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.
We consider an extension of classical stochastic reaction-diffusion (RD) dynamics to open quantum systems. We study a class of models of hard-core particles on a one-dimensional lattice whose dynamics is generated by a quantum master operator. Particle hopping is coherent while reactions, such as pair annihilation or pair coalescence, are dissipative. These are quantum open generalizations of the A+A→⌀ and A+A→A classical RD models. We characterize the relaxation of the state towards the stationary regime via a decomposition of the system Hilbert space into transient and recurrent subspaces. We provide a complete classification of the structure of the recurrent subspace (and the nonequilibrium steady states) in terms of the dark states associated to the quantum master operator and its general spectral properties. We also show that, in one dimension, relaxation towards these absorbing dark states is slower than that predicted by a mean-field analysis due to fluctuation effects, in analogy with what occurs in classical RD systems. Numerical simulations of small systems suggest that the decay of the density in one dimension, in both the open quantum A+A→⌀ and A+A→A systems, behaves asymptotically as t-b with 1/2<b<1.