Mixed-State Quantum Denoising Diffusion Probabilistic Model

TL;DR

MSQuDDPM addresses the challenge of generating mixed-state quantum data without scrambling unitaries by replacing the forward scrambling with depolarizing noise and employing backward parameterized quantum circuits with projective measurements. It introduces a cosine-exponent noise schedule, Haar random ancilla, and superfidelity-based losses to stabilize training and improve convergence, validated across clustered, circular, and many-body phase learning tasks. The results demonstrate faithful ensemble generation and magnetization recovery, with diffusion-step strategies outperforming heavier qubit configurations, highlighting practical advantages for near-term quantum devices. Overall, MSQuDDPM broadens the applicability of quantum diffusion models to mixed states while reducing implementation complexity, enabling more feasible near-term quantum generative modeling.

Abstract

Generative quantum machine learning has gained significant attention for its ability to produce quantum states with desired distributions. Among various quantum generative models, quantum denoising diffusion probabilistic models (QuDDPMs) [Phys. Rev. Lett. 132, 100602 (2024)] provide a promising approach with stepwise learning that resolves the training issues. However, the requirement of high-fidelity scrambling unitaries in QuDDPM poses a challenge in near-term implementation. We propose the \textit{mixed-state quantum denoising diffusion probabilistic model} (MSQuDDPM) to eliminate the need for scrambling unitaries. Our approach focuses on adapting the quantum noise channels to the model architecture, which integrates depolarizing noise channels in the forward diffusion process and parameterized quantum circuits with projective measurements in the backward denoising steps. We also introduce several techniques to improve MSQuDDPM, including a cosine-exponent schedule of noise interpolation, the use of single-qubit random ancilla, and superfidelity-based cost functions to enhance the convergence. We evaluate MSQuDDPM on quantum ensemble generation tasks, demonstrating its successful performance.

Figures (8)

• Figure 1: The model architecture and training strategy of MSQuDDPM are described in (b). The forward process utilizes depolarizing channels to inject noise, while the backward process in the inverse order is constructed with PQCs to learn the noise removal. (a) and (c) show MSQuDDPM's forward process samples from $t=0$ to $t=6$ and its backward procedure samples from $t = 6$ to $t=0$ in a single-qubit learning task. See Table \ref{['tab:expresult']} for details.
• Figure 2: Bloch sphere representations of (a) forward and (b) backward testing samples for the circular state ensemble generation. (c) displays the mean purity decay for the forward (blue) and the backward (red) results. The Wasserstein distance decay, $\text{Wass}(\{\tilde{\rho}_t\}, \{\rho_0\})$, for the forward (blue) and the backward (red) outcomes is plotted in (d). The Wasserstein distance is computed between the model outputs $\{\rho_t\}$ or $\{\tilde{\rho}_t\}$ and the initial input samples $\{\rho_0\}$.
• Figure 3: Distribution of magnetization along the X-axis ($M_x$) for the forward and backward results in the many-body phase learning task. The red line represents the distribution of backward test samples at $t = 0$. The blue and orange lines denote the X-axis magnetization distribution of the forward initial ensemble ($t = 0$) and the forward final ensemble ($t = T$), respectively.
• Figure 4: The MMD distance decay of models with different diffusion steps in the many-body phase learning task, computed between the training outputs $\{\tilde{\rho}_t\}$ and the samples from initial ensemble $\{\rho_0\}$. The purple line denotes the proposed model, which has a total of 864 parameters. The gray line represents the benchmark model with 840 parameters in total. Both models are trained with $n_{\rm train} = 50$ data. The periodic spikes are caused by the randomized initialization at each backward step.
• Figure 5: Comparison of average purity decay for forward samples and MMD distance decay between the output ensemble of each training iteration and the initial state ensemble in many-body phase learning task with different forward schedulings. The gray, green, and purple lines in (a) represent the decay of the mean purity throughout the forward diffusion process of the 4-qubit many-body phase learning task with linear, cosine, and cosine square scheduling, respectively. The final mean purity for the $n=4$ qubits is $1/(2^4) = 0.0625$. In (b), the models utilize $n=4$ qubits, $n_a = 2$ ancillary qubits with $|00\rangle$ state, $T=6$, $L=12$, and MMD distance with superfidelity as a cost function. The gray, green, and purple lines indicate the MMD distance decay for linear, cosine, and cosine square scheduling, respectively.