Renormalization group and approximate error correction

TL;DR

This paper explores the connection between renormalization group flows and approximate quantum error correction, demonstrating how low-energy states in RG form code subspaces protected against local errors, with applications in classical and quantum field theories.

Contribution

It introduces a framework linking RG flow to approximate error correction, using examples from classical spin models, quantum field theory, and holography.

Findings

Low-energy states in RG are approximately protected from local errors.

cMERA states exhibit error resilience against high-energy localized operators.

Holographic RG shows phase transitions affecting entanglement wedge reconstructability.

Abstract

In renormalization group (RG) flow, the low energy states form a code subspace that is approximately protected against the local short-distance errors. We motivate this connection with an example of spin-blocking RG in classical spin models. We consider the continuous multi-scale renormalization ansatz (cMERA) for massive free fields as a concrete example of real-space RG in quantum field theory (QFT) and show that the low-energy coherent states are approximately protected from the 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.