Renormalization Group Circuits for Weakly Interacting Continuum Field Theories

TL;DR

This paper introduces a systematic method to construct local quantum circuits that approximate ground states of weakly interacting continuum quantum field theories, extending cMERA to interacting cases and linking renormalization techniques.

Contribution

It develops a perturbative approach to build local unitaries for weakly interacting theories and generalizes cMERA to include interactions, providing new tools for quantum field theory analysis.

Findings

Constructed 1-loop cMERA circuit for scalar  theory

Demonstrated equivalence of Wilsonian renormalization and local cMERA in position space

Provided insights into the connection between position and momentum space RG methods

Abstract

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.