The first numerical study of continuous-time flow and diffusion models is presented, in which time-dependent potentials and states are represented as tensor networks, and a coherent amplitude encoding is prepared that can be post-processed by quantum algorithms offering a quadratic advantage over Monte Carlo sampling.
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.
Quantum diffusion models provide a physics-consistent route to generative learning by formulating noising and denoising directly on quantum states. However, applying such models to classical high-dimensional data is constrained by the qubit cost of state encoding and the computational burden of simulating large density operators. We propose a scalable hybrid generative pipeline that combines a classical autoencoder for dimensionality reduction with a mixed-state quantum denoising diffusion probabilistic model (MSQuDDPM) operating in the learned latent space. The autoencoder compresses data into compact latent codes that can be embedded into a small-qubit Hilbert space, after which the quantum diffusion model learns a generative distribution over latent density operators and decodes samples back to the original domain. Algorithmically, we simplify the reverse dynamics by predicting an estimate of the clean state $\rho_0$ at timestep $t$ and computing the one-step reverse update via an analytic backward propagation rule, rather than learning an explicit predictor for $\rho_{t-1}$. We demonstrate the proposed approach on MNIST image generation and discuss how mixed-state quantum diffusion can serve as a practical backbone for hybrid quantum--classical generative modeling under realistic qubit budgets.
Learning dissipation rates in large-scale open quantum systems is a major obstacle for near-term quantum technologies, as existing Lindblad estimation methods are typically limited to small system sizes due to the computational complexity of repeatedly solving the Lindblad equation during optimization. Here, we propose a scalable noise-learning framework for Lindblad dissipation rates that combines a stochastic simulation method, the Tensor Jump Method (TJM), with gradient-free optimization of a least-squares cost-function defined on time series of local-observable expectation values. We demonstrate the approach on two noise models in the Ising model: a site-resolved (local) model, in which independent dissipation rates are learned for each site up to $N_{\mathrm{site}}=16$, and a spatially homogeneous (global) model with only seven parameters, scaled to $N_{\mathrm{site}}=160$ sites.We complement these numerical results with a series of exact, provable guarantees: the Frobenius variance of the TJM density-matrix estimator is shown to equal $(1-\mathrm{Tr}[\rho^2])/N_{\mathrm{traj}}$, an exact purity-based characterization of the stochastic estimation error; the corresponding purity evolution is proven to be monotonically non-increasing for Hermitian jump operators; and, under a finite covariance distance assumption, the standard deviation of the cost-function is shown to decrease with system size, so that fewer trajectories are needed to reach a fixed target accuracy as the system grows. Together, this combination of scalable numerics and rigorous theoretical guarantees positions TJM-based noise learning as a practical foundation for characterizing dissipation in large quantum devices and for guiding future work on error mitigation and quantum error correction.
A. R. Ramos Ramos, Maximilian Fröhlich, Aaron Sander et al.· 0 citations
Predicting molecular self-diffusion coefficients (D*) across chemical space remains challenging due to sparse experimental data and the high computational cost of molecular simulations. We present a data-centric machine learning framework that integrates experimental diffusion measurements with molecular dynamics simulations through a semisupervised distillation strategy. Unlike conventional approaches that treat limited experiments or simulations as direct ground truth, our method selectively incorporates simulation-derived D* only when they align with the uncertainty bounds of a Random Forest model initially trained on experimental D*. This enables controlled data set expansion, from 130 unique experimentally measured molecules to over 1,000 unique substances, while preserving label reliability. We further employ pretrained large language model embeddings to encode transferable chemical context beyond conventional descriptors, reducing predictive variance and improving generalization across chemically diverse systems. The resulting model achieves improved accuracy, with a held-out test set R2 of 0.87, an overall R2 of 0.92, and a test set mean absolute error of 0.13 × 10-9 m2 s-1 after iterative distillation across more than 1,000 chemically diverse molecules under near-ambient conditions (295-300 K). We further demonstrate that learned D* act as transferable, physically interpretable descriptors in skin permeability modeling, where their inclusion reduces test set error and improves correlation relative to models relying solely on conventional physicochemical descriptors. This work establishes a scalable framework for bridging sparse experimental measurements with broadly generalizable predictions, enabling interpretable and transferable modeling across distinct molecular transport phenomena.
