Relation between spectra of Narain CFTs and properties of associated boolean functions

TL;DR

This paper explores the link between the spectral properties of Narain CFTs derived from quantum error correcting codes and the properties of associated boolean functions, revealing new ways to optimize spectral gaps.

Contribution

It establishes that the spectral gap of Narain CFTs correlates with the EPC distance of associated boolean functions, connecting quantum codes, boolean function properties, and spectral analysis.

Findings

Binary distance equals EPC distance of boolean functions

High spectral gap relates to boolean functions with high EPC distance

Codes with high distance are linked to graphs with low independence numbers

Abstract

Recently, the construction of Narain CFT from a certain class of quantum error correcting codes has been discovered. In particular, the spectral gap of Narain CFT corresponds to the binary distance of the code, not the genuine Hamming distance. In this paper, we show that the binary distance is identical to the so-called EPC distance of the boolean function uniquely associated with the quantum code. Therefore, seeking Narain CFT with high spectral gap is equivalent to getting a boolean function with high EPC distance. Furthermore, this problem can be addressed by finding lower Peak-to-Average Power ratio (PAR) with respect to the binary truth table of the boolean function. Though this is neither sufficient nor necessary condition for high EPC distance, we construct some examples of relatively high EPC distances referring to the constructions for lower PAR. We also see that codes with high distance are related to induced graphs with low independence numbers.