Abstract
We consider circuit complexity in certain interacting scalar quantum field theories, mainly focusing on the φ⁴ theory. We work out the circuit complexity for evolving from a nearly Gaussian unentangled reference state to the entangled ground state of the theory. Our approach uses Nielsen's geometric method, which translates into working out the geodesic equation arising from a certain cost functional. We present a general method, making use of integral transforms, to do the required lattice sums analytically and give explicit expressions for the d=2,3 cases. Our method enables a study of circuit complexity in the epsilon expansion for the Wilson-Fisher fixed point. We find that with increasing dimensionality the circuit depth increases in the presence of the φ⁴ interaction eventually causing the perturbative calculation to breakdown. We discuss how circuit complexity relates with the renormalization group.