Hojin Jung, Su-min Song, Sabari Kumar et al.· Journal of Chemical Informat...· 0 citations
We investigate generative diffusion models as denoising samplers for two- and three-dimensional lattice $\phi^4$ theory across the symmetric, near-critical, and broken phases. Validated against ensembles generated by Fourier-accelerated HMC combined with Wolff cluster updates, the reverse-SDE sampler reproduces scalar observables and the momentum-space propagator $G(|k|)$, with residual bias concentrated in the zero-mode and, in three dimensions, the action density. We introduce two local diagnostics and an HMC-referenced effective sample size (ESS), which probe the learned drift directly, through a Metropolis-adjusted Langevin acceptance rate, and through observable-level bias and variance. Exploiting a fully convolutional architecture with weights shared across different volumes ($V=L^D$), we show that cross-volume training transfers to unseen sizes, matching or slightly improving in-distribution training in the two-dimensional symmetric and broken phases. A three-dimensional model trained on $L \in \{4, 8, 16, 32\}$ reproduces the propagator and most scalar observables at the unseen lattice size $L = 64$ across the phase diagram, with the residual susceptibility excess in the broken phase as the main exception, and improves several critical observables relative to in-distribution $L = 64$ training. This establishes cross-volume generalization as a viable mechanism for large-volume sampling, and the score learned from many cheap small-lattice configurations transfers to the target volume without retraining.
Yang-yang Tan, G. Aarts, Diaa E. Habibi et al.· 3 citations
This work introduces implicit machine learning force fields, which replace explicit stacks of neural network layers with self-consistent fixed-point equations, and demonstrates this across three major classes of graph neural networks: invariant, equivariant Cartesian tensor, and SO(3)-equivariant spherical-tensor architectures.
J. Maess, Leon Werner, J. Frank et al.· 0 citations
Machine-learning surrogates of physical systems face a paradox: explainable models facing the challenge of expressivity to capture complex nonlinear flows, whereas expressive deep surrogates match high-fidelity simulations only through massive parameterisations that turn the learned dynamics into a black box. Here, we introduce quantum-compressed machine learning (QCML), which resolves this tension by compressing the latent propagator of a flow surrogate from $524{,}288$ trainable parameters to no more than $8$. This parameter reduction brings the learned dynamical law to the parameter scale of a physical constitutive relation rather than a black-box neural network, making the surrogate directly interpretable and controllable without sacrificing expressivity. The compression is realised by a structured quantum circuit whose unitary propagator constrains the latent spectrum to the unit circle exactly and by construction, replacing exponential error growth with linear accumulation over autoregressive rollouts. Classical regularisation only approximates this constraint: even a quantum-inspired classical baseline penalised towards unitarity collapses within one Lyapunov time on turbulent channel flow, whereas QCML remains stable over the full rollout. Shared phase and coupling angles parameterising the circuit correspond directly to modal frequencies and inter-mode interactions, giving the learned dynamics a physical interpretation in spectral space. On two patient-specific cardiovascular benchmarks, the structured QCML propagator matches the predictive accuracy of its classical counterpart on surface pressure spectra, pressure drop, and wall shear stress. These results establish QCML as a working component of scientific machine learning and a concrete contribution towards practical quantum advantage in real-world prediction.
Xiao Xue, Maida Wang, Mingyang Gao et al.· 0 citations
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MIT News · Artificial Intelligence· news.mit.eduAug 27, 2026
A new machine-learning framework aims to improve the success rate of computational protein design while moving away from results that reproduce sequences found in nature.
