Quantum simulation of fermionic non-Abelian lattice gauge theories in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> dimensions with built-in gauge protection

Recent advancements in the field of quantum simulation have significantly expanded the potential for applications, particularly in the context of lattice gauge theories (LGTs). Maintaining gauge invariance throughout a simulation remains a central challenge, especially for large-scale non-Abelian LGTs with dynamical matter, which are particularly complex in terms of engineering for experiments. Gauge-symmetry breaking is inevitable in established rishon-based schemes for alkaline-earth-like atoms (AELAs) and controlling the magnitude of its effect is an open challenge. Here we first construct a minimal model to quantum simulate non-Abelian LGTs ensuring that the gauge constraints are met and explicitly derive their unambiguous non-Abelian nature. Second, we present a proposal for a gauge protection scheme using native interactions in AELAs enabling the simulation of toy models of non-Abelian U ( 2 ) LGTs with dynamical fermionic matter in ( 2 + 1 ) dimensions on large scales. Due to the simplicity of the gauge protection mechanism, based on a Zeeman shift in combination with superexchange interactions, our scheme can be naturally included in other rishon-based quantum simulation protocols. Third, we extend our approach to a fully scalable, hybrid digital-analog simulator for U ( N ) LGTs based on Rydberg AELA with variable rishon number. The proposed general mechanism for gauge protection provides a promising path towards the simulation of non-Abelian LGTs relevant to particle physics.