Hybrid Quantum-Classical Approaches to NP-complete Problems

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.

Fei Li, Arul Rhik Mazumder

In this paper, we design, analyze, and evaluate a hybrid quantum algorithm for the metric traveling salesman problem (TSP). TSP is a well-studied NP-complete problem that many algorithmic techniques have been developed for, on both classic computers and quantum computers. The existing literature of algorithms for TSP are neither adaptive to input data nor suitable for processing medium-size data on the modern classic and quantum machines. In this work, we leverage the classic computers’ power (large memory) and the quantum computers’ power (quantum parallelism), based on the input data, to fasten the hybrid algorithm’s overall running time. Our algorithmic ideas include trimming the input data efficiently using a classic algorithm, finding an optimal solution for the post-processed data using a quantum-only algorithm, and constructing an optimal solution for the untrimmed data input efficiently using a classic algorithm. We conduct experiments to compare our hybrid algorithm against the state-of-the-art classic and quantum algorithms on real data sets. The experimental results show that our solution truly outperforms the others and thus confirm our theoretical analysis. This work provides insightful quantitative tools for people and compilers to choose appropriate quantum or classical or hybrid algorithms, especially in the NISQ (noisy intermediate-scale quantum) era, for NP-complete problems such as TSP.

Takashi Mihara, Tetsuro Nishino

Tetsuro Nishino

Michail A. Zak

The challenge of this paper is to relate artificial intuition-based intelligence, represented by self-supervised systems, to solutions of NP-complete problems. By self-supervised systems we understand systems that are capable to move from disorder to order without external effort, i.e. in violation of the second law of thermodynamics. It has been demonstrated, [1], that such systems exist in the mathematical world: they are presented by ODE coupled with their Liouville equation, but they belong neither to Newtonian nor to quantum physics since they are capable to violate the second law of thermodynamics. That suggests that machines could not simulate intuition-based intelligence if they are composed only of physical parts, but without digital components. Nevertheless it was found such quantum-classical hybrids, [1], that simulates some of self-supervised systems. The main achievement of this work is a demonstration that self-supervised systems can solve NP-complete problems in polynomial time by replacing an enumeration of exponentially large number of possible choices with a short cut provided by a non-Newtonian and non-quantum nature of self-supervised systems.