Circuit complexity in interacting QFTs and RG flows

TL;DR

This paper investigates the circuit complexity of interacting scalar quantum field theories, especially $^4$ theory, using Nielsen's geometric approach to understand how complexity evolves with RG flows and dimensionality.

Contribution

It introduces a general analytical method using integral transforms to compute circuit complexity in lattice scalar QFTs, including the epsilon expansion near the Wilson-Fisher fixed point.

Findings

Circuit complexity increases with dimensionality in $^4$ theory.

The perturbative calculation breaks down at high dimensions due to increased complexity.

The study links circuit complexity with renormalization group flows.

Abstract

We consider circuit complexity in certain interacting scalar quantum field theories, mainly focusing on the $\phi^4$ 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 $\phi^4$ interaction eventually causing the perturbative calculation to breakdown. We discuss how circuit complexity relates with the renormalization group.