QDiffusion-TS: Hybrid Quantum Time Series Synthesis

• QDiffusion-TS is a hybrid quantum-classical diffusion model that replaces transformer FFNNs with quantum neural networks for time series synthesis.
• It achieves significant parameter reduction and improves distribution reproduction, reducing Wasserstein distance by up to 51% on financial datasets.
• Empirical results show that synthetic data from QDiffusion-TS enhances forecasting accuracy and performs reliably on both simulators and real quantum hardware.

QDiffusion-TS is a hybrid quantum-classical diffusion architecture for time-series synthesis that extends the classical Diffusion-TS framework by replacing the transformer feed-forward neural network submodules in the denoising model with quantum neural networks while retaining the rest of the diffusion and transformer pipeline in classical form. It is presented as the first quantum generative diffusion model for time series synthesis and is evaluated on financial time series from Apple and Amazon, with additional validation on the IQM Emerald quantum processor via Amazon Braket (Waller et al., 25 Jun 2026).

1. Definition, scope, and motivation

QDiffusion-TS is designed for generative modeling of real-world time series, especially financial series whose distributions are described as non-stationary, heavy-tailed, and characterized by volatility clustering and long-range dependencies. The method is motivated by two claims: classical generative models become expensive as model scale increases, and quantum neural networks may represent complex functions more compactly because an $n$-qubit system lives in a $2^n$-dimensional Hilbert space and can exploit entanglement (Waller et al., 25 Jun 2026).

Within that framing, QDiffusion-TS does not replace the diffusion paradigm itself. The forward process remains standard diffusion, and the denoising model remains transformer-based. The intervention is architectural rather than probabilistic: the learnable nonlinear feed-forward transformations inside the denoising transformer are replaced by parameterized quantum circuits. This makes QDiffusion-TS a hybrid quantum transformer rather than a fully quantum diffusion model.

The method targets time-series synthesis rather than forecasting alone. Its reported evaluation therefore emphasizes two distinct questions: whether synthetic sequences reproduce the real distributional structure of financial log-returns, and whether those sequences are useful as augmentation data in a downstream forecasting task. This dual framing is central to the paper’s positioning of the model (Waller et al., 25 Jun 2026).

2. Hybrid denoising transformer architecture

QDiffusion-TS extends the encoder-decoder transformer used in Diffusion-TS. The retained classical components include self-attention for temporal dependencies, a trend synthetic layer for global patterns, and a Fourier synthetic layer for periodic and irregular frequency structure. The architectural change is that every FFNN block inside the transformer encoder and decoder is replaced by a QNN (Waller et al., 25 Jun 2026).

Component	Classical version	QDiffusion-TS version
Parameters per replaced block	33,088	36
Total model parameters	589,030	461,254

The per-block reduction is described as nearly three orders of magnitude, while the full-model reduction is roughly 20%. The paper therefore makes a sharp distinction between local and global compression effects: the quantum substitution is extremely compact at the level of the replaced nonlinear submodule, but the surrounding attention and diffusion machinery remains classical and contributes substantially to the total parameter count.

This architectural boundary is important for interpretation. The “quantum” component is specifically the nonlinear transformation inside the transformer blocks, not the entire denoising pipeline. A common misconception is to read the model as a fully quantum generator; the reported system is instead a classical diffusion model whose FFNN sublayers are replaced by QNNs (Waller et al., 25 Jun 2026).

3. Diffusion formulation and data pipeline

The forward diffusion process follows the standard Gaussian Markov chain: $$q(x_t|x_{t-1})=\mathcal{N}(x_t;\sqrt{1-\beta_t}x_{t-1}, \beta_tI),$$

$q(x_{1:T}|x_0) = \prod_{t=1}^T q(x_t|x_{t-1}),$

where $\beta_t \in (0,1)$ is the noise schedule (Waller et al., 25 Jun 2026).

A notable modeling choice is that the reverse model directly predicts the clean sample estimate $\hat{x}_0(x_t,t,\theta)$ rather than predicting only noise. The main reconstruction objective is

$$\mathcal{L}=\mathbb{E}_{t, x_0}\left[\omega_t\|x_0-\hat{x}_0(x_t, t, \theta)\|^2\right],$$

and the model also uses a Fourier-based auxiliary loss

$$\mathcal{L}_{Fourier}=\mathbb{E}_{t,x_0}\left[\omega_t\left[\lambda_1\|x_0-\hat{x}_0(x_t, t, \theta)\|^2+\lambda_2\|\mathcal{FFT}(x_0)-\mathcal{FFT}(\hat{x}_0(x_t, t, \theta))\|^2\right]\right].$$

The auxiliary term is intended to preserve temporal structure and seasonal or frequency characteristics of the series (Waller et al., 25 Jun 2026).

The experiments use daily historical price data for Apple and Amazon from Yahoo Finance with features open, close, high, low, and volume. The model is trained on log-returns,

$r_i = \ln\left(\frac{y_i}{y_{i-1}}\right),$

after standardization. The series are partitioned into overlapping windows of fixed length $256$ with stride $2^n$0, and a weighted sampling strategy is used to reduce bias from overlapping windows. For the hardware experiment, the sequence length is reduced to $2^n$1 to fit hardware constraints (Waller et al., 25 Jun 2026).

