  We develop techniques to systematically construct local unitaries which map
scale-invariant, product state wavefunctionals to the ground states of weakly
interacting, continuum quantum field theories. More broadly, we devise a
"quantum circuit perturbation theory" to construct local unitaries which map
between any pair of wavefunctionals which are each Gaussian with arbitrary
perturbative corrections. Further, we generalize cMERA to interacting continuum
field theories, which requires reworking the existing formalism which is
tailored to non-interacting examples. Our methods enable the systematic
perturbative calculation of cMERA circuits for weakly interacting theories, and
as a demonstration we compute the 1-loop cMERA circuit for scalar $\varphi^4$
theory and analyze its properties. In this case, we show that Wilsonian
renormalization of the spatial momentum modes is equivalent to a local position
space cMERA circuit. This example provides new insights into the connection
between position space and momentum space renormalization group methods in
quantum field theory. The form of cMERA circuits derived from perturbation
theory suggests useful ansatzes for numerical variational calculations.
