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A major driver of quantum-simulator technology is the prospect of probinghigh-energy phenomena in synthetic quantum matter setups at a high level ofcontrol and tunability. Here, we propose an experimentally feasible realizationof a large-scale $2+1$D $\mathrm{U}(1)$ gauge theory with dynamical matter andgauge fields in a cold-atom quantum simulator with spinless bosons. We presentthe full mapping of the corresponding Gauss's law onto the bosoniccomputational basis. We then show that the target gauge theory can befaithfully realized and stabilized by an emergent gauge protection term in atwo-dimensional single-species Bose--Hubbard optical Lieb superlattice with twospatial periods along either direction, thereby requiring only moderateexperimental resources already available in current cold-atom setups. Usinginfinite matrix product states, we calculate numerical benchmarks for adiabaticsweeps and global quench dynamics that further confirm the fidelity of themapping. Our work brings quantum simulators of gauge theories a significantstep forward in terms of investigating particle physics in higher spatialdimensions, and is readily implementable in existing cold-atom platforms.

Large-Scale $2+1$D $\mathrm{U}(1)$ Gauge Theory with Dynamical Matter in a Cold-Atom Quantum Simulator