Performance Gains in Quantum SAT Solvers Using ESOP Encoding

Performance Gains in Quantum SAT Solvers Using ESOP Encoding

Overview and Motivation

The Boolean Satisfiability (SAT) problem remains central in computational complexity, underlining its significance across diverse domains including formal verification, hardware equivalence checking, cryptanalysis, and combinatorial optimization. Although classical SAT solvers utilizing CNF encodings have reached considerable maturity, the exponential worst-case complexity persists. Quantum algorithms, specifically Grover's search and related oracle-based strategies, offer an asymptotic quadratic speedup for such unstructured search tasks. However, the construction of reversible Boolean oracles, critical to Grover-based SAT solvers, often contributes prohibitive qubit overhead and gate complexity, predominantly when conventional CNF encodings are employed.

This work analyzes and demonstrates explicit performance improvements in quantum SAT solvers by leveraging ESOP-based CNF (e-CNF) encodings for oracle construction, targeting reduced qubit usage and Clifford+T gate counts. The authors offer both theoretical resource bounds and empirical results across a broad suite of SAT benchmarks, quantifying substantial reductions in quantum circuit complexity versus CNF-based oracle synthesis.

Quantum SAT Oracle Construction: Encoding Impact

Grover's algorithm, serving as a generic quantum search protocol, requires a phase oracle that reflects the underlying Boolean formula's satisfiability. Classical CNF representations, especially when derived via Tseitin transformations, introduce numerous auxiliary variables and yield multi-controlled Toffoli gates with high ancilla demand and T-gate overhead. The ESOP encoding, by contrast, replaces disjunctions with exclusive ORs, offering concise conjunctions and a direct translation to XOR/AND gates, thus alleviating depth and ancilla complexity.

Specifically, the e-CNF approach reformulates equivalence relations, prevalent in hardware verification tasks, into structurally compact ESOP expressions. The quantum realization of such constraints requires far fewer ancillas and gates: e.g., for a proposition of the form $p_1 \leftrightarrow (a_1 \land \neg a_2)$, the CNF-based quantum circuit requires five ancilla qubits and 99 Clifford+T gates, while its e-CNF counterpart consumes no additional ancillas and merely 17 gates.

Resource Bound Analysis

The paper presents explicit upper bounds for qubit and gate requirements in Grover-based SAT solvers under e-CNF encoding. For $m$ equivalence constraints, the e-CNF realization leads to:

• Ancilla requirement: zero additional ancilla qubits for each equivalence or XOR clause.
• Clifford+T gate count: $17m$ for equivalence constraints and $4m$ for XOR clauses, compared to $99m$ and $187m$ respectively in CNF-based circuits.
• Conjunctions of ESOP expressions require a single $C_mX$ gate, further minimizing depth.

For arbitrary Boolean formulas featuring equivalence ($=$), the total Clifford+T gate cost is reduced by a factor exceeding $2.5$ ($252m-61$ to $m$0 as per the detailed analysis), with circuit depth and ancilla overhead scaling linearly but with significantly lower constants.

Experimental Evaluation

The empirical evaluation spans SAT instances typical of hardware verification: simple logic gates, arithmetic circuits (carry-ripple and carry-lookahead adders), and various multiplier architectures (USP, AR, WT). Using the qCheck and qSAT frameworks, both CNF and e-CNF oracle circuits were synthesized and decomposed, employing established Clifford+T realizations for multi-controlled operations.

Measured improvements using e-CNF encoding include:

• Up to 56% reduction in qubit count
• Up to 81% reduction in CNOT gates
• Up to 85% reduction in T gates
• Up to 95% reduction in circuit depth

Performance gains were consistently observed across all tested benchmarks, especially pronounced in equivalence and XOR-heavy SAT instances. Both theory and experiment confirm that e-CNF-based oracles require fewer circuit resources, making them substantially more practical for scalable quantum SAT solving.

Theoretical and Practical Implications

The research substantiates that the structural choice of Boolean encoding dramatically affects quantum resource efficiency. The e-CNF encoding, particularly for equivalence checking tasks, provides lower bounds on ancilla and gate requirements, facilitating more scalable circuit synthesis. This is especially relevant for future fault-tolerant quantum architectures, where resource overheads in oracles remain the dominant bottleneck.

Practically, these reductions pave the way for quantum SAT solvers to handle larger industrial-scale instances, previously infeasible due to qubit and gate constraints. Theoretically, the results highlight that classical encoding heuristics (optimized for DPLL/CDCL-style solvers) may not translate to optimal quantum circuit efficiency, initiating the need for quantum-aware representations and circuit synthesis paradigms.

The elimination of Tseitin-auxiliary variables when using e-CNF exposes further optimization avenues: automated ESOP minimization, improved gate decompositions for high-degree XORs, and potential extension to other quantum oracle-based algorithms like quantum model checking and QAOA variants.

Speculative Future Directions

Exploring tighter automated resource estimation frameworks for e-CNF-based synthesis is warranted, particularly as quantum hardware matures. Further research into optimizing ESOP minimization and tailored synthesis for multi-controlled Toffoli gates can unlock additional savings. Extending e-CNF applicability beyond SAT to general constraint satisfaction and model checking, as well as hybrid quantum-classical solvers, may yield broader practical impact. Benchmarking on larger and more complex SAT instances, especially those with cryptographic or combinatorial structure, could validate scalability claims and drive the next generation of quantum SAT solvers.

Conclusion

The study establishes that ESOP-based CNF encoding (e-CNF) substantively enhances quantum resource efficiency in Grover-based SAT solvers, predominantly for equivalence-rich formulas. Theoretical analysis and experimental results demonstrate marked improvements in qubit demand, T-gate complexity, CNOT count, and circuit depth. These findings suggest that e-CNF encoding is a promising candidate for quantum-aware SAT representations, directly impacting the practicality of quantum SAT solving in both current and future fault-tolerant quantum architectures. Further progress in quantum SAT research will likely stem from continued optimization of encoding strategies and circuit synthesis frameworks tailored to quantum resource constraints.