Performance Gains in Quantum SAT Solvers Using ESOP Encoding
利用ESOP编码提升量子SAT求解器性能
Majd Assaad, Abhoy Kole, Rolf Drechsler
AI总结 本文研究了专为量子SAT求解设计的ESOP-CNF编码(e-CNF),通过减少量子资源消耗提升求解效率。
Comments 18 pages, 6 figures
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布尔可满足性(SAT)问题是一个经典的NP完全问题,是通过基于搜索的算法实现量子加速的自然候选者。在基于Grover的量子SAT求解器中,主要的计算开销来自于可逆 oracle 的构造,这使得 SAT 编码的选择对整体量子资源效率至关重要。尽管 SAT 实例通常以析取范式(CNF)表示,但此类编码通常会转化为具有显著量子比特开销和高非Clifford门复杂度的量子电路。在本工作中,我们研究了专为量子SAT求解设计的Exclusive-Sum-of-Products(ESOP)基于CNF(e-CNF)表示,并分析了其对oracle构造的影响。我们推导了在使用e-CNF编码代替标准CNF时,基于Grover的SAT求解器的量子比特需求和Clifford+T门数量的更紧上界。此外,我们提出了一种可扩展的从布尔公式到e-CNF的转换方法,并展示了将e-CNF表示解释为可逆量子电路的系统性程序,适用于oracle实现。在具有代表性的SAT基准测试中的实验评估表明,所提出的基于e-CNF的方法在与基于CNF的oracle构造相比时,能够显著且一致地减少量子资源,包括量子比特数量、T门复杂度和电路深度。这些结果确立了e-CNF作为有效的量子-aware SAT 编码,显著提高了基于oracle的量子SAT求解的实用性。
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.