Quantum Generative Diffusion Model: A Fully Quantum-Mechanical Model for Generating Quantum State Ensemble

TL;DR

The paper introduces a fully quantum diffusion framework, QGDM, for generating quantum state ensembles by diffusing any target state $\rho_0$ to the completely mixed state $\mathbb{I}/d$ via a non-unitary forward process and then denoising with a trainable non-unitary backward process. It achieves parameter-efficient learning through timestep embedding and shared backward-process parameters, and further provides a resource-efficient variant (RQGDM) that reduces qubit overhead. Empirical results show that QGDM and RQGDM outperform quantum GAN baselines (QuGAN, EQ-GAN) in both pure and mixed-state generation, with up to 53.02% relative fidelity improvements in mixed-state tasks across 1–8 qubits, and high single-qubit fidelity approaching 0.999 in small systems. The work highlights the potential of quantum diffusion models for scalable quantum state generation and suggests directions for noise schedules, denoiser design, and theoretical convergence analyses.

Abstract

Classical diffusion models have shown superior generative results. Exploring them in the quantum domain can advance the field of quantum generative learning. This work introduces Quantum Generative Diffusion Model (QGDM) as their simple and elegant quantum counterpart. Through a non-unitary forward process, any target quantum state can be transformed into a completely mixed state that has the highest entropy and maximum uncertainty about the system. A trainable backward process is used to recover the former from the latter. The design requirements for its backward process includes non-unitarity and small parameter count. We introduce partial trace operations to enforce non-unitary and reduce the number of trainable parameters by using a parameter-sharing strategy and incorporating temporal information as an input in the backward process. We present QGDM's resource-efficient version to reduce auxiliary qubits while preserving generative capabilities. QGDM exhibits faster convergence than Quantum Generative Adversarial Network (QGAN) because its adopted convex-based optimization can result in better convergence. The results of comparing it with QGAN demonstrate its effectiveness in generating both pure and mixed quantum states. It can achieve 53.02% higher fidelity in mixed-state generation than QGAN. The results highlight its great potential to tackle challenging quantum generation tasks.

Figures (16)

• Figure 1: The core idea of the proposed QGDM.
• Figure 1: The conventional ansatz lloyd2020 is used for our timestep embedding with trainable parameter $\boldsymbol{\omega}$, and given parameter $t' = t\pi/T$. The circuit in the dashed lines is repeated for $L_{\mathcal{T}}$ times. Where $l\in\{1,\dots,L_{\mathcal{T}}\}$.
• Figure 2: The framework of the proposed Quantum Generative Diffusion Model (QGDM).
• Figure 2: The architecture of a 4-qubit denoising circuit $U(\boldsymbol{\theta})$, where $\boldsymbol{\theta}$ denotes trainable parameters. To enhance its learning capability, the dashed-boxed part is repeated $L$ times. $l\in \{1,\dots,L\}$.
• Figure 3: The denoising process framework, $f_{\boldsymbol{\Theta}}(\rho_t,t)$, is composed of a timestep embedding circuit $\mathcal{T}(\boldsymbol{\omega},t')$ and a denoising circuit $U(\boldsymbol{\theta})$, where $\boldsymbol{\Theta}=\{\boldsymbol{\omega},\boldsymbol{\theta}\}$.