quant-phFeb 25, 2026

Learning Quantum Data Distribution via Chaotic Quantum Diffusion Model

Authors: Quoc Hoan Tran, Koki Chinzei, Yasuhiro Endo, Hirotaka Oshima

Organizations: Quantum Laboratory, Fujitsu Research, Fujitsu Limited, Kawasaki, Kanagawa 211-8588, Japan

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Abstract

Generative models for quantum data pose significant challenges but hold immense potential in fields such as chemoinformatics and quantum physics. Quantum denoising diffusion probabilistic models (QuDDPMs) enable efficient learning of quantum data distributions by progressively scrambling and denoising quantum states. However, existing implementations typically rely on circuit-based random unitary dynamics, which can be costly to implement and sensitive to control imperfections, particularly on analog quantum hardware. We propose the chaotic quantum diffusion model, a framework that generates projected ensembles via chaotic Hamiltonian time evolution, providing a flexible and hardware-compatible diffusion mechanism. Requiring only global, time-independent control, our approach substantially reduces implementation overhead across diverse analog quantum platforms while achieving accuracy comparable to QuDDPMs. This method improves trainability and robustness, broadening the applicability of quantum generative modeling.

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Figure 1: (a) General scheme of the quantum denoising diffusion probabilistic model (QuDDPM). The forward process applies quantum operations to initial quantum states ∣ψj(0)⟩|\psi^{(0)}_{j}\rangle in the training dataset S0S_{0} , generating progressively noisier ensembles SkS_{k} over KK steps. The backward process starts from random states {∣ψ~j(K)⟩}\{|\tilde{\psi}^{(K)}_{j}\rangle\} and trains parameterized quantum circuits VkV_{k} on the data system D\mathcal{D} ( ndn_{d} qubits) coupled with an ancilla system A\mathcal{A} ( nan_{a} qubits), followed by measurements on A\mathcal{A} to iteratively reconstruct the target distribution. (b) Random unitary circuit diffusion (RUCD), where sequences of random unitary gates are applied to each training instance. (c) Chaotic quantum diffusion (our proposal) via the projected ensemble framework. Here, a chaotic and fixed Hamiltonian HH governs time evolution on a bipartite system comprising the data subsystem M{\mathcal{M}} ( ndn_{d} qubits) and the complement subsystem F{\mathcal{F}} (initialized by random computational basis states with nfn_{f} qubits). Projective measurements on F{\mathcal{F}} yield a projected ensemble on M{\mathcal{M}} . In creating the diffused state ensemble Sk{\mathcal{S}}_{k} , applying cumulative evolution kΔtk\Delta t to S0{\mathcal{S}}_{0} defines cumulative time evolution diffusion (CTED), while applying repeated time evolution Δt\Delta t to Sk−1{\mathcal{S}}_{k-1} defines repeated time evolution diffusion (RTED).
Figure 2: The schematic circuit to compute the fidelity between the forward state ∣ψj(k−1)⟩\ket{\psi_{j}^{(k-1)}} and the denoised state ∣ψ~j(k−1)⟩\ket{\tilde{\psi}_{j}^{(k-1)}} using the SWAP test.
Figure 3: Average adjacent-gap ratio ⟨r⟩\braket{r} of the mixed-field Ising Hamiltonian in Eq. ( 6 ) across the (hx,hy)(h^{x},h^{y}) plane at J=1J=1 and the total system size nd+nf=14n_{d}+n_{f}=14 . ⟨r⟩≈0.53\braket{r}\approx 0.53 indicates chaotic and Wigner-Dyson statistics, where ⟨r⟩≈0.39\braket{r}\approx 0.39 indicates integrable and Poisson statistics. The star marks the operating point (hx,hy)=(0.8090,0.9045)(h^{x},h^{y})=(0.8090,0.9045) , well inside the chaotic region.
Figure 4: 1-Wasserstein distance and normalized moment distance ΔHaar(m)(k)\Delta_{\textrm{Haar}}^{(m)}(k) between the diffused ensemble and the Haar-random ensemble versus diffusion step kk , for CTED and RTED. Results are shown for different complement sizes nf∈{2,4,6,8}n_{f}\in\{2,4,6,8\} and moment orders mm . The dashed horizontal line in each panel is the finite-sample Haar floor. It depends only on the sample size NN and the moment order mm and is independent of nfn_{f} .
Figure 5: 1-Wasserstein distance and normalized moment distance ΔHaar(m)(k)\Delta_{\textrm{Haar}}^{(m)}(k) between the diffused ensemble and the target multi-cluster ensemble versus diffusion step kk , for CTED and RTED. Results are shown for different complement sizes nf∈{2,4,6,8}n_{f}\in\{2,4,6,8\} and moment orders mm . Inset plots display values at diffusion step k≤50k\leq 50 .
Figure 6: 1-Wasserstein distances between the generated ensembles and the true ensemble over backward steps, sampled from (a) multi-cluster and (b) circular datasets of quantum states, for CTED (left), RTED (center), and RUCD (right). The backward process (solid lines with circle markers) is compared to the forward diffusion (dotted lines). Lines and shaded areas represent the mean and standard deviation over ten trials. Inset plots display values at diffusion step k≤20k\leq 20 .
Figure 7: 1-Wasserstein distance between the generated ensemble and the true ensemble of the QM9 subset dataset as a function of diffusion step kk , comparing backward diffusion performed in the full Hilbert space (“Full”, dashed) and in the latent space learned by a quantum autoencoder (“Latent”, solid). Results are shown for CTED (left), RTED (center), and RUCD (right). Shaded regions denote the standard deviation over ten trials.
Figure 8: Wasserstein distance between the generated ensemble and the target ensemble as a function of the noise level for CTED (left), RTED (center), and RUCD (right). Each marker corresponds to an independent run. CTED and RTED exhibit gradual degradation as noise increases, whereas RUCD shows a sharp loss of performance at higher noise levels. Here, we consider different physical noise mechanisms for the gate-based ( p1p_{1} with RUCD) and analog ( p2p_{2} with CTED and RTED) models.

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