Thermal Quantum Annealing (tQuA) is an open‑system quantum computing architecture in which an Ising‑type continuous variable array coupled non‑adiabatically to a thermal reservoir evolves toward the ground state of a target Hamiltonian by dissipating energy to its environment. This paper provides a rigorous mathematical foundation for tQuA and its architectural extension to Quantum Intelligent Matrix Methods (QIMM). We enforce strict mathematical honesty by explicitly distinguishing among Theorems (proved in cited literature), Propositions (proof sketches with identified gaps), and Conjectures (claims with open steps). The work rests on a five‑pillar proof chain establishing global Gibbs convergence: (1) Classical‑Quantum Equivalence (Werbos 2002, 2004) ; (2) Lindblad CPTP dynamics (Lindblad 1976) ; (3) Eigenstate‑Free Lindblad engineering (Cattaneo et al. 2021) ; (4) Global convergence to the unique stationary state (Spohn 1977) ; and (5) Mixing time bounds via the Liouvillian spectral gap (Kastoryano & Temme 2013). Within this framework we deliver three new contributions. First, we report honest numerical scaling benchmarks for the Liouvillian spectral gap Δ on gopherhole landscapes. By migrating from earlier classical Metropolis step‑heuristics to a physically accurate continuous Ohmic bath satisfying Kubo‑Martin‑Schwinger detailed balance, we uncover an unexpected inversion: the "Hard" regime (where local minima grow proportionally with system size) scales better than the sparse regime, exhibiting a pre‑asymptotic power‑law decay Δ ∼ d^−0. 72 vs. d^−0. 88. We identify this as a collective thermal assistance effect—the dense landscape provides an effective energy ladder—and provide the full numerical dataset up to 24 qubits (d = 2^24). Second, we strengthen the earlier Proposition 1 into a full Theorem 7, proving a rigorous asymptotic lower bound Δ ≥ C / log d for the dense gopherhole ensemble. The proof proceeds by reduction to a classical Markov chain, comparison to a complete graph, and application of a modified logarithmic Sobolev inequality. The resulting mixing time τₘix = O (log d · log (1/ε) ) provides a provable polynomial quantum advantage over classical methods that suffer exponential slowdown on the same landscapes. Third, we formalize tQuA as an adaptive policy system, where temperature schedules β (t) and coupling strengths γ (t) are runtime‑adjustable degrees of freedom, and we outline the architecture of Quantum Intelligent Matrix Methods (QIMM) —brain‑inspired matrix algorithms, including quantum associative memory and quantum Decoupled Extended Kalman Filtering, that leverage tQuA's nonconvex optimization power for energy‑efficient AI and astronomical imaging. The paper is structured as a hybrid document: front sections provide a manager‑accessible overview grounded in the six core claims of US Patent 12, 118, 434 B2, while later sections and appendices contain full mathematical rigor for specialists. All code, data, and a portable CUDA‑Q demonstration are publicly available.
Claude et al. (Wed,) studied this question.