Generalized Color Codes Supporting Non-Abelian Anyons

We propose a generalization of the color codes based on finite groups $G$. For non-abelian groups, the resulting model supports non-abelian anyonic quasiparticles and topological order. We examine the properties of these models such as their relationship to Kitaev quantum double models, quasiparticle spectrum, and boundary structure.Comment: 17 pages, 8 figures; references added, typos remove

Generalized Cluster States Based on Finite Groups

We define generalized cluster states based on finite group algebras in analogy to the generalization of the toric code to the Kitaev quantum double models. We do this by showing a general correspondence between systems with CSS structure and finite group algebras, and applying this to the cluster states to derive their generalization. We then investigate properties of these states including their PEPS representations, global symmetries, and relationship to the Kitaev quantum double models. We also discuss possible applications of these states.Comment: 23 pages, 4 figure

A proposal for self-correcting stabilizer quantum memories in 3 dimensions (or slightly less)

We propose a family of local CSS stabilizer codes as possible candidates for self-correcting quantum memories in 3D. The construction is inspired by the classical Ising model on a Sierpinski carpet fractal, which acts as a classical self-correcting memory. Our models are naturally defined on fractal subsets of a 4D hypercubic lattice with Hausdorff dimension less than 3. Though this does not imply that these models can be realised with local interactions in 3D Euclidean space, we also discuss this possibility. The X and Z sectors of the code are dual to one another, and we show that there exists a finite temperature phase transition associated with each of these sectors, providing evidence that the system may robustly store quantum information at finite temperature.Comment: 16 pages, 6 figures. In v2, erroneous argument about embeddability into R3 was removed. In v3, minor changes to match journal versio

Perturbative 2-body Parent Hamiltonians for Projected Entangled Pair States

We construct parent Hamiltonians involving only local 2-body interactions for a broad class of Projected Entangled Pair States (PEPS). Making use of perturbation gadget techniques, we define a perturbative Hamiltonian acting on the virtual PEPS space with a finite order low energy effective Hamiltonian that is a gapped, frustration-free parent Hamiltonian for an encoded version of a desired PEPS. For topologically ordered PEPS, the ground space of the low energy effective Hamiltonian is shown to be in the same phase as the desired state to all orders of perturbation theory. An encoded parent Hamiltonian for the double semion string net ground state is explicitly constructed as a concrete example.Comment: 26 pages, 4 figures, v2 published versio

Toric codes and quantum doubles from two-body Hamiltonians

We present here a procedure to obtain the Hamiltonians of the toric code and Kitaev quantum double models as the low-energy limits of entirely two-body Hamiltonians. Our construction makes use of a new type of perturbation gadget based on error-detecting subsystem codes. The procedure is motivated by a projected entangled pair states (PEPS) description of the target models, and reproduces the target models' behavior using only couplings that are natural in terms of the original Hamiltonians. This allows our construction to capture the symmetries of the target models

Thermalization, Error-Correction, and Memory Lifetime for Ising Anyon Systems

We consider two-dimensional lattice models that support Ising anyonic excitations and are coupled to a thermal bath. We propose a phenomenological model for the resulting short-time dynamics that includes pair-creation, hopping, braiding, and fusion of anyons. By explicitly constructing topological quantum error-correcting codes for this class of system, we use our thermalization model to estimate the lifetime of the quantum information stored in the encoded spaces. To decode and correct errors in these codes, we adapt several existing topological decoders to the non-Abelian setting. We perform large-scale numerical simulations of these two-dimensional Ising anyon systems and find that the thresholds of these models range between 13% to 25%. To our knowledge, these are the first numerical threshold estimates for quantum codes without explicit additive structure.Comment: 34 pages, 9 figures; v2 matches the journal version and corrects a misstatement about the detailed balance condition of our Metropolis simulations. All conclusions from v1 are unaffected by this correctio