AI 中文总结
研究在有限辅助预算下为SAT公式合成深度高效预言机的问题，提出基于HRSE模型的聚类合成树（CST）框架，由SeedGrow、ClausePack和CST-Map组成，在随机4-CNF和SATLIB基准测试中大幅降低预言机电路深度，优势延续到完整算法。
AI 中文摘要
量子预言机是许多量子算法的常见构建块，电路深度是直接影响整体性能的主要成本。在有限辅助预算下为SAT（CNF）公式合成预言机往往会产生深度电路，因为现有方法未充分利用子句级并行性。本文提出了聚类合成树（CST），这是一个面向深度的框架，其核心思想是将分层合成树的单个子句叶子分组为簇，在辅助约束下暴露实例相关的子句级并行性。CST由三部分组成：通过多项式时间\(O(m^2 k)\)启发式算法SeedGrow解决其引发的子句分组问题，该问题被公式化为辅助约束调度问题并证明一般为NP完全问题；ClausePack，一个可逆预言机，仅以对数深度开销并行评估簇的子句；以及CST-Map，将聚类树编译为可执行的SAT预言机。在相同辅助预算下，对于随机四元合取范式，CST将预言机的电路深度比现有技术基线降低了68%至94%。在标准SATLIB基准测试中，CST比现有技术基线减少了约2.6倍至43.2倍，在密集变量共享下增益最大，并且仅使用其辅助量子比特的3.7%至20%就达到了基线的最大预算深度。格罗弗搜索资源估计表明，这种优势延续到了完整算法中，将总电路深度降低了70%至89%。
英文摘要
Quantum oracles are a common building block of many quantum algorithms, where circuit depth is a primary cost that directly affects overall performance. Synthesizing oracles for SAT (CNF) formulas under a limited ancilla budget, however, tends to yield deep circuits, as existing methods underexploit clause-level parallelism. In this work, we present the Clustered Synthesis Tree (CST), a depth-oriented framework whose core idea is to group the individual clause leaves of a hierarchical synthesis tree into clusters, exposing instance-dependent clause-level parallelism under ancilla constraints. CST comprises three parts: the clause-grouping problem it induces, which we formulate as an ancilla-constrained scheduling problem and prove NP-complete in general, is addressed by SeedGrow, a polynomial-time $O(m^2 k)$ heuristic; ClausePack, a reversible oracle that evaluates a cluster's clauses in parallel at only a logarithmic-depth overhead; and CST-Map, which compiles the clustered tree into an executable SAT-oracle. On random $4$-CNF under the same ancilla budgets, CST reduces the oracle's circuit depth over the state-of-the-art (SOTA) baseline by $68\%$--$94\%$. On the standard SATLIB benchmarks, CST achieves about a $2.6\times$--$43.2\times$ reduction over the SOTA baseline, with the largest gains under dense variable sharing, and matches the baseline's maximum-budget depth using only $3.7\%$--$20\%$ of its ancilla qubits. A Grover-search resource estimate shows the advantage carries over to the full algorithm, reducing total circuit depth by $70\%$--$89\%$.
Comments17 pages, 8 figures, 6 tables
