Topological stabilizer models on continuous variables

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.

Figures (4)

• Figure 1: The logical operators of the $K_{2,-4}$ stabilizer code on a torus. The logical operators $\bar{X}_2$ and $\bar{Z}_2$ square to a product of stabilizers in $\mathcal{S}_2$ and anticommute. Therefore, they define an encoding of a qubit. Likewise the fourth powers of $\bar{X}_{-4}$ and $\bar{Z}_{-4}$ are products of stabilizers in $\mathcal{S}_{-4}$ and they fail to commute by a fourth root of unity. Hence, they define an encoding of a four-dimensional qudit.
• Figure 2: Anyonic excitations of the $K_{2,-4}$ stabilizer Hamiltonian. The $K_{2,-4}$ stabilizer Hamiltonian admits point-like excitations that cannot be created or destroyed with operators localized near the excitations. The excitations are instead created at the endpoints of string operators (purple) along oriented paths (red dashed). Only the vertex terms $A_v^{(2)}$ and $A_v^{(-4)}$ are violated at the endpoints of the strings.
• Figure 3: The topological excitations of the $\mathbb{R}$ gauge theory Hamiltonian. The $\mathbb{R}$ gauge theory Hamiltonian in Eq. \ref{['eq:H-R-gauge']} admits point-like excitations that are topological, in the sense that they cannot be created by operators supported solely in the vicinity of the excitation. The topological excitations are generated by the gauge charges, labeled by $c \in \mathbb{R}$, and the gauge fluxes, labeled by $\varphi \in \mathbb{R}$. The gauge charges and gauge fluxes are created by string operators on the direct lattice and dual lattice, respectively. Here, the notation $\vec{X}$ and $\vec{Z}$ denotes that the operator should be Hermitian conjugated if the orientation of the path (red dashed) points to the left or downward.
• Figure 4: Effect of condensing a charge-flux composite in $\mathbb{R}$ gauge theory. The $x$ axis corresponds to the flux label and the $y$ axis to the charge label of a deconfined excitation in the parent $\mathbb{R}$ theory. The deconfined excitations can be identified with a $\mathbb{Z}\times \mathbb{R}$ subgroup (indicated with dark blue lines), after condensing a $\mathbb{Z}$ subgroup of bosons (indicated with red dots). Additionally, condensation compactifies the space by adding equivalences among the excitations. In the case shown here, this leads to an effective description within the cylinder-shaped light blue region. The discrete part of the deconfined excitations (indicated with dark blue dots) gets compactified to a cyclic group while the continuous part is unaffected by the compactification since the $\mathbb{R}$ parameter runs along the cylinder.

Theorems & Definitions (2)

• Definition 1: topological CV stabilizer code
• Definition 2: homogeneous subalgebra