A local automaton for the 2D toric code

TL;DR

This paper introduces a local quantum decoder for the 2D toric code, enabling stable quantum memory through local operations without non-local classical communication, advancing topological quantum error correction.

Contribution

It presents the first local decoder for the 2D toric code that maintains logical information for exponential time using only local quantum operations.

Findings

Decoder operates for exponential time under circuit noise

No need for non-local classical communication

Can be extended to 3D stacks of 2D codes

Abstract

We construct a local decoder for the 2D toric code using ideas from the hierarchical classical cellular automata of Tsirelson and G\'acs. Our decoder is a circuit of strictly local quantum operations preserving a logical state for exponential time in the presence of circuit-level noise without the need for non-local classical computation or communication. Our construction is not translation invariant in spacetime, but can be made time-translation invariant in 3D with stacks of 2D toric codes. This solves the open problem of constructing a local topological quantum memory below four dimensions.