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arXiv:2608.21700

Scalable quantum simulation of continuous-time generative models via tensor networks

Nathan X. Kodama, L. Andrew Wray, Sam Cochran, Chad Rigetti, Shravan Veerapaneni, Michael J. Keiser

quant-phcs.AIcs.LGSubmitted Aug 22, 2026• Announced Aug 25, 2026• 0 citations
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Abstract

Continuous-time flow and diffusion models are widely used across many application domains, from large-scale deployment in computer vision and protein folding to emerging adoption for modeling language, time series, and quantum states. After training, inferring statistical properties from continuous-time models is costly. Wavefunction flows target this cost by recasting learned transport as unitary evolution, whose final Born distribution approximates the target distribution. This prepares a coherent amplitude encoding (a qsample) that can be post-processed by quantum algorithms offering a quadratic advantage over Monte Carlo sampling. We present the first numerical study of these flows, in which we represent time-dependent potentials and states as tensor networks. At spatial dimension $d=8$, storage falls by $\sim 10^7\times$ relative to the dense grid of $N^d$ points, and evolution wall-clock time falls by $\gtrsim 10^3\times$ against a baseline extrapolated from the measured $d\le 5$ scaling. We validate our pipeline by reproducing the $O(1/\sqrt{p_{\rm rare}})$ scaling of rare-event sampling.

