Quantum stabilizer codes, lattices, and CFTs

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 $c=n\le 12$, 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 $n\le 8$ 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.

99 pages

References in corpus (5)

• Universal Spectrum of 2d Conformal Field Theory in the Large c Limit
• Averaging Over Narain Moduli Space
• Free partition functions and an averaged holographic duality
• Edge Local Complementation and Equivalence of Binary Linear Codes
• Non-chiral 2d CFT with integer energy levels

Cited by in corpus (13)

• Bootstrapping Non-Invertible Symmetries
• ${\mathbb{Z}}_N$ Symmetries, Anomalies, and the Modular Bootstrap
• Solutions of modular bootstrap constraints from quantum codes
• Comments on the holographic description of Narain theories
• Relation between spectra of Narain CFTs and properties of associated boolean functions
• Bootstrapping Lieb-Schultz-Mattis anomalies
• Global Symmetries, Code Ensembles, and Sums Over Geometries
• Non-Chiral Vertex Operator Algebra Associated To Lorentzian Lattices And Narain CFTs
• Operator-state correspondence in simple current extended conformal field theories: Toward a general understanding of chiral conformal field theories and topological orders
• Constructing quantum codes from any classical code and their embedding in ground space of local Hamiltonians
• Algebraic constructions of code lattices in Narain conformal field theories
• Optimal Narain CFTs from Codes
• Code CFTs and Topological Matter