Circuit Complexity in ${\mathcal{Z}}_2$ $\mathcal{EEF}\mathcal{T}$

Abstract

Motivated by recent studies of circuit complexity in weakly interacting scalar field theory, we explore the computation of circuit complexity in ${\mathcal{Z}}_2$ Even Effective Field Theories (${\mathcal{Z}}_2$ EEFTs). We consider a massive free field theory with higher-order Wilsonian operators such as ${\varphi }^4$, ${\varphi }^6$ and ${\varphi }^8.$ To facilitate our computation we regularize the theory by putting it on a lattice. First, we consider a simple case of two oscillators and later generalize the results to $N$ oscillators. The study has been carried out for nearly Gaussian states. In our computation, the reference state is an approximately Gaussian unentangled state, and the corresponding target state, calculated from our theory, is an approximately Gaussian entangled state. We compute the complexity using the geometric approach developed by Nielsen, parameterizing the path ordered unitary transformation and minimizing the geodesic in the space of unitaries. The contribution of higher-order operators, to the circuit complexity, in our theory has been discussed. We also explore the dependency of complexity with other parameters in our theory for various cases.

Comments: 35 pages, 5 figures, 3 tables, reference list updated and version modified, Accepted for publication in Symmetry (section: Physics and Symmetry/Asymmetry, Special issue: Symmetry and Geometry in Physics II)