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Computing Performance and Communication systems

Procedure of Solving 3-SAT Problem by Combining Quantum Search Algorithm and DPLL Algorithm

Runkai Zhang, Jing Chen, Huiling Zhao

Published Dec 4, 2020• 4 citations
Publisher

Abstract

Although some classical algorithms have been applied to solve the satisfiability problem, more effective methods are still explored because existing algorithms are constrained by inadequate computing capability of traditional computers. The parallelism of quantum computation makes quantum algorithms with promising potential to improve the computing ability, but existing quantum algorithms still require too large number of qubits to solve a simple problem effectively. In this paper, an optimized data structure was structured to solve Boolean satisfiability problem by utilizing Grover's algorithm, and then the corresponding formula was proposed to balance variables in consideration of complexity. With reasonable simplification, quantum circuits were built to decrease the number of qubits required in Grover's algorithm. The result of verification experiment further demonstrated that the proposed approach is simple, reliable and of a certain practical value.

DOI
10.23977/cpcs.2020.41003

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Although some classical algorithms have been applied to solve the satisfiability problem, more effective methods are still explored because existing algorithms are constrained by inadequate computing capability of traditional computers. The parallelism of quantum computation makes quantum algorithms with promising potential to improve the computing ability, but existing quantum algorithms still require too large number of qubits to solve a simple problem effectively. In this paper, an optimized data structure was structured to solve Boolean satisfiability problem by utilizing Grover's algorithm, and then the corresponding formula was proposed to balance variables in consideration of complexity. With reasonable simplification, quantum circuits were built to decrease the number of qubits required in Grover's algorithm. The result of verification experiment further demonstrated that the proposed approach is simple, reliable and of a certain practical value.
This study explores how well Grover's Algorithm performs in solving the Boolean Satisfiability Problem (SAT) using quantum circuits. The algorithm is implemented with IBM's Qiskit framework and compared to classical brute-force methods. Experiments focus on 3-SAT, 4-SAT, and 5-SAT problems, using quantum simulators and IBM quantum hardware. The results show that Grover's Algorithm is more efficient, offering a theoretical quadratic speedup over classical methods. However, practical issues like limited qubit availability, hardware noise, and optimization challenges impact its current performance. The data highlights the potential for quantum computing to scale and solve NP-complete problems. This research shows how quantum computing can improve problem-solving and lays the groundwork for future studies on more complex SAT problems and advanced quantum hardware.
This study explores how well Grover's Algorithm performs in solving the Boolean Satisfiability Problem (SAT) using quantum circuits. The algorithm is implemented with IBM's Qiskit framework and compared to classical brute-force methods. Experiments focus on 3-SAT, 4-SAT, and 5-SAT problems, using quantum simulators and IBM quantum hardware. The results show that Grover's Algorithm is more efficient, offering a theoretical quadratic speedup over classical methods. However, practical issues like limited qubit availability, hardware noise, and optimization challenges impact its current performance. The data highlights the potential for quantum computing to scale and solve NP-complete problems. This research shows how quantum computing can improve problem-solving and lays the groundwork for future studies on more complex SAT problems and advanced quantum hardware.
This study investigates the application of quantum computing to the Boolean satisfaction problem (SAT), with a special focus on the special case of the Exactly-1 3-SAT problem. Quantum computing has shown significant advantages in solving various computational problems and achieving exponential accelerations. This study designs and implements a quantum algorithm to identify assignments that prove Boolean formulas of bound form using the Grover search algorithm. We tested the implementation of three quantum circuits using as many quantum techniques as possible using the IBM quantum simulator. Only the second implementation showed consistent results due to the number of qubits, decoherence time, and circuit complexity. This study demonstrates that the design of algorithms meets hardware constraints and lays the foundation for future research as quantum devices improve.
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.
In this paper, we proposed a novel quantum algorithm for the maximum satisfiability problem. Satisfiability (SAT) is to find the set of assignment values of input variables for the given Boolean function that evaluates this function as TRUE or prove that such satisfying values do not exist. For a POS SAT problem, we proposed a novel quantum algorithm for the maximum satisfiability (MAX-SAT), which returns the maximum number of OR terms that are satisfied for the SAT-unsatisfiable function, providing us with information on how far the given Boolean function is from the SAT satisfaction. We used Grover’s algorithm with a new block called quantum counter in the oracle circuit. The proposed circuit can be adapted for various forms of satisfiability expressions and several satisfiability-like problems. Using the quantum counter and mirrors for SAT terms reduces the need for ancilla qubits and realizes a large Toffoli gate that is then not needed. Our circuit reduces the number of ancilla qubits for the terms T of the Boolean function from T of ancilla qubits to ≈log2T+1. We analyzed and compared the quantum cost of the traditional oracle design with our design which gives a low quantum cost.
Constraint satisfiability problems, crucial to several applications, are solved on a quantum computer using the Grover's search algorithm, leading to a quadratic improvement over the classical case. The solutions are obtained with high probability for several cases and are illustrated for the cases involving 2 variables for both 3-bit and 4-bit numbers. Methods are defined for inequality comparisons and these are combined according to the form of the satisfiability formula, to form the oracle for the algorithm. The circuit is constructed using IBM Qiskit and is verified on an IBM simulator. It is further executed on one of the Noisy Intermediate-Scale Quantum (NISQ) processors from IBM on the cloud. Noise levels in the processor at present are found to be too high for successful execution. Running the algorithm on the simulator with a custom noise model lets us identify the noise threshold for successful execution.