We construct a discrete subset of Narain CFTs from quantum stabilizer codes with qudit (including qubit) systems whose dimension is a prime number. Our construction exploits three important relations. The first relation is between qudit stabilizer codes and classical codes. The second is between classical codes and Lorentzian lattices. The third is between Lorentzian lattices and Narain CFTs. In particular, we study qudit Calderbank-Shor-Steane (CSS) codes as a special class of qudit stabilizer codes and the ensembles of the Narain code CFTs constructed from CSS codes. We obtain exact results for the averaged partition functions over the ensembles and discuss their implications for holographic duality.
We construct a discrete subset of Narain CFTs from quantum stabilizer codes with qudit (including qubit) systems whose dimension is a prime number. Our construction exploits three important relations. The first relation is between qudit stabilizer codes and classical codes. The second is between classical codes and Lorentzian lattices. The third is between Lorentzian lattices and Narain CFTs. In particular, we study qudit Calderbank-Shor-Steane (CSS) codes as a special class of qudit stabilizer codes and the ensembles of the Narain code CFTs constructed from CSS codes. We obtain exact results for the averaged partition functions over the ensembles and discuss their implications for holographic duality.
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 ≤ 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.
Abstract 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 ( $$ {\mathbbm{F}}_{p^m} $$ F p m with p prime and m ≥ 1) or over the ring ℤk with k > 1. 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.
A bstract We construct Narain conformal field theories (CFTs) from quantum subsystem codes, a more comprehensive class of quantum error-correcting codes than quantum stabilizer codes, for qudit systems of prime dimensions. The resulting code CFTs exhibit a global ℤ 2 symmetry, enabling us to perform the ℤ 2 -gauging to derive their orbifolded and fermionized theories when the symmetry is non-anomalous. We classify a subset of these subsystem code CFTs using weighted oriented graphs and enumerate those with small central charges. Consequently, we identify several bosonic code CFTs self-dual under the ℤ 2 -orbifold, new supersymmetric code CFTs, and a few fermionic code CFTs with spontaneously broken supersymmetry.
Quantum subsystem codes, CFTs and their Z2-gaugings
We construct Narain conformal field theories (CFTs) from quantum subsystem codes, a more comprehensive class of quantum error-correcting codes than quantum stabilizer codes, for qudit systems of prime dimensions. The resulting code CFTs exhibit a global Z2 symmetry, enabling us to perform the Z2-gauging to derive their orbifolded and fermionized theories when the symmetry is non-anomalous. We classify a subset of these subsystem code CFTs using weighted oriented graphs and enumerate those with small central charges. Consequently, we identify several bosonic code CFTs self-dual under the Z2-orbifold, new supersymmetric code CFTs, and a few fermionic code CFTs with spontaneously broken supersymmetry.
We prove that on any two-dimensional lattice of qudits of a prime dimension, every translation invariant Pauli stabilizer group with local generators and with the code distance being the linear system size is decomposed by a local Clifford circuit of constant depth into a finite number of copies of the toric code stabilizer group (Abelian discrete gauge theory). This means that under local Clifford circuits, the number of toric code copies is the complete invariant of topological Pauli stabilizer codes. Previously, the same conclusion was obtained under the assumption of nonchirality for qubit codes or the Calderbank–Shor–Steane structure for prime qudit codes; we do not assume any of these.
Fp-Linear and Fpm-Linear Qudit Codes From Dual-Containing Classical Codes
Quantum code construction from two classical codes D1[n,k1,d1] and D2[n,k2,d2] over the field Fpm(p is prime and m is an integer) satisfying the dual containing criteria D1⊥⊂ D2using the Calderbank-Shor-Steane (CSS) framework is well-studied. We show that the generalization of the CSS framework for qubits to qudits yields two different classes of codes, namely, the Fp-linear CSS codes and the well-known Fpm-linear CSS codes based on the check matrix-based definition and the coset-based definition of CSS codes over qubits. Our contribution to this article are three-folds. 1) We study the properties of the \mathbb Fp-linear and Fpm-linear CSS codes and demonstrate the tradeoff for designing codes with higher rates or better error detection and correction capability, useful for quantum systems. 2) For Fpm-linear CSS codes, we provide the explicit form of the check matrix and show that the minimum distances dxand dzare equal to d2and d1, respectively, if and only if the code is nondegenerate. 3) We propose two classes of quantum codes obtained from the codes D1and D2, where one code is an Fpl-linear code (l divides m) and the other code is obtained from a particular subgroup of the stabilizer group of the Fpm-linear CSS code. Within each class of codes, we demonstrate the tradeoff between higher rates and better error detection and correction capability.