Quantum and classical spin network algorithms for $q$-deformed Kogut-Susskind gauge theories

Abstract

Treating the infinite-dimensional Hilbert space of non-abelian gauge theories is an outstanding challenge for classical and quantum simulations. Here, we introduce $q$-deformed Kogut-Susskind lattice gauge theories, obtained by deforming the defining symmetry algebra to a quantum group. In contrast to other formulations, our proposal 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 (SNAQs). To be explicit, we focus on SU(2)${}_k$ gauge theories, that are controlled by the deformation parameter $k$ 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 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.

Comments: 5+4 pages, 4+1 figures; references added, Fig. 3 revised and typos corrected