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Parameterized Quantum Query Algorithms for Graph Problems

Tatsuya Terao, Ryuhei Mori

TL;DR

The paper investigates parameterized quantum query complexity for graph problems, introducing Fixed Parameter Improved (FPI) algorithms and the quantum kernelization framework. It presents tight quantum upper bounds for the k-vertex cover and k-matching problems, showing $O\left(\sqrt{k}\,n + k^{3/2}\sqrt{n}\right)$ and $O\left(\sqrt{k}\,n + k^{2}\right)$ queries respectively, along with matching lower bounds $\Omega\left(\sqrt{k}\,n\right)$ for small $k$, establishing polynomial FPI and optimality in this regime. The approach combines Grover-based search, threshold maximal matching, augmentation-path techniques, and a novel quantum kernelization pipeline, enabling reductions to small kernels of size $O(k^{2})$ edges and enabling efficient quantum querying. These results highlight kernelization as a viable tool for parameterized quantum query complexity in graphs and pave the way for further refinements for larger parameters and broader graph problems. The work has implications for quantum query efficiency in NP-hard graph problems and strengthens the case for parameterized quantum techniques in practical quantum algorithms.

Abstract

In this paper, we consider the parameterized quantum query complexity for graph problems. We design parameterized quantum query algorithms for $k$-vertex cover and $k$-matching problems, and present lower bounds on the parameterized quantum query complexity. Then, we show that our quantum query algorithms are optimal up to a constant factor when the parameters are small.

Parameterized Quantum Query Algorithms for Graph Problems

TL;DR

The paper investigates parameterized quantum query complexity for graph problems, introducing Fixed Parameter Improved (FPI) algorithms and the quantum kernelization framework. It presents tight quantum upper bounds for the k-vertex cover and k-matching problems, showing and queries respectively, along with matching lower bounds for small , establishing polynomial FPI and optimality in this regime. The approach combines Grover-based search, threshold maximal matching, augmentation-path techniques, and a novel quantum kernelization pipeline, enabling reductions to small kernels of size edges and enabling efficient quantum querying. These results highlight kernelization as a viable tool for parameterized quantum query complexity in graphs and pave the way for further refinements for larger parameters and broader graph problems. The work has implications for quantum query efficiency in NP-hard graph problems and strengthens the case for parameterized quantum techniques in practical quantum algorithms.

Abstract

In this paper, we consider the parameterized quantum query complexity for graph problems. We design parameterized quantum query algorithms for -vertex cover and -matching problems, and present lower bounds on the parameterized quantum query complexity. Then, we show that our quantum query algorithms are optimal up to a constant factor when the parameters are small.
Paper Structure (27 sections, 26 theorems, 10 equations, 4 figures, 1 table, 7 algorithms)

This paper contains 27 sections, 26 theorems, 10 equations, 4 figures, 1 table, 7 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Parameterized Quantum Query Complexity of Vertex Cover.
  4. Parameterized Quantum Query Complexity of Matching.
  5. Fixed Parameter Improved Algorithms
  6. Overview of Our Upper Bounds
  7. Grover's Search Algorithm
  8. Overview of Our Parameterized Quantum Query Algorithm for Vertex Cover
  9. Overview of Our Parameterized Quantum Query Algorithm for Matching
  10. Overview of Our Lower Bounds
  11. Additional Related Work
  12. Preliminaries
  13. Basic Notation
  14. Query Complexity
  15. Kernelization and Quantum Query Kernelization
  16. ...and 12 more sections

Key Result

Theorem 1.1

The quantum query complexity for finding a vertex cover of size at most $k$ or determining that there does not exist a vertex cover of size at most $k$ with bounded error is $O(\sqrt{k}n + k^{3/2}\sqrt{n})$ in the adjacency matrix model.

Figures (4)

  • Figure 1: The quantum query complexity of the $k$-vertex cover problem.
  • Figure 2: The quantum query complexity of the $k$-matching problem.
  • Figure 3: The kernel for the $k$-vertex cover problem consists of maximal matching (with some missing vertices) and independent set. The size of the maximal matching is at most $k$. All non-missing vertices in the maximal matching have degree at most $k$.
  • Figure 4: Example of an $M$-augmenting path and a candidate path

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Proposition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Proposition 1.6
  • Theorem 1.7
  • Definition 1.8: Parameterized graph problem
  • ...and 35 more