Quantum diffusions are non-commutative generalisations of classical diffusions on manifolds, in which the manifold i s ‘quantised’ by deforming the algebra of smooth functions thereon to a non-commutative algebra, and in which quantum stochastic calculus replaces the Itô calculus of Brownian motion. Quantum diffusions fall into two categories; inner diffusions, which are in a certain sense cohomologically trivial, and which act by conjugating by a unitary process, and outer diffusions, which do not. A natural perturbation procedure is described for linking two cohomologically equivalent quantum diffusions; it involves the construction of a new type of intertwining unitary process.
We present a channel-constrained Markovian quantum diffusion (CCMQD) model that prepares quantum states by rigorously framing the generative process within the dynamics of open quantum systems. Our model interprets the forward diffusion process as natural decoherence using quantum master equations, whereas the reverse denoising is achieved by learning inverse quantum channels. Our core innovation is a comprehensive channel-constrained framework: we model the diffusion and denoising steps as quantum channels defined by Kraus operators, ensure their physical validity through optimization on the Stiefel manifold, and introduce tailored training strategies and loss functions that leverage this constrained structure for high-fidelity state reconstruction. Experimental validation on systems ranging from single qubits to entangled states
7 -qubits demonstrates high-fidelity state generation, achieving fidelities exceeding
0.998 under both random and depolarizing noise conditions. This work confirms that quantum diffusion can be characterized as a controlled Markov evolution, demonstrating that environmental interactions are not limited to being a source of decoherence but can also be utilized to achieve high-fidelity quantum state synthesis.