Channel-Constrained Markovian Quantum Diffusion Model from Open System Perspective

TL;DR

The paper introduces the Channel-Constrained Markovian Quantum Diffusion (CCMQD) model, a physically grounded approach to quantum state generation that treats diffusion as open-system decoherence and denoising as learning inverse quantum channels. By enforcing CPTP constraints through Kraus representations and Stiefel-manifold optimization, CCMQD achieves high-fidelity state reconstruction across up to 7 qubits, even under Haar-random noise. A holistic training strategy (HQTO) with path-constrained loss (PC-loss) is shown to substantially outperform modular optimization, preserving nonlocal quantum correlations throughout the diffusion trajectory. The results demonstrate that environmental interactions can be harnessed for robust quantum state synthesis and suggest practical pathways for implementing quantum diffusion on near-term hardware, with implications for quantum error mitigation and state preparation.

Abstract

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.

Figures (10)

• Figure 1: Channel-based framework for quantum diffusion. Forward diffusion (top) progressively achieve diffusion through sequential quantum channels $\mathcal{E}_i$ , while backward denoising (bottom) reconstructs the initial state through learned inverse channels $\mathcal{E}^{-1}_i$. Each channel is characterized by its Kraus operator decomposition.
• Figure 2: Quantum diffusion framework with forward noising and backward denoising. Framework showing forward diffusion via noise channels (upper, left-to-right) and backward denoising via learnable channels (lower, right-to-left). Colors indicate entropy: red (low) to blue (high). Asymmetric time steps reflect different dynamics of noising and denoising processes.
• Figure 3: Schematic representation of $(L_f, k_f)$ and $(L_b, k_b)$ noise configurations
• Figure 4: Scalability of channel-constrained reconstruction across Quantum system scale. Performance comparison between SQCO and HQTO+PC-loss for systems ranging from single qubits to 7-qubit entangled states. While SQCO shows rapid degradation with increasing system size, HQTO+PC-loss maintains fidelity $>0.99$ across systems under both noise types. Random noise conditions exhibit slightly reduced robustness compared to depolarizing noise, but still achieve reconstruction quality with fidelity $>0.997$.
• Figure 5: SQCO optimization targets: step-wise quantum state reconstruction. The optimization goal is to minimize the sum of differences between reconstructed states $\hat{\rho}_i$ and original states $\rho_i$ as $\sum_{i=0}^{L-1} [1 - F(\rho_i, \hat{\rho}_i)]$, where $F$ denotes the fidelity. This follows the principle of modular optimization where individual denoising operations are specialized for their corresponding forward noise channels.