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Learning Quantum Data Distribution via Chaotic Quantum Diffusion Model

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

TL;DR

This work proposes the chaotic quantum diffusion model, a framework that generates projected ensembles via chaotic Hamiltonian time evolution, providing a flexible and hardware-compatible diffusion mechanism and improves trainability and robustness, broadening the applicability of quantum generative modeling.

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 that can be costly to realize 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.

Learning Quantum Data Distribution via Chaotic Quantum Diffusion Model

TL;DR

This work proposes the chaotic quantum diffusion model, a framework that generates projected ensembles via chaotic Hamiltonian time evolution, providing a flexible and hardware-compatible diffusion mechanism and improves trainability and robustness, broadening the applicability of quantum generative modeling.

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 that can be costly to realize 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.
Paper Structure (15 sections, 2 theorems, 9 equations, 7 figures)

This paper contains 15 sections, 2 theorems, 9 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Quantum Denoising Diffusion Probabilistic Model (QuDDPM)
  3. Forward diffusion process
  4. Backward denoising process
  5. Proposal: Chaotic Quantum Diffusion
  6. Projected Ensemble
  7. Cumulative Time Evolution Diffusion (CTED)
  8. Repeated Time Evolution Diffusion (RTED)
  9. Universal Ensemble
  10. Results
  11. Learning Multi-cluster and Circular Distribution
  12. Learning Quantum Chemistry Distribution
  13. Resource Comparison
  14. Noise Robustness
  15. Summary and Discussion

Key Result

Lemma 1

Let ${\mathcal{N}}_{\mathcal{F}}$ be an arbitrary CPTP noise channel acting on ${\mathcal{F}}$ immediately before the measurement. Then the measurement statistics under ${\mathcal{N}}_{\mathcal{F}}$ are identical to those obtained by measuring the noiseless state $\ket{\Phi}$ with a POVM $\{E_{\bold where ${\mathcal{N}}_{\mathcal{F}}^\dagger$ is the adjoint (Heisenberg-picture) map. In particular,

Figures (7)

  • Figure 1: (a) The general scheme of quantum denoising diffusion probabilistic model. (b) The implementation of the random unitary circuit diffusion (RUCD) model. (c) The chaotic quantum diffusion model in our proposal.
  • Figure 2: The schematic circuit to compute the fidelity between the forward state $\ket{\psi_j^{(k-1)}}$ and the denoised state $\ket{\tilde{\psi}_j^{(k-1)}}$ using the SWAP test.
  • Figure 3: 1-Wasserstein distance and normalized moment distance $\Delta_{\textrm{Haar}}^{(m)}(k)$ between the diffused ensemble and the Haar-random ensemble versus diffusion step $k$, for CTED and RTED. Results are shown for different complement sizes $n_f \in \{2, 4, 6, 8\}$ and moment orders $m$.
  • Figure 4: 1-Wasserstein distance and normalized moment distance $\Delta_{\textrm{Haar}}^{(m)}(k)$ between the diffused ensemble and the target multi-cluster ensemble versus diffusion step $k$, for CTED and RTED. Results are shown for different complement sizes $n_f \in \{2, 4, 6, 8\}$ and moment orders $m$.
  • Figure 5: 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.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Lemma 1: Noise becomes POVM
  • proof
  • Corollary 1: Pauli relabeling invariance
  • proof