Topologically protected quantum states and quantum computing in Josephson junctions arrays

We review recent results on a new class of Josephson arrays which have non-trivial topology and exhibit a novel quantum states at low temperatures. One of these states is characterized by long range order in a two Cooper pair condensate and by a discrete topological order parameter. The second state is insulating and can be considered as a result of evolution of the former state due to Bose-condensation of usual superconductive vortices with a flux quantum Φ0. Quantum phase transition between these two states is controlled by variation of external magnetic field. Both the superconductive and insulating states are characterized by the presence of 2K-degenerate ground states, with K being the number of topologically different cycles existing in the plane of the array. This degeneracy is 'protected' from the external perturbations (and noise) by the topological order parameter and spectral gap. We show that in ideal conditions the low order effect of the external perturbations on this degeneracy is exactly zero and that deviations from ideality lead to only exponentially small effects of perturbations. We argue that this system provides a physical implementation of an ideal quantum computer with a built in error correction. A number of relatively simple 'echo-like' experiments possible on small-size arrays are discussed