Thermal Quantum Annealing (tQuA) is an open-system quantum architecture for solving the Quantum Quadratic Optimization (QQO) problem: finding the vector that minimizes a quadratic form , where is a Hermitian matrix. This is a continuous-variable optimization problem, with aleph-1 possible solutions, and it is the problem class that underlies nonconvex optimization in deep learning, cognitive prediction, and reinforcement learning for decision and control.

This paper addresses a defect in the current literature on quantum annealing: almost all published work, including our own prior versions, treats the QUBO problem — binary variables with configurations — as if it were the same problem. It is not. A QUBO solver cannot perform the continuous optimization that deep learning requires. The distinction is shown in the figure below (Section 1).

We present a corrected architecture, ED-tQuA for continuous variables, in which:

• the core dynamics are governed by the Ising equation on continuous complex vectors ;
• thermal exchange with a reservoir at temperature drives the distribution of toward the Boltzmann equilibrium;
• engineered dissipation — programmable non-unitary terms acting on the continuous modes — accelerates convergence by increasing the effective spectral gap of the continuous Fokker-Planck/Lindblad operator;
• the architecture maps naturally to neutral-atom hardware (Pasqal, Infleqtion) via collective spin encodings of continuous variables.

We prove three results for the continuous-variable architecture:

1. Convergence: the generalized Boltzmann distribution over is the unique stationary state, and the system converges to it from any initial distribution.
2. Positive escape rate: the thermal reservoir provides a strictly positive transition rate out of every local minimum of the continuous landscape at any .
3. Engineered acceleration: appropriately designed non-local damping terms increase the spectral gap of the evolution operator relative to the natural-bath case, yielding a provable speedup in the continuous-variable setting.

No claims are made here that depend on the discrete QUBO analysis of v1.