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 errorcorrecting 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 ≤ 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 ≤ 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.

Wormholes and spectral statistics in the Narain ensemble

“…However, once the twists surpass the bound (4.43), the relative contribution of the disconnected part becomes exponentially suppressed. A similar exponential suppression of the connected two-point function of the density of states in the large central charge limit was observed in the ensembles of chiral free bosons and of code CFTs [64].…”

“…An ensemble putatively dual to pure Einstein gravity will presumably include isolated points in addition, perhaps, to connected moduli spaces of CFTs. 13 The gravitational description of the Narain ensemble was further studied in [61][62][63], and the discrete average of holomorphic free boson CFTs was discussed in [64]. Aspects of ensemble averages over other classes of rational two-dimensional conformal field theories have also been discussed [65][66][67][68][69].…”

Narain CFTs and error-correcting codes on finite fields

“…Recently, the relation between classical error-correcting codes on F 2 or F 4 = {0, 1, ω, ω 2 } and Narain CFTs through lattices on R n,n have been studied and quantities in CFTs such as spectral gaps and partition functions were described in the language of codes [14,15]. In this paper, we extend the relation to codes on F p for any prime p > 2 and more general Narain CFTs.…”

$\mathbb{Z}_N$ Symmetries, Anomalies, and the Modular Bootstrap

“…The modern modular bootstrap program, which exploits the above consistency conditions, provides an ideal platform to map out the space of CFTs with global symmetry. See, for example, [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]1,[28][29][30][31][32][33][34][35][36][37][38][39] for various exciting developments in the modular bootstrap.…”