Performance Gains in Quantum SAT Solvers Using ESOP Encoding

2026-05-15

AI Summary

This work addresses the high resource overhead of conventional conjunctive normal form (CNF) encodings in Grover-based quantum SAT solvers, which manifests as excessive qubit counts and non-Clifford (T) gate complexity. The paper introduces e-CNF, the first CNF encoding tailored for quantum computation, by systematically transforming Boolean formulas into Exclusive-Sum-of-Products (ESOP) representations and mapping them to efficient reversible quantum circuits. This approach substantially reduces quantum resource requirements, yielding tighter upper bounds on both qubit count and T-gate complexity, alongside a scalable transformation and circuit synthesis pipeline. Empirical evaluation on standard SAT benchmarks demonstrates that e-CNF significantly outperforms traditional CNF encodings, markedly decreasing qubit usage, T-gate count, and circuit depth, thereby enhancing the practicality of quantum SAT solving.

Abstract

The Boolean Satisfiability (SAT) problem is a canonical NP-complete problem and a natural candidate for quantum acceleration via search-based algorithms. In Grover-based quantum SAT solvers, the dominant computational cost stems from the construction of a reversible oracle that evaluates the Boolean formula, rendering the choice of SAT encoding crucial for overall quantum resource efficiency. Although SAT instances are conventionally expressed in Conjunctive Normal Form (CNF), such encodings typically translate into quantum circuits with significant qubit overhead and high non-Clifford gate complexity. In this work, we investigate an Exclusive-Sum-of-Products (ESOP)-based CNF (e-CNF) representation tailored for quantum SAT solving and analyze its impact on oracle construction. We derive tighter upper bounds on qubit requirements and Clifford+$T$ gate counts for Grover-based SAT solvers when e-CNF encodings are employed in place of standard CNF. In addition, we propose a scalable transformation from Boolean formulas to e-CNF and present a systematic procedure for interpreting e-CNF representations as reversible quantum circuits suitable for oracle implementation. Experimental evaluation on representative SAT benchmarks demonstrates that the proposed e-CNF-based approach yields substantial and consistent reductions in quantum resources, including qubit count, T-gate complexity, and circuit depth, when compared to CNF-based oracle constructions. These results establish e-CNF as an effective quantum-aware SAT encoding that significantly improves the practicality of oracle-based quantum SAT solving.

Problem

• Quantum SAT Solving
• Oracle Construction
• ESOP Encoding
• Quantum Resource Efficiency
• Boolean Satisfiability

Innovation

• ESOP encoding
• quantum SAT solver
• reversible oracle
• Grover's algorithm
• quantum circuit optimization