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Towards efficient quantum algorithms for diffusion probabilistic models

Yunfei Wang, Ruoxi Jiang, Yingda Fan, Xiaowei Jia, Jens Eisert, Junyu Liu, Jin-Peng Liu

TL;DR

The paper tackles the high computational cost of diffusion probabilistic models (DPMs) on large-scale data by introducing efficient quantum algorithms that implement DPMs through quantum ODE solvers. It develops a Carleman linearization framework to embed nonlinear diffusion dynamics into linear quantum systems, yielding two architectures: DPM-solver-$k$ (exact $k$-th derivatives) and UniPC (finite-difference derivatives), with backends based on quantum linear system solvers ($QLSS$) or linear combination of Hamiltonian simulations ($LCHS$). The authors provide informal complexity theorems and numerical experiments on latent representations of ImageNet-100 to validate dissipativity and practical feasibility. Overall, the work presents a concrete, pragmatic path toward quantum utility in large-scale generative modeling, with potential to impact inference and synthesis tasks on future fault-tolerant quantum hardware.

Abstract

A diffusion probabilistic model (DPM) is a generative model renowned for its ability to produce high-quality outputs in tasks such as image and audio generation. However, training DPMs on large, high-dimensional datasets such as high-resolution images or audio incurs significant computational, energy, and hardware costs. In this work, we introduce efficient quantum algorithms for implementing DPMs through various quantum ODE solvers. These algorithms highlight the potential of quantum Carleman linearization for diverse mathematical structures, leveraging state-of-the-art quantum linear system solvers (QLSS) or linear combination of Hamiltonian simulations (LCHS). Specifically, we focus on two approaches: DPM-solver-$k$ which employs exact $k$-th order derivatives to compute a polynomial approximation of $ε_θ(x_λ,λ)$; and UniPC which uses finite difference of $ε_θ(x_λ,λ)$ at different points $(x_{s_m}, λ_{s_m})$ to approximate higher-order derivatives. As such, this work represents one of the most direct and pragmatic applications of quantum algorithms to large-scale machine learning models, presumably taking substantial steps towards demonstrating the practical utility of quantum computing.

Towards efficient quantum algorithms for diffusion probabilistic models

TL;DR

The paper tackles the high computational cost of diffusion probabilistic models (DPMs) on large-scale data by introducing efficient quantum algorithms that implement DPMs through quantum ODE solvers. It develops a Carleman linearization framework to embed nonlinear diffusion dynamics into linear quantum systems, yielding two architectures: DPM-solver- (exact -th derivatives) and UniPC (finite-difference derivatives), with backends based on quantum linear system solvers () or linear combination of Hamiltonian simulations (). The authors provide informal complexity theorems and numerical experiments on latent representations of ImageNet-100 to validate dissipativity and practical feasibility. Overall, the work presents a concrete, pragmatic path toward quantum utility in large-scale generative modeling, with potential to impact inference and synthesis tasks on future fault-tolerant quantum hardware.

Abstract

A diffusion probabilistic model (DPM) is a generative model renowned for its ability to produce high-quality outputs in tasks such as image and audio generation. However, training DPMs on large, high-dimensional datasets such as high-resolution images or audio incurs significant computational, energy, and hardware costs. In this work, we introduce efficient quantum algorithms for implementing DPMs through various quantum ODE solvers. These algorithms highlight the potential of quantum Carleman linearization for diverse mathematical structures, leveraging state-of-the-art quantum linear system solvers (QLSS) or linear combination of Hamiltonian simulations (LCHS). Specifically, we focus on two approaches: DPM-solver- which employs exact -th order derivatives to compute a polynomial approximation of ; and UniPC which uses finite difference of at different points to approximate higher-order derivatives. As such, this work represents one of the most direct and pragmatic applications of quantum algorithms to large-scale machine learning models, presumably taking substantial steps towards demonstrating the practical utility of quantum computing.

Paper Structure

This paper contains 9 sections, 2 theorems, 11 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Embedding classical neural networks
  4. Carleman linearization for DPM-solvers
  5. Carleman linearization for UniPC
  6. End-to-end quantum ODE solver
  7. Theorems
  8. Numerical analysis and experiments
  9. Discussion

Key Result

Theorem 1

For a DPM-solver with a denoising function expanded up to the $J$-th order Taylor approximation and truncated at $N$, the system of linear equations $M \boldsymbol{Y} = \boldsymbol{b}$ can be solved using a quantum algorithm. If $M$ is sparse and block-encoded efficiently, the quantum state proporti where $\kappa>0$ is an upper bound on the condition number of $M$. The algorithm achieves a success

Figures (2)

  • Figure 1: Sketch of classical (1.) diffusion probability models (DPM) and the quantum algorithms proposed here for quantum analogs (2.-4.). We pursue two approaches: This is on the one hand Carleman linearization for DPM solvers (2.), on the other Carleman linearization for UniPC (3.). For both approaches, quantum ODE solvers are instrumental in the step before measurement. QLSS stands for quantum linear system solvers, LCHS for linear combination of Hamiltonian simulation.
  • Figure 2: Results of numerical experiments. 1. Spectrum dynamics of the reverse diffusion process on ImageNet-100. Using the DPM-Solver lu2022dpm, we show the eigenvalue trajectories of the Jacobian $\mathbf{J}(t)$ for dimensions $d=1024,4096$, and $16384$, each representing image generation at a different resolution, ranging from coarse to fine-grained. Across all three cases, the eigenvalues start positive and progressively develop a larger spectral gap with extremely positive values, highlighting the system’s consistent dissipative behavior. 2. Dissipative measure of the reverse diffusion process on ImageNet-100. The shaded area represents the [25th, 75th] percentile of the statistics. Across three models with $d=$ 1024, 4096, and 16384, $P(t)$ consistently decreases with $t$.

Theorems & Definitions (2)

  • Theorem 1: Informal complexity of the DPM-solver
  • Theorem 2: Informal complexity of the UniPC-$p$ framework