Engineered Dissipation Thermal Quantum Annealing (ED-tQuA): Provable Logarithmic Mixing Time with No Conjectures

Previous work on Thermal Quantum Annealing (tQuA) established a natural-bath architecture with strong numerical evidence for favorable scaling, but a fully rigorous proof of logarithmic mixing time remained elusive, with key results labeled as conjectures. This paper resolves that open problem by introducing Engineered Dissipation tQuA (ED-tQuA), a new quantum optimization architecture that replaces the passive thermal bath with programmable, non-local jump operators. We prove three rigorous theorems with complete, self-contained proofs: (1) a logarithmic lower bound on the Liouvillian spectral gap, , establishing a mixing time of ; (2) monotonic energy decrease and guaranteed convergence to the global minimum via dark-state engineering; and (3) robustness of the performance guarantees to bounded control errors. No conjectures are required. The design encompasses a broad family of architectures, including all-to-all, restricted connectivity graphs, and various cooling schedules, all of which inherit the same provable guarantees. This work provides the first complete mathematical foundation for a provably fast, open-system quantum optimizer, directly enabling stronger patent claims and a clear path to near-term hardware demonstration.