Renormalization group and approximate error correction

Abstract

We show that in renormalization group (RG) flow the low-energy states form a code subspace that is approximately protected against the local short-distance errors. To demonstrate how general this connection is, we consider three examples: the classical Ising model in one dimension, free relativistic scalar quantum field theory (QFT) in two spacetime dimensions, and holographic field theories as examples of strongly coupled systems. As a concrete example of real-space RG in QFT, we consider the continuous multiscale renormalization ansatz for massive free fields and show that the low-energy coherent states are approximately protected from the quantum errors caused by the high-energy localized coherent operators. In holographic RG flows, we study the phase transition in the entanglement wedge of a single region and argue that one needs to define the price and the distance of the code with respect to the reconstructable wedge.