Trivial Quantum Search Algorithm
  10. A Quantum Random Walk Algorithm for the CSP under Hamming Distance
  11. A Quantum Dynamic Programming Algorithm for the CSP under Levenshtein Distance
  12. Quantum Algorithms for the Closest Substring Problem
  13. Discussion and Open Questions
  14. Quantum Search via Random Walk
  15. Markov Chains, Eigenvalue Gaps and Congestion
  16. ...and 2 more sections

Key Result

Proposition 1

There exists a quantum algorithm for solving the CSP under any desired distance metric, with runtime $\tilde{\mathcal{O}}\lparen*\rparen{\sigma^{n/2}\cdot\sqrt{k}\cdot D(n)}$, where $D(n)$ is the classical time required to compute the distance between 2 strings of length $n$. In particular, $D(n) =

Theorems & Definitions (25)

  • Proposition 1: CSP, any distance
  • Proposition 2: CSP, Hamming distance
  • Proposition 3: CSP, Levenshtein distance
  • Proposition 4: CSSP, Hamming distance
  • Theorem 1: Grover search grover_fast_1996
  • Theorem 2: Quantum minimum/maximum finding durr_quantum_1999
  • Theorem 3: Classical Search via Random Walk magniez_search_2011
  • Theorem 4: Quantum Search via Random Walk magniez_search_2011
  • Theorem 5: Quantum Dynamic Programming glos_quantum_2021
  • Proposition 4: CSP, any distance
  • ...and 15 more