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1. Introduction 1.1 Research Background 1.2 Problem Statement 1.3 Objective of the Study 1.4 Scope and Limitations 2. Quantum Computing Basics 2.1 Historical Development 2.2 Fundamental Concepts 2.3 Quantum vs Classical Computing 3. Overview of Computational Complexity 3.1 Complexity Classes 3.2 P vs NP Problem 3.3 Importance in Mathematics 4. Quantum Algorithms and Complexity 4.1 Shor's Algorithm 4.2 Grover's Algorithm 4.3 Quantum Speedup 5. Impact on Theoretical Models 5.1 Quantum Complexity Classes 5.2 BQP and Related Classes 5.3 Implications for Existing Theories 6. Applications in Modern Mathematics 6.1 Cryptography 6.2 Algorithm Optimization 6.3 Mathematical Problem Solving 7. Challenges and Limitations 7.1 Technological Constraints 7.2 Theoretical Counterarguments 7.3 Ethical and Practical Concerns 8. Conclusion 8.1 Summary of Findings 8.2 Future Research Directions 8.3 Final Thoughts on Impact
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