  We present an operational procedure to transform global symmetries into local
symmetries at the level of individual quantum states, as opposed to typical
gauging prescriptions for Hamiltonians or Lagrangians. We then construct a
compatible gauging map for operators, which preserves locality and reproduces
the minimal coupling scheme for simple operators. By combining this
construction with the formalism of projected entangled-pair states (PEPS), we
can show that an injective PEPS for the matter fields is gauged into a
G-injective PEPS for the combined gauge-matter system, which potentially has
topological order. We derive the corresponding parent Hamiltonian, which is a
frustration free gauge theory Hamiltonian closely related to the Kogut-Susskind
Hamiltonian at zero coupling constant. We can then introduce gauge dynamics at
finite values of the coupling constant by applying a local filtering operation.
This scheme results in a low-parameter family of gauge invariant states of
which we can accurately probe the phase diagram, as we illustrate by studying a
Z2 gauge theory with Higgs matter.
