  We adapt the techniques of entanglement renormalization tensor networks to
weakly interacting quantum field theories in the continuum. A key tool is
"quantum circuit perturbation theory," which enables us to systematically
construct unitaries that map between wavefunctionals which are Gaussian with
arbitrary perturbative corrections. As an application, we construct a local,
continuous MERA (cMERA) circuit that maps an unentangled scale-invariant state
to the ground state of $\varphi^4$ theory to 1-loop. Our local cMERA circuit
corresponds exactly to 1-loop Wilsonian RG on the spatial momentum modes. In
other words, we establish that perturbative Wilsonian RG on spatial momentum
modes can be equivalently recast as a local cMERA circuit in $\varphi^4$
theory, and argue that this correspondence holds more generally. Our analysis
also suggests useful numerical ansatzes for cMERA in the non-perturbative
regime.
