19 research outputs found
Codes over rings of size p2 and lattices over imaginary quadratic fields
AbstractLet ℓ>0 be a square-free integer congruent to 3 mod 4 and OK the ring of integers of the imaginary quadratic field K=Q(−ℓ). Codes C over rings OK/pOK determine lattices Λℓ(C) over K. If p∤ℓ then the ring R:=OK/pOK is isomorphic to Fp2 or Fp×Fp. Given a code C over R, theta functions on the corresponding lattices are defined. These theta series θΛℓ(C)(q) can be written in terms of the complete weight enumerators of C. We show that for any two ℓ<ℓ′ the first ℓ+14 terms of their corresponding theta functions are the same. Moreover, we conjecture that for ℓ>p(n+1)(n+2)2 there is a unique symmetric weight enumerator corresponding to a given theta function. We verify the conjecture for primes p<7, ℓ⩽59, and small n
Comments on the holographic description of Narain theories
We discuss the holographic description of Narain
conformal field theories, and their potential similarity to conventional weakly
coupled gravity in the bulk, in the sense that the effective IR bulk
description includes " gravity" amended with additional light degrees of
freedom. Starting from this picture, we formulate the hypothesis that in the
large central charge limit the density of states of any Narain theory is
bounded by below by the density of states of gravity. This immediately
implies that the maximal value of the spectral gap for primary fields is
. To test the self-consistency of this proposal, we study
its implications using chiral lattice CFTs and CFTs based on quantum stabilizer
codes. First we notice that the conjecture yields a new bound on quantum
stabilizer codes, which is compatible with previously known bounds in the
literature. We proceed to discuss the variance of the density of states, which
for consistency must be vanishingly small in the large- limit. We consider
ensembles of code and chiral theories and show that in both cases the density
variance is exponentially small in the central charge
Narain CFTs from nonbinary stabilizer codes
We generalize the construction of Narain conformal field theories (CFTs) from
qudit stabilizer codes to the construction from quantum stabilizer codes over
the finite field of prime power order ( with prime and
) or over the ring with . Our construction results
in rational CFTs, which cover a larger set of points in the moduli space of
Narain CFTs than the previous one. We also propose a correspondence between a
quantum stabilizer code with non-zero logical qubits and a finite set of Narain
CFTs. We illustrate the correspondence with well-known stabilizer codes.Comment: 38 page
Quantum Stabilizer Codes, Lattices, and CFTs
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 ≤ 12, and find many interesting examples. Among them is a non-chiral E8 theory, which is based on the root lattice of E8 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