A New Efficient Quantum Algorithm for Solving NP-complete Problems

Masanori Ohya, Igor V. Volovich

Masanori Ohya, Igor Vasil'evich Volovich

Ordinary approach to quantum algorithm is based on quantum Turing machine or quantum circuits. It is known that this approach is not powerful enough to solve NP-complete problems. In this paper we study a new approach to quantum algorithm which is a combination of the ordinary quantum algorithm with a chaotic dynamical system. We consider the satisfiability problem as an example of NP-complete problems and argue that the problem, in principle, can be solved in polynomial time by using our new quantum algorithm. keywords: Quantum Algorithm, NP-complete problem, Chaotic Dynamics 1

Makoto Ohya

We review the quantum chaos algorithm solving the NP-complete problems in polynomial time. This work has been done in the series of papers with Professor Igor Volovich for nearly ten years.

Masanori Ohya, Igor Vasil'evich Volovich

Colin P. Williams

Iago Gaspar

The P vs. NP problem is a central open question in theoretical computer science, asking whether every problem whose solution can be verified in polynomial time can also be solved in polynomial time. This paper presents an interdisciplinary exploration of this problem by integrating mathematical analysis, quantum computing, and machine learning.We analyze the limitations of polynomial-time reductions among NP-complete problems under the Exponential Time Hypothesis (ETH), providing detailed proofs that offer new insights into the structural complexities within NP. Additionally, we develop a novel quantum-classical hybrid algorithm that combines Grover's algorithm with classical heuristics to solve specific instances of NP-complete problems more efficiently. This includes a comprehensive complexity analysis, scalability discussions, and detailed practical implementation aspects such as oracle constructions.Furthermore, we propose a machine learning framework based on Graph Neural Networks (GNNs) to capture patterns within NP-complete problems. This approach is theoretically justified, enhances heuristic methods, and demonstrates superior performance through extensive experimental comparisons with classical heuristics and exact solvers.By integrating these methods, this work offers fresh perspectives on the P vs. NP problem and contributes to the broader understanding of computational complexity.