Generative Quantum Machine Learning via Denoising Diffusion Probabilistic Models

Deep generative models are key-enabling technology to computer vision, text generation, and large language models. Denoising diffusion probabilistic models (DDPMs) have recently gained much attention due to their ability to generate diverse and high-quality samples in many computer vision tasks, as well as to incorporate flexible model architectures and a relatively simple training scheme. Quantum generative models, empowered by entanglement and superposition, have brought new insight to learning classical and quantum data. Inspired by the classical counterpart, we propose the quantum denoising diffusion probabilistic model (QuDDPM) to enable efficiently trainable generative learning of quantum data. QuDDPM adopts sufficient layers of circuits to guarantee expressivity, while it introduces multiple intermediate training tasks as interpolation between the target distribution and noise to avoid barren plateau and guarantee efficient training. We provide bounds on the learning error and demonstrate QuDDPM’s capability in learning correlated quantum noise model, quantum many-body phases, and topological structure of quantum data. The results provide a paradigm for versatile and efficient quantum generative learning.

Figure 1

Schematic of QuDDPM. The forward noisy process is implemented by a quantum scrambling circuit (QSC) in (a), while in the backward denoising process is achieved via measurement enabled by ancilla and PQC in (d). Subplots (b1)–(b5) and (c1)–(c5) present the Bloch sphere dynamics in generation of states clustering around $|0⟩$, where convergence can be seen despite sample fluctuations, as shown in the Supplemental Material [33].

Figure 2

The training of QuDDPM at each step $t=k$. Pairwise distance between states in generated ensemble ${\tilde{\psi }}_i^{(k)}\in {\tilde{\mathcal{S}}}_k$ and true diffusion ensemble ${\psi }_j^{(k)}\in {\mathcal{S}}_k$ is measured and utilized in the evaluation of the loss function $\mathcal{L}$.

Figure 3

The decay of MMD distance $\mathcal{D}$ between generated ensemble ${\tilde{\mathcal{S}}}_t$ using different models and target ensemble of states ${\mathcal{E}}_0$ clustered around $|0,0⟩$ versus training steps. The converged value is $\mathcal{D}\simeq 0.002$ for QuDDPM, showing an advantage of 2 orders of magnitude over QuDT and QuGAN.

Figure 4

The generalization error of QuDDPM in generating cluster states versus (a) diffusion steps $T$ and (b) training dataset size $N$. Dots are numerical results, and orange dashed line is linear fitting results with both exponents equal to 1 within the numerical precision.

Figure 5

Generation of states with probabilistic correlated noise on a specific state in (a) and (b) states with ferromagnetic phase. In (a), average fidelity $\overline{F_{10}}$ between states at step $t$ and $|10⟩$ for diffusion (red), training (blue), and testing (green) are plotted. In (b), we show the distribution of magnetization for generated data from training (blue) and testing (green) dataset, and compared to true data (red) and full noise (orange). Four qubits are considered in (b).

Figure 6

Bloch visualization of the forward (a1)–(a3) and backward (b1)–(b3),(c1)–(c3) process. (d),(e) deviation of generated states from unit circle in $X\text{−}Z$ plane. The deviation $⟨Y{⟩}^2$ for forward diffusion (red), backward training (blue), and backward test (green) are plotted. The shaded area shows the sample standard deviation.