Solving sign problems with physics-informed kernels

TL;DR

Key to the complex PIK-architecture is its probability-weight preserving property, which allows it to map the sampling task to one on a sign-problem free manifold with a simple distribution and efficient sampling.

Abstract

In the present work we construct a novel generative architecture for systems with complex probability distributions. In general, these sampling tasks come with two challenges: resolving sign problems and efficient sampling. The architecture is based on physics-informed kernels (PIKs) introduced in arXiv:2510.26678, and aims at resolving both challenges. Key to the complex PIK-architecture is its probability-weight preserving property, which allows us to map the sampling task to one on a sign-problem free manifold with a simple distribution and efficient sampling. The potential of this novel architecture is demonstrated within applications to zero-dimensional field theories with complex couplings, as well as the real-time evolution of the quantum-mechanical harmonic oscillator.

Figures (10)

• Figure 1: Final sampling manifold $\mathcal{M}_1$(black) for the zero-dimensional target action \ref{['eq:Action0d']}. The thimbles and anti-thimbles are shown in red and grey, respectively. The shaded areas indicate: $\Re[S(\phi)]>0$(blue) and $0 \leq \pm \Im[S(\phi)] \leq \frac{\pi}{2}$(orange/red).
• Figure 2: RG-time dependent 10th cumulant of the model with the action \ref{['eq:Action0d']}. The cumulant is computed using $10^5$ Monte Carlo samples at $t=0$, which are subsequently transported to $t>0$ using \ref{['eq:IntegrateMap']} and the respective $\dot{\phi}$. As a visual guide we depict exact solutions computed by a direct numerical evaluation of the integral.
• Figure 3: Two-point function $\langle \varphi_0\,\varphi_{x_0} \rangle$ of the real-time harmonic oscillator in 1+0 dimensions \ref{['eq:QMaction']}. The computation uses $N_{\textrm{max}}=30$ and $N_{\textrm{max}}=75$ (in-lay) lattice sites with couplings $m^2 = 0.3$, $\epsilon = \frac{2m}{N_{\textrm{max}}}$.
• Figure 4: Illustration of different scenarios for the change in the manifolds ${\cal M}_t$ under the application of the PIK ${\dot \phi}_t$.
• Figure 5: PIKfold (black) in the complex plane for the action \ref{['eq:actionLT']} with different values of $m^2$. In all cases we use $m^2_0 = 1$, $\lambda = 1$. We also depict the existing thimble solutions: The thimble with a real saddle at $\phi = 0$(light red), the thimbles with complex saddles (dark red, dashed) as well as the corresponding anti-thimbles (grey). Furthermore, the blue areas indicate $\Re[S(\phi)]>0$, i.e. where the contour at infinity can be closed safely. The (orange) areas indicate $0 \leq \Im[S(\phi)] \leq \frac{\pi}{2}$.