Potential for Polynomial Solution for NP-Complete Problems using Quantum Computation

TL;DR

This work targets Set Constraint Problems (SCP), a framework that generalizes NP-Complete Set Cover. It introduces a quantum-inspired matrix method (QIMM) and a quantum matrix method (QMM) that incorporate uncertainty via a tri-state representation $M \in \{-1,0,1\}^{n\times k}$ and a dedicated operator $\RL{ن}$, moving beyond classical binary logic. A further variant, Quantum Matrix Method with Separate States (QMMSS), assigns distinct quantum states to each element across sets, achieving $O(m\times n)$ time for state preparation and measurement and arguing potential polynomial-time solvability for NP-Complete problems under quantum computation, though outcomes remain probabilistic. The paper emphasizes how uncertainty can be naturally modeled in quantum frameworks and demonstrates a full translation of SCP constraints into quantum representations, offering a pathway to leverage quantum computation for combinatorial constraint problems. Overall, the work presents a structured progression from classical SCP methods to quantum-augmented approaches, highlighting both theoretical potential and the need to address probabilistic measurement effects in practice.

Abstract

In this paper, we propose two new methods for solving Set Constraint Problems, as well as a potential polynomial solution for NP-Complete problems using quantum computation. While current methods of solving Set Constraint Problems focus on classical techniques, we offer both a quantum-inspired matrix method and a quantum matrix method that neutralizes common contradictions and inconsistencies that appear in these types of problems. We then use our new method to show how a potential polynomial solution for NP-Complete problems could be found using quantum computation. We state this as a potential solution, rather than an actual solution, as the outcome of any quantum computation may not be the same as the expected outcome. We start by formally defining a Set Constraint Problem. We then explain current, classical methods that are used to solve these problems and the drawbacks of such methods. After this, we explain a new quantum-inspired matrix method that allows us to solve these problems, with classical limitations. Then, we explain a new quantum matrix method that solves these problems using quantum information science. Finally, we describe how we can extend this method to potentially solve NP-Complete problems in polynomial time using quantum computation.