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 projected entangled pair state representations, global symmetries, and relationship to the Kitaev quantum double models. We also discuss possible applications of these states.

The D(D-3)-anyon chain: integrable boundary conditions and excitation spectra

Chains of interacting non-Abelian anyons with local interactions invariant under the action of the Drinfeld double of the dihedral group D-3 are constructed. Formulated as a spin chain the Hamiltonians are generated from commuting transfer matrices of an integrable vertex model for periodic and braided as well as open boundaries. A different anyonic model with the same local Hamiltonian is obtained within the fusion path formulation. This model is shown to be related to an integrable fusion interaction round the face model. Bulk and surface properties of the anyon chain are computed from the Bethe equations for the spin chain. The low-energy effective theories and operator content of the models (in both the spin chain and fusion path formulation) are identified from analytical and numerical studies of the finite-size spectra. For all boundary conditions considered the continuum theory is found to be a product of two conformal field theories. Depending on the coupling constants the factors can be a Z(4) parafermion or a M-(5,M-6) minimal model.

Generating a state t-design by diagonal quantum circuits

We investigate protocols for generating a state t-design by using a fixed separable initial state and a diagonal-unitary t-design in the computational basis, which is a t-design of an ensemble of diagonal unitary matrices with random phases as their eigenvalues. We first show that a diagonal-unitary t-design generates a O (1/2(N))-approximate state t-design, where N is the number of qubits. We then discuss a way of improving the degree of approximation by exploiting non-diagonal gates after applying a diagonal-unitary t-design. We also show that it is necessary and sufficient to use O (log(2)(t)) -qubit gates with random phases to generate a diagonal-unitary t-design by diagonal quantum circuits, and that each multi-qubit diagonal gate can be replaced by a sequence of multi-qubit controlled-phase-type gates with discrete-valued random phases. Finally, we analyze the number of gates for implementing a diagonal-unitary t-design by non-diagonal two- and one-qubit gates. Our results provide a concrete application of diagonal quantum circuits in quantum informational tasks.