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Quantum Computing for solving the 3SAT problem by reduction to the MIS combinatorial optimization problem

Samuel Deleplanque

Published Apr 17, 2023• 0 citations
OpenAlex

Abstract

This work tries to anticipate the main issues related to the D-Wave machines topology for solving the 3Sat problem. The polynomial reduction to the Maximum Independent Set Problem (MIS) is employed for solving a 3Sat instance to furnish a less dense graph to the machine. These analog quantum machines, based on the quantum annealing process, takes a Quadratic Unconstrained Binary Optimization model (QUBO) as input, which is a specific mathematical program easy to build considering the MIS.

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This work tries to anticipate the main issues related to the D-Wave machines topology for solving the 3Sat problem. The polynomial reduction to the Maximum Independent Set Problem (MIS) is employed for solving a 3Sat instance to furnish a less dense graph to the machine. These analog quantum machines, based on the quantum annealing process, takes a Quadratic Unconstrained Binary Optimization model (QUBO) as input, which is a specific mathematical program easy to build considering the MIS.
Commercial quantum processing units (QPUs) such as those made by D-Wave Systems are being increasingly used for solving complex combinatorial optimization problems. In this paper, we review a canonical NP-hard problem, the Maximum Independent Set (MIS) problem. We show how to map MIS problems to quadratic unconstrained binary optimization (QUBO) problems, and use a D-Wave 2000Q QPU to solve them. We compare the results from the D-Wave system to classical algorithms such as simulated thermal annealing and the graphical networks package NetworkX. To our knowledge, these are the first results of experiments involving arbitrarily-structured MIS inputs using a D-Wave QPU. We find that the QPU can be used as a heuristic optimizer for randomly generated inputs, but due to physical control errors, can be outperformed by simulated thermal annealing.
As contemporary quantum computers do not possess error correction, any calculation performed by these devices can be considered an involuntary approximation. To solve a problem on a quantum annealer, it has to be expressed as an instance of Quadratic Unconstrained Binary Optimization (QUBO). In this work, we thus study whether systematically approximating QUBO representations of the MAX-3SAT problem can improve the solution quality when solved on contemporary quantum hardware, compared to using exact, non-approximated QUBO representations. For a MAX-3SAT instance consisting of a 3SAT formula withnnnvariables andmmmclauses, we propose a method of systematically creating approximate QUBO representations of dimension(n×n)(n\times n)(n×n), which is significantly smaller than the QUBO matrices of any exact, non-approximated MAX-3SAT QUBO transformation. In an empirical evaluation, we demonstrate that using our QUBO approximations for solving MAX-3SAT problems on D-Wave's quantum annealer Advantage_System6.4 can yield better results than using state-of-the-art exact QUBO transformations. Furthermore, we demonstrate that using naive QUBO approximation methods, based on removing values from exact(n+m)×(n+m)−dimensional(n+m)\times(n+m)-\mathbf{dimensional}(n+m)×(n+m)−dimensionalQUBO representations of MAX-3SAT instances, is ineffective.
Quantum annealing is getting increasing attention in combinatorial optimization. The quantum processing unit by D-Wave is constructed to approximately solve Ising models on so-called Chimera graphs. Ising models are equivalent to quadratic unconstrained binary optimization (QUBO) problems and maximum cut problems on the associated graphs. We have tailored branch-and-cut as well as semidefinite programming algorithms for solving Ising models for Chimera graphs to provable optimality and use the strength of these approaches for comparing our solution values to those obtained on the current quantum annealing machine, D-Wave 2000Q. This allows for the assessment of the quality of solutions produced by the D-Wave hardware. In addition, we also evaluate the performance of a heuristic by Selby. It has been a matter of discussion in the literature how well the D-Wave hardware performs at its native task, and our experiments shed some more light on this issue. In particular, we examine how reliably the D-Wave computer can deliver true optimum solutions and present some surprising results.
We present experiments in solving combinatorial optimization and constraint satisfaction problems by means of Quantum Annealing. We describe how to model classic constraint problems such as N-queens and Magic Square as well as hard combinatorial problems such as the Costas Array Problem or the Quadratic Assignment Problem in terms of QUBO (Quadratic Unconstrained Binary Optimization). QUBO is the input language of quantum computers based on quantum annealing such as the D-Wave systems and of the ”quantum-inspired” but classical devices such as Fujitsu’s Digital Annealing Unit or Hitachi’s CMOS Annealing Machine. We present preliminary results for solving these combinatorial optimization and constraint satisfaction problems by implementation on the D-Wave quantum computer.
The increasing complexity of computational problems in various fields entails the development of efficient methods to solve them. Thus, the emerging technology of quantum computers comes with algorithms, such as quantum annealing, that could solve optimization problems exponentially faster. In particular, to solve the 3SAT problem on quantum annealers, it must be transformed into an instance of Quadratic Unconstrained Binary Optimization (QUBO). Given that the choice of 3SAT-to-QUBO transformation affects the solution quality on quantum annealing, an algorithmic method for creating numerous 3SAT-to-QUBO transformations has recently been proposed, aiming to investigate more deeply such effect. This represents a challenge when deciding which is the best performing 3SAT-to-QUBO transformation that should be used when solving a 3SAT instance on quantum annealing. To overcome this issue, this work investigates the impact of different 3SAT-to-QUBO transformations on the solution quality of simulated annealing, as a classical analogue to quantum annealing. We demonstrate that different QUBO encodings of the same 3SAT instance can lead to different outcomes in terms of energy and clause satisfaction. We also reveal some correlations between 3SAT features, QUBO structural properties, and solution quality. These findings underscore the importance of the choice of 3SAT-to-QUBO transformation in simulated annealing, and provide a foundation for the future development of machine learning models for guiding encoding choices in both classical and quantum environments.
One way of solving 3sat instances on a quantum computer is to transform the 3sat instances into instances of Quadratic Unconstrained Binary Optimizations (QUBOs), which can be used as an input for the QAOA algorithm on quantum gate systems or as an input for quantum annealers. This mapping is performed by a 3sat-to-QUBO transformation. Recently, it has been shown that the choice of the 3sat-to-QUBO transformation can significantly impact the solution quality of quantum annealing. It has been shown that the solution quality can vary up to an order of magnitude difference in the number of correct solutions received, depending solely on the 3sat-to-QUBO transformation. An open question is: what causes these differences in the solution quality when solving 3sat-instances with different 3sat-to-QUBO transformations? To be able to conduct meaningful studies that assess the reasons for the differences in the performance, a larger number of different 3sat-to-QUBO transformations would be needed. However, currently, there are only a few known 3sat-to-QUBO transformations, and all of them were created manually by experts, who used time and clever reasoning to create these transformations. In this paper, we will solve this problem by proposing an algorithmic method that is able to create thousands of new and different 3sat-to-QUBO transformations, and thus enables researchers to systematically study the reasons for the significant difference in the performance of different 3sat-to-QUBO transformations. Our algorithmic method is an exhaustive search procedure that exploits properties of 4×4 dimensional pattern QUBOs, a concept which has been used implicitly in the creation of 3sat-to-QUBO transformations before, but was never described explicitly. We will thus also formally and explicitly introduce the concept of pattern QUBOs in this paper.