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arXiv:2607.11401

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

Zhihang Li, Wei Zi, Shuai Yang, Bei Zhou, Zongjiang Yi, Woji He, Yingjie Lin, Kaixiang Ji, Heru Du, Jinchen Xu

quant-phSubmitted Jul 13, 2026• Announced Jul 14, 2026• 0 citations• Revised Oct 27, 2026 (v2)
v1 · Jul 13, 2026v2 · Oct 27, 2026
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Abstract

Quantum oracles play a fundamental role in many quantum algorithms, such as Grover's search and quantum counting. In oracle synthesis, reducing the circuit depth under a limited number of ancilla qubits is essential, given the constraints of current quantum devices. However, for satisfiability oracles (SAT-oracles) under a limited ancilla budget, existing synthesis methods fail to fully exploit clause-level parallelism, leading to deep circuits. In practice, evaluating clauses in parallel can quickly exhaust the ancilla budget. To address this issue, we introduce the Clustered Synthesis Tree (CST), a depth-oriented SAT-oracle synthesis framework that exploits clause-level parallelism while respecting the ancilla budget. CST leverages a recursive synthesis tree as the backbone for scheduling clause evaluations and groups its clause leaves into clusters, whose clauses are evaluated in parallel by a reversible oracle called ClausePack. To form the clusters, CST introduces a polynomial-time heuristic called SeedGrow for the NP-hard clause-grouping problem, and the resulting tree is compiled into an executable circuit by CST-Map. Compared with the depth-oriented pAND baseline, CST can reduce the circuit depth by an average of 10.2x and 8.8x-11.0x on SATLIB and random 4-CNF benchmarks, respectively, under the same ancilla budgets, while also outperforming the width-oriented pAND, clause-storing, and qSAT baselines.

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13 pages, 7 figures, 4 tables

