Computing stable configurations of confined smectic liquid crystals with a deep variational framework

The Vicsek model is a minimal model of collective motion, capturing how local alignment interactions can generate macroscopic nonequilibrium order in systems such as bird flocks. In this work, we use active learning to map the Vicsek phase diagram as a function of noise strength, density, and particle speed. A neural-network classifier is trained on global polar-order labels, and classifier entropy is used to select new simulations near uncertain crossover regions. The resulting phase map resolves a high-noise disordered gas, a low-noise polar ordered regime, and an intermediate coexistence-candidate regime whose noise window shifts upward and broadens with increasing density. Independent density and local-order diagnostics indicate that the intermediate regime contains dense, locally ordered bands coexisting with a dilute, weakly ordered background. Comparison with the ordered regime shows that banded coexistence is identified by the joint enhancement of band contrast, local-order heterogeneity, and positive density-order correlation. Overall, these results establish a machine-learning-guided workflow for active matter, in which active learning constructs an operational phase map and independent spatial diagnostics convert classifier-defined regimes into physically interpretable nonequilibrium morphologies.

Variational Monte Carlo (VMC) is a central algorithm in electronic structure theory and has gained renewed importance through modern neural-network ansätze such as FermiNet. At its core, VMC seeks ground states by minimizing the Rayleigh quotient by stochastic optimization. In this work, we show that the resulting stochastic optimization problem is intrinsically governed by the nodal geometry of the underlying wave function. More precisely, we establish that properties of the nodal set determine the integrability of the local energy and gradient estimators that drive VMC. For broad and practically relevant ansatz classes, including Slater-Jastrow wave functions with variable-exponent Slater-type orbitals, we prove that these estimators are generically heavy-tailed and fail to admit higher moments. At the same time, for general analytic ansätze, we prove weak moment bounds for the relevant estimators and identify precise low-moment regimes, showing how generic and degenerate nodal structures lead to different integrability thresholds. Building on this analysis, we introduce a new robust variant of VMC $\unicode{x2013}$ coined PS-Clip-VMC $\unicode{x2013}$ which is based on clipping both the local energy and the gradient random variable. We prove that PS-Clip-VMC converges both in expectation and with high probability in the weak moment regime of VMC. Preliminary experiments for training FermiNet on Atoms with up to 18 electrons suggest that PS-Clip-VMC is significantly more robust than standard methods.

Variational quantum circuits have been central to many proposed near-term applications of quantum computing, but a growing body of evidence suggests that trainability and quantum advantage are fundamentally at odds: ansätze expressive enough to resist efficient classical simulation tend to exhibit barren plateaus, while structures that provably rule out barren plateaus typically render them classically simulable. We propose a stacked linear combination of unitaries (S-LCU) as a variational ansatz which provides a tunable trade-off between barren plateaus and classical simulability. Using a diagrammatic analysis, we bound the loss-landscape variance of the Free Fermion S-LCU, whose elements are fermionic Gaussian unitaries. We prove a variance lower bound of $Ω(1/(n k^{3l}))$, with a simulation cost of $O(k^{2l} n^3)$ using the best known classical algorithm, compared to a quantum gate complexity of only $O(lkn^2)$. The number of layers $l$ serves as a single dial that trades computational complexity against the rate of cost concentration. This offers practitioners a systematic method for constructing ansätze with a complexity-trainability trade-off that best suits their application and hardware.