Tensor networks for lattice gauge theories beyond one dimension

Tensor network methods are numerical tools and algorithms to study many-body quantum systems in and out of equilibrium, based on tailored variational wave functions. They have found significant applications in simulating lattice gauge theories that approach relevant problems in high-energy physics. Compared to Monte Carlo methods, they do not suffer from the sign problem, allowing them to explore challenging regimes such as finite chemical potentials and real-time dynamics. Further development is required to tackle fundamental challenges, such as accessing continuum limits or computations of large-scale quantum chromodynamics. This work reviews the state-of-the-art tensor network methods and discusses a possible roadmap for algorithmic development and strategies to enhance their capabilities and extend their applicability to open high-energy problems. We provide tailored estimates of the theoretical and computational resource scaling for attacking large-scale lattice gauge theories. Tensor networks are a powerful complementary approach to Monte Carlo methods for simulating lattice gauge theories, enabling access to challenging regimes such as real-time dynamics and finite-density systems. This work reviews the state of the art, outlines a roadmap for algorithmic developments, and provides resource estimates to guide large-scale applications in high-energy physics.