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Continuous-time flow and diffusion models are widely used across many application domains, from large-scale deployment in computer vision and protein folding to emerging adoption for modeling language, time series, and quantum states. After training, inferring statistical properties from continuous-time models is costly. Wavefunction flows target this cost by recasting learned transport as unitary evolution, whose final Born distribution approximates the target distribution. This prepares a coherent amplitude encoding (a qsample) that can be post-processed by quantum algorithms offering a quadratic advantage over Monte Carlo sampling. We present the first numerical study of these flows, in which we represent time-dependent potentials and states as tensor networks. At spatial dimension d=8d=8d=8, storage falls by ∼107×\sim 10^7\times∼107× relative to the dense grid of NdN^dNd points, and evolution wall-clock time falls by ≳103×\gtrsim 10^3\times≳103× against a baseline extrapolated from the measured d≤5d\le 5d≤5 scaling. We validate our pipeline by reproducing the O(1/prare)O(1/\sqrt{p_{\rm rare}})O(1/prare​​) scaling of rare-event sampling.
Flow models are a cornerstone of modern machine learning. They are generative models that progressively transform probability distributions according to learned dynamics. Specifically, they learn a continuous-time Markov process that efficiently maps samples from a simple source distribution into samples from a complex target distribution. We show that these models are naturally related to the Schrödinger equation, for an unusual Hamiltonian on continuous variables. Moreover, we prove that the dynamics generated by this Hamiltonian can be efficiently simulated on a quantum computer. Together, these results give a quantum algorithm for preparing coherent encodings (a.k.a., qsamples) for a vast family of probability distributions--namely, those expressible by flow models--by reducing the task to an existing classical learning problem, plus Hamiltonian simulation. For statistical problems defined by flow models, such as mean estimation and property testing, this enables the use of quantum algorithms tailored to qsamples, which may offer advantages over classical algorithms based only on samples from a flow model. More broadly, these results reveal a close connection between state-of-the-art machine learning models, such as flow matching and diffusion models, and one of the main expected capabilities of quantum computers: simulating quantum dynamics.
The rapid growth of entanglement under unitary time evolution is the primary bottleneck for modern tensor-network techniques--such as Matrix Product States (MPS)--when computing time-dependent expectation values. This {entanglement barrier} restricts classical simulations and, conversely, underpins the quantum advantage anticipated from future devices. Here we show that, for one-dimensional Hamiltonian dynamics, the spatio-temporal tensor network encoding the evolved wave function amplitudes can be contracted efficiently along the left-right (spatial) direction. Exploiting this structure, we develop a hybrid Tensor-Network/Monte-Carlo (TN-MC) algorithm that samples the wave function and evaluates expectation values of generic local operators with computational cost that scales only polynomially in time. The accurate contraction of the wave function amplitudes is a consequence of the favorable scaling with time of the generalised temporal entropies. We find that their real part either saturates or, at most, grows logarithmically with time, revealing new instances of continuous dynamical quantum phase transitions (DQPTs) which we characterize. Our results therefore show that, when computing expectation values of local operators, the entanglement barrier in one-dimensional Hamiltonian evolution can be bypassed with a TN-MC blend.
Generative models have achieved remarkable success in data synthesis, though recent advances driven by increasing model scale have introduced challenges in computational cost and efficiency. Quantum machine learning offers a promising alternative, representing complex data distributions using compact, highly expressive models. Here, we propose QDiffusion-TS, the first quantum generative diffusion model for time series synthesis, and validate it on the IQM quantum processor. The framework extends a classical diffusion architecture by replacing feed-forward components within the denoising transformer with quantum neural networks, yielding a hybrid quantum transformer that reduces the number of trainable parameters in each replaced component by nearly three orders of magnitude. Evaluated on financial time series from Apple and Amazon, the model generates synthetic data that more accurately reproduces the real distributions, reducing Wasserstein distance by approximately 44% relative to its classical counterpart across both datasets. In a downstream forecasting task, augmentation with the generated data improves predictive performance by up to 71% in RMSE over a baseline trained solely on real data. These results show that quantum enhanced architectures can consistently match and frequently surpass classical performance with substantially fewer parameters, establishing a practical framework towards more efficient and scalable data-driven generative modelling.
Recently developed tools such as neural-network based generative models and incremental tensor networks are promising methods for simplifying the learning of functions in high-dimensional space. We consider the application of these tools in two particular areas: modeling quantum states and imitation learning from high-dimensional data. In the first application, we consider the use of generative models such as normalizing flows as neural quantum states. Because quantum states are functions ψ such that |ψ|2 is a probability density, using normalizing flows as neural quantum states is a natural choice. Neural quantum states are commonly used in Variational Monte Carlo (VMC), where the parameters θ of a parameterized quantum state ψθ are varied in order to minimize some objective function. Due to the ability to exactly sample from and estimate probability densities from normalizing flows, the use of normalizing flows as neural quantum states simplifies the calculation of integrals used in Variational Monte Carlo. This is applied in two settings: first we consider using VMC for minimizing the ground state energy of high-dimensional continuous quantum states. Second, we explore using VMC for numerically solving time-dependent partial differential equations. For the second application, we explore using generative models and incremental tensor networks for extracting a low-dimensional latent space for use in downstream tasks like imitation learning. We first explore the use of neural-network architectures like Variational Autoencoders for learning a latent space that can be sampled from. Doing this allows for synthetic states to be generated, as well as predicted actions for those states. However, the use of these neural network-based feature extractors can be expensive, due to hyperparameter tuning and the gradient-based optimization methods used to find a good latent space representation. We explore the use of incremental tensor network approaches, which can extract a latent space in one pass through the training data and requires minimal hyperparameter tuning. We find that these incremental tensor network approaches learn a latent space representation that leads to similar imitation learning accuracy, but provides significant computational speedups over the neural network approaches.
We propose an algorithmic framework for building and training quantum generative models corresponding to multivariate probability distributions. Our model uses parallel Fourier feature maps for embedding continuous-valued variables combined with a forrelation-type quantum circuit for tuning Fourier coefficients of the quantum model. Crucially, we develop a distinct training strategy where training is enabled at large scale by log-likelihood loss with unbiased Monte Carlo estimator based on Parseval's identity. Unlike prior work that relied on maximal mean discrepancy (MMD) loss, our approach goes beyond matching just low frequency moments, while enabling efficient classical training. Once the model is trained, we use inverse quantum Fourier transforms to map it into a separate sampling circuit in the computational basis. We demonstrate the efficiency of the suggested framework by validating loss estimation at the scale of over 1000 qubits on a single GPU. We show that univariate and bivariate models with highly non-trivial structure can be trained to low total variation distance, while fine-tuned IQP models with MMD loss show poor performance. Comparing to classical baselines represented by normalizing flow and diffusion models, we show that our approach avoids oversmoothing and preserves multi-modal structure of the target. Finally, we have deployed the trained models on superconducting quantum devices, successfully sampling distributions with per-sample execution times of approximately 300 μs300\,μ\mathrm{s}300μs. Our work shows that quantum generative models with the train-on-classical deploy-on-quantum approach can provide both high-quality structure at increased scale and fast sampling access needed for inference.
Flow models generate trajectories from an initial distribution to a target distribution by solving an ordinary differential equation defined by a velocity field. Flow matching learns this velocity field by modeling the transport dynamics between the two distributions. Wavefunction flow establishes a formal connection between flow models and quantum dynamics by introducing a continuity Hamiltonian, which drives the Schrödinger evolution of quantum states. In this paper, we investigate accurate and efficient quantum simulation of the wavefunction flow, thereby realizing the efficient implementation of flow models on quantum computers. We first leverage a quantum read-only memory (QROM)-based phase kickback framework for the wavefunction flow simulation, generating probability densities that closely match those produced by the corresponding conventional flow model. To address the high circuit-resource cost, we further incorporate a trained quantum neural network (QNN) into the phase kickback framework, replacing QROM for data encoding. Numerical experiments demonstrate that our proposed method implements flow models on quantum computers more efficiently, since it maintains the accuracy of wavefunction flow simulation compared with the QROM-based framework, and significantly reduces the circuit resources.