Explicit reversible quantum oracles for bounded Diophantine systems achieve quadratic speedup with qubit count O((n + d²) log₂ N) and Toffoli depth O(q²).
Matiyasevich, Enumerable sets are diophantine, in Mathematical Logic in the 20th Century(2003) p
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From Hilbert's Tenth Problem to Quantum Speedup: Explicit Oracles for Bounded Diophantine Systems
Explicit reversible quantum oracles for bounded Diophantine systems achieve quadratic speedup with qubit count O((n + d²) log₂ N) and Toffoli depth O(q²).