From Leaves to Clusters: Depth-Efficient SAT-Oracle Synthesis Based on the HRSE Model

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 $70\%$--$89\%$.