4 research outputs found
Generalized Color Codes Supporting Non-Abelian Anyons
We propose a generalization of the color codes based on finite groups .
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
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
Qudit color codes and gauge color codes in all spatial dimensions
The surface code is one of the most promising candidates for combating errors in large scale fault-tolerant
quantum computation. A fault-tolerant decoder is a vital part of the error correction processâit is the algorithm
which computes the operations needed to correct or compensate for the errors according to the measured
syndrome, even when the measurement itself is error prone. Previously decoders based on minimum-weight
perfect matching have been studied. However, these are not immediately generalizable from qubit to qudit codes.
In this work, we develop a fault-tolerant decoder for the surface code, capable of efficient operation for qubits
and qudits of any dimension, generalizing the decoder first introduced by Bravyi and Haah [Phys. Rev. Lett. 111,
200501 (2013)]. We study its performance when both the physical qudits and the syndromes measurements are
subject to generalized uncorrelated bit-flip noise (and the higher-dimensional equivalent). We show that, with
appropriate enhancements to the decoder and a high enough qudit dimension, a threshold at an error rate of more
than 8% can be achieved
Many-body models for topological quantum information
We develop and investigate several quantum many-body spin models of use for topological quantum information processing and storage. These models fall into two categories: those that are designed to be more realistic than alternative models with similar phenomenology, and those that are designed to have richer phenomenology than related models. In the first category, we present a procedure to obtain the Hamiltonians of the toric code and Kitaev quantum double models as the perturbative low-energy limits of entirely two-body Hamiltonians. This construction reproduces the target models' behavior using only couplings which are natural in terms of the original Hamiltonians. As an extension of this work, we construct parent Hamiltonians involving only local 2-body interactions for a broad class of Projected Entangled Pair States (PEPS). We define a perturbative Hamiltonian 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. We then move on to define models that generalize the phenomenology of several well-known systems. We first define generalized cluster states based on finite group algebras, and investigate properties of these states including their PEPS representations, global symmetries, relationship to the Kitaev quantum double models, and possible applications. Finally, we propose a generalization of the color codes based on finite groups. 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