Qudit stabiliser codes for $\mathbb{Z}_N$ lattice gauge theories with matter

TL;DR

This paper establishes a connection between $ ext{Z}_N$ lattice gauge theories with matter and qudit stabilizer codes, enabling dual descriptions and fault-tolerant gate implementation through quantum error correction.

Contribution

It extends the stabilizer code framework to $ ext{Z}_N$ gauge theories with matter, providing exact mappings and demonstrating universal fault-tolerant gates for qudits.

Findings

Mapped $ ext{Z}_N$ gauge theories onto bosonic models

Uncovered a logical duality via error correction

Demonstrated fault-tolerant gates through state injection

Abstract

In this work we extend the connection between Quantum Error Correction (QEC) and Lattice Gauge Theories (LGTs) by showing that a $\mathbb{Z}_N$ gauge theory with prime dimension $N$ coupled to dynamical matter can be expressed as a qudit stabilizer code. Using the stabilizer formalism we show how to formulate an exact mapping of the encoded $\mathbb{Z}_N$ gauge theory onto two different bosonic models, uncovering a logical duality generated by error correction itself. From this perspective, quantum error correction provides a unifying language to expose dual descriptions of lattice gauge theories. In addition, we generalize earlier $\mathbb{Z}_2$ constructions on qubits to $\mathbb{Z}_N$ on $N$-level qudits and demonstrate how universal fault-tolerant gates can be implemented via state injection between compatible qudit codes.