  Currently, there are intense experimental efforts to realize lattice gauge
theories in quantum simulators. Except for specific models, however, practical
quantum simulators can never be fine-tuned to perfect local gauge invariance.
There is thus a strong need for a rigorous understanding of gauge-invariance
violation and how to reliably protect against it. As we show through analytic
and numerical evidence, in the presence of a gauge invariance-breaking term the
gauge violation accumulates only perturbatively at short times before
proliferating only at very long times. This proliferation can be suppressed up
to infinite times by energetically penalizing processes that drive the dynamics
away from the initial gauge-invariant sector. Our results provide a theoretical
basis that highlights a surprising robustness of gauge-theory quantum
simulators.
