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Quantum Complexity vs Classical Complexity: A Survey

Arash Vaezi, Ali Movaghar, Mohammad Ghodsi, Seyed Mohammad Hussein Kazemi, Negin Bagheri Noghrehy, Seyed Mohsen Kazemi

TL;DR

This paper concentrates on aggregating prior research efforts dedicated to solving intricate classical computational problems through quantum computing to systematically compile an exhaustive inventory of these solutions and categorize a collection of demanding open problems that await further exploration.

Abstract

Scientists have demonstrated that quantum computing has presented novel approaches to address computational challenges, each varying in complexity. Adapting problem-solving strategies is crucial to harness the full potential of quantum computing. Nonetheless, there are defined boundaries to the capabilities of quantum computing. This paper concentrates on aggregating prior research efforts dedicated to solving intricate classical computational problems through quantum computing. The objective is to systematically compile an exhaustive inventory of these solutions and categorize a collection of demanding open problems that await further exploration. Through statistical analysis, we help the researchers with their further investigations.

Quantum Complexity vs Classical Complexity: A Survey

TL;DR

This paper concentrates on aggregating prior research efforts dedicated to solving intricate classical computational problems through quantum computing to systematically compile an exhaustive inventory of these solutions and categorize a collection of demanding open problems that await further exploration.

Abstract

Scientists have demonstrated that quantum computing has presented novel approaches to address computational challenges, each varying in complexity. Adapting problem-solving strategies is crucial to harness the full potential of quantum computing. Nonetheless, there are defined boundaries to the capabilities of quantum computing. This paper concentrates on aggregating prior research efforts dedicated to solving intricate classical computational problems through quantum computing. The objective is to systematically compile an exhaustive inventory of these solutions and categorize a collection of demanding open problems that await further exploration. Through statistical analysis, we help the researchers with their further investigations.
Paper Structure (53 sections, 17 equations, 5 figures, 5 tables)

This paper contains 53 sections, 17 equations, 5 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Overview of this Paper
  3. Statistical Analysis
  4. Descriptive Analysis
  5. Inferential Analysis
  6. Quantum Algorithms Complexity
  7. Models of Quantum Complexity Growth
  8. Quantum Randomness and Classical Complexity
  9. Approximation for Quantum Problems
  10. Quantum Query Complexity
  11. Quantum Random Walk
  12. Quantum Solutions to Classical Problems
  13. Satisfiability Problem
  14. Maximum Independent Set
  15. Graph Traversal Problems
  16. ...and 38 more sections

Figures (5)

  • Figure 1: Number of Open Problems Corresponding to Well-known Problems in Computer Science
  • Figure 2: Number of Open Problems Corresponding to Well-known Problems in Computer Science
  • Figure 3: Number of Open Problems Corresponding to Well-known Problems in Computer Science
  • Figure 4: The figure is taken from jeong2023quantum. The MIS graphs reduced from the 3-SAT instance, where vertices represent literals $(x_1, \cdots, x_6 \text{ and negations})$, solid edges intra-clause logics, and dashed edges the inter-clause logics (between literals and their negations).
  • Figure 5: Taken from kim2022rydberg a) $K_{3, 3}$ Graph, b) experimental $K_{3,3}$ in 3D (blue circles) making by quantum wires. c) The experimental probability distribution of $K_{3,3}$ be the solution of MIS

Theorems & Definitions (29)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • definition 5: SAT problem
  • definition 6: Internal Set Problem
  • definition 7: Maximum Independent Set (MIS) Problem
  • definition 8
  • definition 9: Hamiltonian Cycle Problem
  • definition 10: Matching
  • ...and 19 more