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Quantum oracles play a fundamental role in many quantum algorithms, such as Grover's search and quantum counting. In oracle synthesis, reducing the circuit depth under a limited number of ancilla qubits is essential, given the constraints of current quantum devices. However, for satisfiability oracles (SAT-oracles) under a limited ancilla budget, existing synthesis methods fail to fully exploit clause-level parallelism, leading to deep circuits. In practice, evaluating clauses in parallel can quickly exhaust the ancilla budget. To address this issue, we introduce the Clustered Synthesis Tree (CST), a depth-oriented SAT-oracle synthesis framework that exploits clause-level parallelism while respecting the ancilla budget. CST leverages a recursive synthesis tree as the backbone for scheduling clause evaluations and groups its clause leaves into clusters, whose clauses are evaluated in parallel by a reversible oracle called ClausePack. To form the clusters, CST introduces a polynomial-time heuristic called SeedGrow for the NP-hard clause-grouping problem, and the resulting tree is compiled into an executable circuit by CST-Map. Compared with the depth-oriented pAND baseline, CST can reduce the circuit depth by an average of 10.2x and 8.8x-11.0x on SATLIB and random 4-CNF benchmarks, respectively, under the same ancilla budgets, while also outperforming the width-oriented pAND, clause-storing, and qSAT baselines.
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Quantum oracles are fundamental building blocks of many quantum algorithms, and their resource consumption directly affects performance, yet structured description and complexity analysis for their composition are still lacking. In this paper, we introduce the Framework for Oracle Recursion Modeling (FORM), a unified formal abstraction of multi-function composition in quantum oracles: it provides a structured description of the composition layer, makes its gate complexity exactly computable, and turns oracle design into an optimizable tree-construction problem. Based on this model, we propose the ShallowGrow algorithm, which constructs an oracle structure under a given ancilla budget and provably minimizes the number of function evaluations. On Boolean quadratic equation systems, ShallowGrow reduces Qiskit-measured circuit depth by 54.1% on average relative to the state-of-the-art W-cycle construction, with consistent reductions on the EPFL and ISCAS85 combinational logic networks under scarce ancilla budgets. Furthermore, pebbling-based syntheses trade space against time within the logic network of a function; ShallowGrow extends this trade-off across functions, and integrated with their published circuits it reduces the ancillary qubits of a complete oracle from one per constraint function to logarithmically many. With half as many ancillas as constraint functions, the T-count falls by a factor of 7.9 to 32 relative to the W-cycle-based construction.
To demonstrate the advantage of quantum computation, many attempts have been made to attack classically intractable problems, such as the satisfiability problem (SAT), with quantum computer. To use quantum algorithms to solve these NP-hard problems, a quantum oracle with quantum circuit implementation is usually required. In this manuscript, we first introduce a novel algorithm to synthesize quantum logic in the Conjunctive Normal Form (CNF) model. Compared with linear ancillary qubits in the implementation of Qiskit open-source framework, our algorithm can synthesize an m clauses n variables k-CNF with O(k2m2/n)O(k^2 m^2/n)O(k2m2/n) quantum gates by only using three ancillary qubits. Both the size and depth of the circuit can be further compressed with the increase in the number of ancillary qubits. When the number of ancillary qubits is Ω(mϵ)\Omega(m^\epsilon)Ω(mϵ) (for any ϵ>0\epsilon > 0ϵ>0), the size of the quantum circuit given by the algorithm is O(km), which is asymptotically optimal. Furthermore, we design another algorithm to optimize the depth of the quantum circuit with only a small increase in the size of the quantum circuit. Experiments show that our algorithms have great improvement in size and depth compared with the previous algorithms.
Clifford circuit optimization is an important step in the quantum compilation pipeline. Major compilers employ heuristic approaches. While they are fast, their results are often suboptimal. Minimization of noisy gates, like 2-qubit CNOT gates, is crucial for practical computing. Exact approaches have been proposed to fill the gap left by heuristic approaches. Among these are SAT based approaches that optimize gate count or depth, but they suffer from scalability issues. Further, they do not guarantee optimality on more important metrics like CNOT count or CNOT depth. A recent work proposed an exhaustive search only on Clifford circuits in a certain normal form to guarantee CNOT count optimality. But an exhaustive approach cannot scale beyond 6 qubits. In this paper, we incorporate search restricted to Clifford normal forms in a SAT encoding to guarantee CNOT count optimality. By allowing parallel plans, we propose a second SAT encoding that optimizes CNOT depth. By taking advantage of flexibility in SAT based approaches, we also handle connectivity restrictions in hardware platforms, and allow for qubit relabeling. We have implemented the above encodings and variations in our open source tool Q-Synth. In experiments, our encodings significantly outperform existing SAT approaches on random Clifford circuits. We consider practical VQE and Feynman benchmarks to compare with TKET and Qiskit compilers. In all-to-all connectivity, we observe reductions up to 32.1% in CNOT count and 48.1% in CNOT depth. Overall, we observe better results than TKET in the CNOT count and depth. We also experiment with connectivity restrictions of major quantum platforms. Compared to Qiskit, we observe up to 30.3% CNOT count and 35.9% CNOT depth further reduction.
Circuit synthesis is the task of decomposing a given logical functionality into a sequence of elementary gates. It is (depth-)optimal if it is impossible to achieve the desired functionality with even shorter circuits. Optimal synthesis is a central problem in both quantum and classical hardware design, but also plagued by complexity-theoretic obstacles. Motivated by fault-tolerant quantum computation, we consider the special case of synthesizing blocks of Clifford unitaries. Leveraging entangling input stimuli and the stabilizer formalism allows us to reduce the Clifford synthesis problem to a family of poly-size satisfiability (SAT) problems - one for each target circuit depth. On a conceptual level, our result showcases that the Clifford synthesis problem is contained in the first level of the polynomial hierarchy (NP), while the classical synthesis problem for logical circuits is known to be complete for the second level of the polynomial hierarchy (Σp2). Based on this theoretical reduction, we formulate a SAT encoding for depth-optimal Clifford synthesis. We then employ SAT solvers to determine a satisfying assignment or to prove that no such assignment exists. From that, the shortest depth for which synthesis is still possible (optimality) as well as the actual circuit (synthesis) can be obtained. Empirical evaluations show that the optimal synthesis approach yields a substantial depth improvement for random Clifford circuits and Clifford+T circuits for Grover search.
Quantum oracles are quantum circuits that implement classical Boolean functions, frequently used as independent black boxes in quantum algorithm design. The hierarchical reversible logic synthesis approach, a scalable method for large-scale oracle synthesis, relies on ancilla qubits to store intermediate results. The sequence in which these results are computed and uncomputed, governed by the reversible pebble game, greatly affects both qubit usage and circuit gate count. This paper introduces a novel divide-and-conquer pebbling strategy that reduces qubit count while maintaining a reasonable circuit gate count. Experimental results show up to a 90% reduction in qubits compared to the most commonly-used strategy, with an average increase of 3 to 4 times in quantum operations, all achieved within a reasonable runtime. In qubit-constrained tasks, our method achieves over 90% reduction in T-count. Additionally, we have incorporated our strategy into the hierarchical synthesis method and implemented it as a potential synthesizer for Qiskit, with experiments demonstrating that it outperforms the default synthesizer. The proposed approach shows promise in optimizing both qubit usage and circuit gate count for large-scale oracle synthesis problems.