Treating the infinite-dimensional Hilbert space of non-Abelian gauge theories is an outstanding challenge for classical and quantum simulations. Here, we employ $q$-deformed Kogut-Susskind lattice gauge theories, obtained by deforming the defining symmetry algebra to a quantum group. In contrast to other formulations, this approach simultaneously provides a controlled regularization of the infinite-dimensional local Hilbert space while preserving essential symmetry-related properties. This enables the development of both quantum as well as quantum-inspired classical spin-network algorithms for $q$-deformed gauge theories. To be explicit, we focus on ${\mathrm{SU}(2)}_k$ gauge theories with $k\in \mathbb{N}$ that are controlled by the deformation parameter $q=e^{2\pi i/(k+2)}$, a root of unity, and converge to the standard SU(2) Kogut-Susskind model as $k\to \infty$. In particular, we demonstrate that this formulation is well suited for efficient tensor network representations by variational ground-state simulations in 2D, providing first evidence that the continuum limit can be reached with $k=\mathcal{O}(10)$. Finally, we develop a scalable quantum algorithm for Trotterized real-time evolution by analytically diagonalizing the ${\mathrm{SU}(2)}_k$ plaquette interactions. Our work gives a new perspective for the application of tensor network methods to high-energy physics and paves the way for quantum simulations of non-Abelian gauge theories far from equilibrium where no other methods are currently available.

• Received 6 May 2023
• Revised 10 August 2023
• Accepted 25 September 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.171902