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Mathematics

Heuristics for Quantum Computing Dealing with 3-SAT

Jose J. Paulet, Luis Llana, Hernán Indíbil de la Cruz Calvo, Mauro Mezzini, Fernando Cuartero, Fernando L. Pelayo

Published Apr 16, 2023• 2 citations
Publisher

Abstract

The SAT problem is maybe one of the most famous NP-complete problems. This paper deals with the 3-SAT problem. We follow a sort of incremental strategy to save computational costs with respect to the classical quantum computing approach. We present an heuristics that leads this strategy, improving the performance of the purely random incremental scheme. We finally validate our approach by means of a thorough empirical study.

DOI
10.3390/math11081888

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This thesis focuses on algorithms over Quantum computing applied over NP hard search problems on domains lacking of any order. The partition problem has been addressed as first stage so providing a piece of quantum computing code which amplifies the amplitude of the solution state noticeably. This work continues addressing the problem of the satisfiability of Boolean formulas, in particular 3-SAT, from hybrid computation perspectives over both classical / quantum and purely quantum computing. There have been proposed strategies which include the definition of own heuristics and own metrics. The application of these strategies has meant a review of Grover's algorithm that has led to propose different methods of modifying it.
The discrete formulation of adiabatic quantum computing is compared with other search methods, classical and quantum, for random satisfiability (SAT) problems. With the number of steps growing only as the cube of the number of variables, the adiabatic method gives solution probabilities close to 1 for problem sizes feasible to evaluate via simulation on current computers. However, for these sizes the minimum energy gaps of most instances are fairly large, so the good performance scaling seen for small problems may not reflect asymptotic behavior where costs are dominated by tiny gaps. Moreover, the resulting search costs are much higher than for other methods. Variants of the quantum algorithm that do not match the adiabatic limit give lower costs, on average, and slower growth than the conventional GSAT heuristic method.
Going as far as possible at SAT problem solving is the main aim of our work. For this sake we have made use of quantum computing from its two, on practice, main models of computation. They have required some reformulations over the former statement of 3-SAT problem in order to accomplish the requirements of both techniques. This paper presents and describes a hybrid quantum computing strategy for solving 3-SAT problems. The performance of this approximation has been tested over a set of representative scenarios when dealing with 3-SAT from the quantum computing perspective.
Quantum computer algorithms can exploit the structure of random satisfiability problems. This paper extends a previous empirical evaluation of such an algorithm and gives an approximate asymptotic analysis accounting for both the average and variation of amplitudes among search states with the same costs. The analysis predicts good performance, on average, for a variety of problems including those near a phase transition associated with a high concentration of hard cases. Based on empirical evaluation for small problems, modifying the algorithm in light of this analysis improves its performance. The algorithm improves on both GSAT, a commonly used conventional heuristic, and quantum algorithms ignoring problem structure.
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.