The paper distinguishes generative and downstream settings. For the forecasting task, the chronological split is 59.5% training, 10.5% validation, and 30% testing. For generative evaluation, the diffusion model is trained on the full dataset because the objective is distribution synthesis rather than forecast generalization (Waller et al., 25 Jun 2026).

4. Quantum neural network realization

The QNN is implemented as a parameterized quantum circuit used in place of each FFNN block. For the simulator-based model, the pipeline includes a learnable dimensionality reduction layer, PCA-based compression of transformer features, and amplitude encoding into a 9-qubit system, which can encode $2^n$2 amplitudes. The circuit ansatz is a real-amplitudes circuit consisting of a Hadamard layer, parameterized $2^n$3 rotations, and CNOT entangling gates, repeated twice for expressivity. Because the simulator has access to the full statevector, the QNN output is taken as the full vector of amplitudes and then projected back to transformer dimension (Waller et al., 25 Jun 2026).

For real quantum hardware, the model is modified because full amplitude readout is unavailable. The hardware version uses angle encoding and Pauli-$2^n$4 expectation values with 128 measurement shots. The encoded state is

$2^n$5

and readout uses

$2^n$6

The resulting expectation values are concatenated into a classical output vector (Waller et al., 25 Jun 2026).

Training is performed classically using the quantum simulator, whereas inference is run on quantum hardware for the hardware validation experiment. Hardware execution is conducted on the IQM Emerald device via Amazon Braket. This division between simulator-based training and hardware-based inference is an operational constraint of the reported implementation rather than a secondary detail: it defines the practical meaning of “validated on real quantum hardware” in the paper (Waller et al., 25 Jun 2026).

5. Empirical results

The primary generative evaluation compares synthetic and real log-return distributions using MAE for central moments, Wasserstein distance, and the Kolmogorov–Smirnov statistic. The moments examined are mean, variance, skewness, and kurtosis. The reported qualitative pattern is that QDiffusion-TS generally reproduces the distributions better than the classical FFNN baseline, with the largest improvement in variance reproduction. Averaged across both datasets, the quantum model reduces Wasserstein distance by about 44% relative to the classical model; the average reduction is about 51% for Amazon and about 36% for Apple (Waller et al., 25 Jun 2026).

The downstream utility experiment trains a BiLSTM to predict the close price using a 30-day lookback window and augments the training data with synthetic sequences at ratios $2^n$7. Synthetic augmentation from both QDiffusion-TS and the classical generator improves forecasting significantly compared to training only on real data, with gains reported as up to 71% improvement in RMSE over the real-only baseline. Performance saturates at about $2^n$8 synthetic augmentation. Quantum and classical synthetic data achieve comparable peak forecasting accuracy, while the quantum model tends to perform slightly better at smaller augmentation ratios. Additional real historical data generally perform worse because of non-stationarity and distribution shift (Waller et al., 25 Jun 2026).

The hardware experiment is particularly specific. It uses a reduced-capacity model and runs only inference on hardware for the Apple dataset. Under this setup, the hardware-executed QDiffusion-TS closely matches the simulated quantum model, outperforms the reduced classical model on statistical distances, reduces Wasserstein distance by about 89% on average relative to the classical baseline, and slightly outperforms the simulator by about 8% on average in Wasserstein distance. The paper interprets this as evidence that near-term quantum noise does not necessarily harm generative performance and may even add helpful stochasticity (Waller et al., 25 Jun 2026).

6. Interpretation, limitations, and relation to adjacent diffusion research

QDiffusion-TS occupies a specific position within diffusion-based sequence modeling. It is not a general framework for irregularly sampled or incomplete time series, nor is it primarily a forecasting model. By contrast, TS-Diffusion addresses irregular sampling, missing values, and high feature-temporal dimensionality through a marked point-process formulation with a neural ODE encoder, latent DDPM, and neural ODE decoder (Li, 2023). DYffusion, in turn, couples diffusion steps to temporal dynamics through a stochastic interpolator and forecaster, replacing the usual Gaussian noising story with a dynamics-informed interpolation-and-forecasting process for probabilistic spatiotemporal forecasting (Cachay et al., 2023). QDiffusion-TS differs from both in that its main innovation is architectural hybridization of a denoising transformer with QNN submodules for synthesis of financial time series (Waller et al., 25 Jun 2026).

Several limitations follow directly from the reported setup. The quantum part is partial rather than end-to-end; training is performed classically on a simulator; the hardware demonstration uses only inference; and the hardware validation requires a reduced-capacity model and shorter sequence length. The empirical domain is also narrow, consisting of Apple and Amazon daily historical price data with five market features. These facts do not negate the reported results, but they delimit what the experiments establish.

At the same time, the reported parameter counts and Wasserstein improvements suggest a particular significance. The per-block replacement from 33,088 parameters to 36 parameters indicates that the QNN substitution is being used as a compact nonlinear module rather than as a wholesale computational substrate. A plausible implication is that the main contribution of QDiffusion-TS lies in demonstrating a workable hybrid design pattern for diffusion-based temporal generation under current hardware constraints, rather than in presenting a fully quantum diffusion pipeline.