Quantum stabilizer codes, lattices, and CFTs

TL;DR

This paper establishes a novel connection between quantum stabilizer codes, Lorentzian lattices, and conformal field theories, introducing code CFTs and exploring their properties, dualities, and examples with small central charge.

Contribution

It introduces the concept of code CFTs derived from stabilizer codes and Lorentzian lattices, and analyzes their properties, dualities, and examples, including non-chiral E8 theories.

Findings

Many pairs and triples of isospectral theories identified.

Constructed modular invariant functions not corresponding to known CFTs.

Ensemble average over code theories suggests potential holographic interpretations.

Abstract

There is a rich connection between classical error-correcting codes, Euclidean lattices, and chiral conformal field theories. Here we show that quantum error-correcting codes, those of the stabilizer type, are related to Lorentzian lattices and non-chiral CFTs. More specifically, real self-dual stabilizer codes can be associated with even self-dual Lorentzian lattices, and thus define Narain CFTs. We dub the resulting theories code CFTs and study their properties. T-duality transformations of a code CFT, at the level of the underlying code, reduce to code equivalences. By means of such equivalences, any stabilizer code can be reduced to a graph code. We can therefore represent code CFTs by graphs. We study code CFTs with small central charge $c=n\leq 12$, and find many interesting examples. Among them is a non-chiral $E_8$ theory, which is based on the root lattice of $E_8$ understood as an even self-dual Lorentzian lattice. By analyzing all graphs with $n\leq 8$ nodes we find many pairs and triples of physically distinct isospectral theories. We also construct numerous modular invariant functions satisfying all the basic properties expected of the CFT partition function, yet which are not partition functions of any known CFTs. We consider the ensemble average over all code theories, calculate the corresponding partition function, and discuss its possible holographic interpretation. The paper is written in a self-contained manner, and includes an extensive pedagogical introduction and many explicit examples.