  The modern description of elementary particles, as formulated in the Standard
Model of particle physics, is built on gauge theories. Gauge theories implement
fundamental laws of physics by local symmetry constraints. For example, in
quantum electrodynamics, Gauss's law introduces an intrinsic local relation
between charged matter and electromagnetic fields, which protects many salient
physical properties including massless photons and a long-ranged Coulomb law.
Solving gauge theories by classical computers is an extremely arduous task,
which has stimulated a vigorous effort to simulate gauge-theory dynamics in
microscopically engineered quantum devices. Previous achievements implemented
density-dependent Peierls phases without defining a local symmetry, realized
mappings onto effective models to integrate out either matter or electric
fields, or were limited to very small systems. The essential gauge symmetry has
not been observed experimentally. Here, we report the quantum simulation of an
extended U(1) lattice gauge theory, and experimentally quantify the gauge
invariance in a many-body system comprising matter and gauge fields. These are
realized in defect-free arrays of bosonic atoms in an optical superlattice of
71 sites. We demonstrate full tunability of the model parameters and benchmark
the matter--gauge interactions by sweeping across a quantum phase transition.
Enabled by high-fidelity manipulation techniques, we measure the degree to
which Gauss's law is violated by extracting probabilities of locally
gauge-invariant states from correlated atom occupations. Our work provides a
way to explore gauge symmetry in the interplay of fundamental particles using
controllable large-scale quantum simulators.
