Majd Assaad

Abhoy Kole, Majd Assaad, Till Schnittka et al.

Multi-controlled Toffoli gates are fundamental building blocks in quantum computation, with applications in quantum arithmetic, simulation, and search algorithms. In fault-tolerant architectures, their realization is constrained by the high cost of non-Clifford resources, particularly in terms of T-count and T-depth. Recent advances have demonstrated that the use of ancillary qubits, relative-phase Toffoli gates, and dynamic circuit techniques can substantially reduce this overhead. In this work, we investigate the decomposition of large Toffoli gates using 3- and 4-input relative-phase Toffoli gates in the presence of a single clean ancilla and conditionally clean ancillas. We derive explicit resource bounds for Clifford+T implementations incorporating dynamic-circuit-based uncomputation and measurement-conditioned corrections. Our analysis emphasizes T-depth reduction under fixed CX and T-count overhead, ensuring relevance for near-term devices. We show that introducing 4-input relative-phase Toffoli gates enables significant T-depth reductions through enhanced parallelism while maintaining favorable ancilla requirements. We further validate our theoretical results through experimental evaluation and comparative analysis with existing approaches.

Majd Assaad, Abhoy Kole, Rolf Drechsler

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.