Topological stabilizer models on continuous variables

TL;DR

This paper introduces a new class of topological stabilizer codes for continuous variable systems, generalizing existing models and exploring their unique anyon theories, with implications for quantum error correction and lattice gauge theories.

Contribution

It constructs a broad family of topological CV stabilizer codes via boson condensation, including non-chiral anyon theories without gapped boundaries, suggesting these codes are intrinsic to CV systems.

Findings

Constructed topological CV stabilizer codes using boson condensation.

Identified codes with anyon theories of $U(1)_{2n}\times U(1)_{-2m}$ Chern-Simons theories.

Demonstrated that associated Hamiltonians can be gapped with quadratic perturbations.

Abstract

We construct a family of two-dimensional topological stabilizer codes on continuous variable (CV) degrees of freedom, which generalize homological rotor codes and the toric-GKP code. Our topological codes are built using the concept of boson condensation -- we start from a parent stabilizer code based on an $\mathbb{R}$ gauge theory and condense various bosonic excitations. This produces a large class of topological CV stabilizer codes, including ones that are characterized by the anyon theories of $U(1)_{2n}\times U(1)_{-2m}$ Chern-Simons theories, for arbitrary pairs of positive integers $(n,m)$. Most notably, this includes anyon theories that are non-chiral and nevertheless do not admit a gapped boundary. It is widely believed that such anyon theories cannot be realized by any stabilizer model on finite-dimensional systems. We conjecture that these CV codes go beyond codes obtained from concatenating a topological qudit code with a local encoding into CVs, and thus, constitute the first example of topological codes that are intrinsic to CV systems. Moreover, we study the Hamiltonians associated to the topological CV stabilizer codes and show that, although they have a gapless spectrum, they can become gapped with the addition of a quadratic perturbation. We show that similar methods can be used to construct a gapped Hamiltonian whose anyon theory agrees with a $U(1)_2$ Chern-Simons theory. Our work initiates the study of scalable stabilizer codes that are intrinsic to CV systems and highlights how error-correcting codes can be used to design and analyze many-body systems of CVs that model lattice gauge theories.