Preconditioning is a key component of MCMC algorithms that improves sampling efficiency by facilitating exploration of geometrically complex target distributions through an invertible map. While linear preconditioners are often sufficient for moderately complex target distributions, recent work has explored nonlinear preconditioning with invertible neural networks as components of normalizing flows (NFs). However, empirical and theoretical studies show that overparameterized NF preconditioners can degrade sampling efficiency and fit quality. Moreover, existing NF-based approaches do not adapt their architectures to the target distribution. Related work outside of MCMC similarly finds that suitably parameterized NFs can achieve comparable or superior performance with substantially less training time or data. We propose a factorized preconditioning architecture that reduces NF complexity by combining a linear component with a conditional NF, improving adaptability to target geometry. The linear preconditioner is applied to dimensions that are approximately Gaussian, as estimated from warmup samples, while the conditional NF models more complex dimensions. Our method yields significantly better tail samples on two complex synthetic distributions and consistently better performance on a sparse logistic regression posterior across varying likelihood and prior strengths. It also achieves higher effective sample sizes on hierarchical Bayesian model posteriors with weak likelihoods and strong funnel geometries. This approach is particularly relevant for hierarchical Bayesian model analyses with limited data and could inform current theoretical and software strides in neural MCMC design.

Recent work has shown that one can efficiently learn fermionic Gaussian unitaries, also commonly known as nearest-neighbor matchcircuits or non-interacting fermionic unitaries. However, one could ask a similar question about unitaries that are near Gaussian: for example, unitaries prepared with a small number of non-Gaussian circuit elements. These operators find significance in quantum chemistry and many-body physics, yet no algorithm exists to learn them. We give the first such result by devising an algorithm which makes queries to an $n$-mode fermionic unitary $U$ prepared by at most $O(t)$ non-Gaussian gates and returns a circuit approximating $U$ to diamond distance $\varepsilon$ in time $\textrm{poly}(n,2^t,1/\varepsilon)$. This resolves a central open question of Mele and Herasymenko under the strongest distance metric. In fact, our algorithm is much more general: we define a property of unitary Gaussianity known as unitary Gaussian dimension and show that our algorithm can learn $n$-mode unitaries of Gaussian dimension at least $2n - O(t)$ in time $\textrm{poly}(n,2^t,1/\varepsilon)$. Indeed, this class subsumes unitaries prepared by at most $O(t)$ non-Gaussian gates but also includes several unitaries that require up to $2^{O(t)}$ non-Gaussian gates to construct. In addition, we give a $\textrm{poly}(n,1/\varepsilon)$-time algorithm to distinguish whether an $n$-mode unitary is of Gaussian dimension at least $k$ or $\varepsilon$-far from all such unitaries in Frobenius distance, promised that one is the case. Along the way, we prove structural results about near-Gaussian fermionic unitaries that are likely to be of independent interest.