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IEEE Access

Efficient Quantum Answer Set Programming Solver Using Partial Diffusion and Entanglement

Esraa Ebdelrehime, Ahmed Younes, Islam Elkabani

Published Jan 1, 2025• 0 citations
Publisher

Abstract

The relation between answer set programming (ASP) and combinatorial problems, especially the satisfiability problem (SAT) has attracted significant interest in academic literature. The aim of this paper is to introduce an efficient quantum answer set programming solver (QSAT) that can handle NP-hard combinatorial search problems by reducing the problem to MAX-3-SAT problem. The paper proposes a quantum algorithm that solves the MAX-3-SAT problem using amplification techniques that exploits entanglement and partial diffusion operator to find a solution with high probability inO(√2n/l), where n is the number of variables and l denotes the number of literals. The proposed algorithm shows a speed-up in solving ASP problems compared to classical approaches.

DOI
10.1109/access.2025.3631582

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The relation between answer set programming (ASP) and combinatorial problems, especially the satisfiability problem (SAT) has attracted significant interest in academic literature. The aim of this paper is to introduce an efficient quantum answer set programming solver (QSAT) that can handle NP-hard combinatorial search problems by reducing the problem to MAX-3-SAT problem. The paper proposes a quantum algorithm that solves the MAX-3-SAT problem using amplification techniques that exploits entanglement and partial diffusion operator to find a solution with high probability inO(√2n/l), where n is the number of variables and l denotes the number of literals. The proposed algorithm shows a speed-up in solving ASP problems compared to classical approaches.
In this paper, a Quantum Answer set programming solver (QASP) will be proposed to solve NP-hard combinatorial search problems. The paper shows that problems such as Hamiltonian cycle problem and N-queen problem encoded as ASP can be reduced to a MAX-3-SAT problem in a 3-CNF form. The 3-CNF Boolean formula will be solved using the proposed quantum solver withO(n+m)O(n+ m)O(n+m)qubits wherennnis the input variables number,mmmis the clauses number inO(2n/m)O(\sqrt{2^{n/m}})O(2n/m​)steps.
Quantum Computing has become a more and more prominent research field in the last few decades. This growth in interest is mainly related to the so-called quantum speed up that some quantum procedure exhibits. The two main examples are Shor and Grover algorithms. The latter will be a key ingredient of this paper. In particular, we propose an attempt to speed up Answer Set Programming (ASP) exploiting Quantum Computing. We rely on two proposals in the literature that use quantum computation for: finding stable models of ASP programs; counting solutions of propositional formulae. For combining such proposals we embed in the quantum framework a third proposal from the literature, namely a purely classical approach for navigating the solution space of ASP models. We end up with a quantum method for counting stable models of ASP programs. After providing the details of our method, we briefly describe a Proof of Concept implementation of these techniques.
The aim of the paper is to answer a long-standing open problem on the relationship between NP and BQP. The paper shows that BQP contains NP by proposing a BQP quantum algorithm for the MAX-E3-SAT problem which is a fundamental NP-hard problem. Given an E3-CNF Boolean formula, the aim of the MAX-E3-SAT problem is to find the variable assignment that maximizes the number of satisfied clauses. The proposed algorithm runs in O(m2)O(m^2)O(m2) for an E3-CNF Boolean formula with mmm clauses and in the worst case runs in O(n6)O(n^6)O(n6) for an E3-CNF Boolean formula with nnn inputs. The proposed algorithm maximizes the set of satisfied clauses using a novel iterative partial negation and partial measurement technique. The algorithm is shown to achieve an arbitrary high probability of success of 1−ε1-ε1−ε for small ε>0ε>0ε>0 using a polynomial resources. In addition to solving the MAX-E3-SAT problem, the proposed algorithm can also be used to decide if an E3-CNF Boolean formula is satisfiable or not, which is an NP-complete problem, based on the maximum number of satisfied clauses.
Satisfiability (SAT) is a central problem in computer science, and advances in SAT-solving algorithms have a far-reaching impact across many fields. Recent works have proposed quantum SAT solvers based on Grover's algorithm, a quantum search technique. However, Grover-based approaches face a key limitation: they typically require prior knowledge of the number of satisfying assignments of the target Boolean formula. This information is unavailable in most practical settings. Quantum counting can be used to estimate this quantity, but it incurs a computational overhead that is several orders of magnitude higher than Grover search. In this paper, we propose a novel quantum SAT solver based on entanglement and equivalence checking. Our method does not assume prior knowledge of the number of solutions and is computationally more efficient than quantum counting. Although the worst case time complexity is inevitably exponential, we prove that the expected time complexity of our approach is only constant time O(1) over random Boolean functions. Experimental results also support our theoretical claim.
The K-SAT problem is a fundamental challenge in computational theory, widely used in artificial intelligence, cryptography, and optimization. Classical algorithms struggle with exponential time complexity, making them inefficient for large-scale instances. In this research, we propose an efficient quantum computing approach based on Grover’s algorithm to enhance the search process for satisfiable solutions in K-SAT problems. Our implementation leverages quantum superposition and amplitude amplification to explore multiple possible solutions simultaneously, reducing the search complexity from O (2ⁿ) in classical methods to O(√2ⁿ) in quantum computing. We design a quantum oracle that encodes the K-SAT clauses and integrates it into Grover’s iterative search framework. The performance is evaluated through Qiskit simulations, demonstrating a significant improvement in search efficiency compared to classical brute-force techniques. The results highlight the potential of quantum algorithms in solving complex combinatorial problems with enhanced speed and accuracy. This study contributes to the development of quantum-accelerated optimization methods, paving the way for real-world applications in machine learning, cryptanalysis, and large-scale data processing.
Boolean satisfiability (SAT) problem is of fundamental importance in computer science and many application domains. For Grover's algorithm, solving the SAT problem requires O(2n)\mathcal{O}(\sqrt{2^n})O(2n​) queries--where n denotes the number of logic variables in the problem. However, Grover's algorithm suffers from the Souffle problem: specifically, when the number of solutions is unknown, terminating the algorithm too early or too late leads to a significant reduction in the probability of obtaining a solution. In this paper, we propose a parallel fixed-point (PFP) search algorithm to solve the SAT problem. By exploiting entanglement, each clause in the conjunctive normal form (CNF) formula can be processed independently, leading to a significant reduction in circuit depth. We also discuss how to perform the algorithm in distributed manner. These make the PFPS algorithm particularly suitable for the noisy intermediate-scale quantum (NISQ) era.