Graphical and algebraic methods for Boolean factoring

2026-06-01

AI Summary

This study addresses the problem of minimizing AND-count in Boolean polynomial factorization to reduce the number of AND and Toffoli gates in classical and quantum circuits. The authors innovatively model this problem as a biclique cover optimization over bipartite graphs and integrate a multivariate Horner scheme to devise efficient algebraic decomposition algorithms tailored for ESOP and SOP representations. Compared to state-of-the-art tools such as EXORCISM-4, the proposed biclique-based approach achieves up to an 80% reduction in AND-count, while the Horner-based algorithm significantly accelerates decomposition for random Boolean functions with up to 12 variables. The work thus advances both optimization quality and computational efficiency in Boolean function synthesis.

Abstract

The problem of factoring Boolean polynomials has significant applications in both classical and quantum computing technology. In this paper we have developed novel algorithms for factoring both ESOP and SOP expressions. Our aim is to optimize the AND-count. The AND-count plays a key role in determining the number of AND and Toffoli gates required to implement a reversible function with classical and quantum circuits, respectively. The first type of algorithms that we develop, are graphical. We reduce the problem of Boolean factoring to covering a bipartite graph with bicliques, and so optimizing the number of bicliques required to cover the bipartite graph, leads to reduced number of factors, and hence AND-count. The second type of algorithm is algebraic, and is derived from multivariate Horner method. We have compared the performances of our algorithms with existing popular methods like EXORCISM-4 and EPOEM2, on random functions of up to 12 variables. We have observed that our multivariate Horner method is substantially faster, while our biclique-based method achieves the maximum AND-count reduction. In fact, compared to EXORCISM-4 our biclique based method achieves up to 5 times reduction in AND-count.

Problem

• Boolean factoring
• AND-count
• reversible logic
• quantum circuits
• ESOP/SOP optimization

Innovation

• Boolean factoring
• biclique covering
• multivariate Horner method
• AND-count optimization
• ESOP/SOP minimization