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Mitigating Barren plateaus in quantum denoising diffusion probabilistic models

Haipeng Cao, Kaining Zhang, Dacheng Tao, Zhaofeng Su

TL;DR

This work identifies a barren plateau problem in quantum denoising diffusion probabilistic models (QuDDPM) caused by Haar-random inputs to the denoising circuits, which eliminates trainability as system size grows. The authors propose an improved QuDDPM that maintains the forward-diffusion output at a controlled distance from Haar, thereby preserving gradient information during training. They provide theoretical analyses showing gradient lower bounds under proper parameter scaling and validate the approach with experiments across multiple qubit counts, demonstrating improved training dynamics and sample quality. The findings advance scalable quantum generative learning by offering a practical mitigation to barren plateaus in diffusion-based quantum models.

Abstract

Quantum generative models leverage quantum superposition and entanglement to enhance learning efficiency for both classical and quantum data. The quantum denoising diffusion probabilistic model (QuDDPM), inspired by its classical counterpart, has been proposed as a promising framework for quantum generative learning. QuDDPM is capable of efficiently learning and generating quantum data, and it demonstrates excellent performance in learning correlated quantum noise models, quantum many-body phases, and the topological structure of quantum data. However, we show that barren plateaus emerge in QuDDPMs due to the use of 2-design states as the input for the denoising process, which severely undermines the performance of QuDDPM. Through theoretical analysis and experimental validation, we confirm the presence of barren plateaus in the original QuDDPM. To address this issue, we introduce an improved QuDDPM that utilizes a distribution maintaining a certain distance from the Haar distribution, ensuring better trainability. Experimental results demonstrate that our approach effectively mitigates the barren plateau problem and generates samples with higher quality, paving the way for scalable and efficient quantum generative learning.

Mitigating Barren plateaus in quantum denoising diffusion probabilistic models

TL;DR

This work identifies a barren plateau problem in quantum denoising diffusion probabilistic models (QuDDPM) caused by Haar-random inputs to the denoising circuits, which eliminates trainability as system size grows. The authors propose an improved QuDDPM that maintains the forward-diffusion output at a controlled distance from Haar, thereby preserving gradient information during training. They provide theoretical analyses showing gradient lower bounds under proper parameter scaling and validate the approach with experiments across multiple qubit counts, demonstrating improved training dynamics and sample quality. The findings advance scalable quantum generative learning by offering a practical mitigation to barren plateaus in diffusion-based quantum models.

Abstract

Quantum generative models leverage quantum superposition and entanglement to enhance learning efficiency for both classical and quantum data. The quantum denoising diffusion probabilistic model (QuDDPM), inspired by its classical counterpart, has been proposed as a promising framework for quantum generative learning. QuDDPM is capable of efficiently learning and generating quantum data, and it demonstrates excellent performance in learning correlated quantum noise models, quantum many-body phases, and the topological structure of quantum data. However, we show that barren plateaus emerge in QuDDPMs due to the use of 2-design states as the input for the denoising process, which severely undermines the performance of QuDDPM. Through theoretical analysis and experimental validation, we confirm the presence of barren plateaus in the original QuDDPM. To address this issue, we introduce an improved QuDDPM that utilizes a distribution maintaining a certain distance from the Haar distribution, ensuring better trainability. Experimental results demonstrate that our approach effectively mitigates the barren plateau problem and generates samples with higher quality, paving the way for scalable and efficient quantum generative learning.

Paper Structure

This paper contains 24 sections, 4 theorems, 48 equations, 9 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Notations and quantum computing knowledge
  4. Notations
  5. Quantum computing
  6. Barren plateaus in quantum machine learning
  7. Overview of QuDDPM
  8. Theoretical results
  9. Theoretical analysis of barren plateau
  10. Improved QuDDPM and Its Effectiveness Analysis
  11. Experimental results
  12. Limitations
  13. Conclusions
  14. Partial derivatives of PQCs
  15. Partial derivatives of loss
  16. ...and 9 more sections

Key Result

Theorem 1

For Eq. eq:L_haar_p and Eq. eq:L_ahaar_p, if $U_{Hi}\in\mathcal{U}_H(N)$ obeying Haar measure, $U_{Ai}\in\mathcal{U}_A(N)$ obeying approximate Haar measure, then: where $n_{data}$ is the number of data qubits, $\varepsilon, \zeta$ denote the deviations of approximate Haar random states from exact Haar random states at different angles. For detailed definitions, see Appendix.

Figures (9)

  • Figure 1: Structure of QuDDPM.The top part of the figure shows the forward noisy diffusion process and the bottom part shows the backward denoising process.
  • Figure 2: Quantum circuit architectures. (a) is the circuit of one step of the forward diffusion process on a system of $n=3$ qubits. (b) is one-layer architecture of L-layer of $\tilde{U}_t(\bm \theta_t)$ on a system of $n=3$ data and $n_A=2$ ancilla qubits.
  • Figure 3: The evolution of the loss function of the original QuDDPM during the first six training cycles of the backward denoising process for different numbers of qubits.
  • Figure 4: During the first six training cycles, the changes in the average training gradients of the improved and original QuDDPM systems as the number of qubits increased.
  • Figure 5: The KL divergence between the sample data generated at each denoising step by the improved QuDDPM and the original QuDDPM and the target data. This quantum system contains 10 qubits, that is, $n_{data}$ = 10
  • ...and 4 more figures

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • proof
  • Lemma 2
  • proof