Computational complexity and applications of quantum algorithm

Jincheng Zhang

Quantum Complexity Theory (QCT) offers a novel framework for algorithm design, predicated on the principle that certain computational problems exhibit inherent quantum complexity. This paper explores the application of QCT to identify problems with a significantly enhanced computational advantage when tackled using quantum algorithms. We argue that by leveraging the unique properties of quantum mechanics, we can potentially develop algorithms that outperform their classical counterparts in solving complex problems, demonstrating a paradigm shift in algorithmic design. This work lays the foundation for a new area of research with significant implications for the future of computational science and engineering.

Marco O. Lanzagorta, Jeffrey K. Uhlmann

Saima Parveen, Waqar Ahmad

Quantum computing (QC) presents transformative potential for solving complex scientific problems that remain intractable for classical computers. Leveraging quantum superposition, entanglement, and interference, quantum algorithms such as Shor’s factoring algorithm, Grover’s search algorithm, variational quantum eigensolvers (VQE), and quantum approximate optimization algorithms (QAOA) enable exponential or polynomial speedups for selected problem classes. This paper explores recent advancements in quantum computing algorithms, their practical implementations, and emerging applications in fields such as molecular simulations, material science, cryptography, optimization, and machine learning. We highlight the strengths and limitations of current approaches, present performance analyses through graphical comparisons, and discuss prospective directions for integrating quantum algorithms into real-world scientific workflows. Our findings demonstrate that quantum computing is gradually transitioning from theoretical frameworks to experimental platforms capable of addressing highly complex computational challenges.

John Watrous

Vasilijs Kravcevs

Hira Fatima, Waqar Ahmad

Quantum computing (QC) presents transformative potential for solving complex scientific problems that remain intractable for classical computers. Leveraging quantum superposition, entanglement, and interference, quantum algorithms such as Shor’s factoring algorithm, Grover’s search algorithm, variational quantum eigensolvers (VQE), and quantum approximate optimization algorithms (QAOA) enable exponential or polynomial speedups for selected problem classes. This paper explores recent advancements in quantum computing algorithms, their practical implementations, and emerging applications in fields such as molecular simulations, material science, cryptography, optimization, and machine learning. We highlight the strengths and limitations of current approaches, present performance analyses through graphical comparisons, and discuss prospective directions for integrating quantum algorithms into real-world scientific workflows. Our findings demonstrate that quantum computing is gradually transitioning from theoretical frameworks to experimental platforms capable of addressing highly complex computational challenges.