The clairvoyant demon has a hard task

Consider the integer lattice L = ℤ2. For some m [ges ] 4, let us colour each column of this lattice independently and uniformly with one of m colours. We do the same for the rows, independently of the columns. A point of L will be called blocked if its row and column have the same colour. We say that this random configuration percolates if there is a path in L starting at the origin, consisting of rightward and upward unit steps, avoiding the blocked points. As a problem arising in distributed computing, it has been conjectured that for m [ges ] 4 the configuration percolates with positive probability. This question remains open, but we prove that the probability that there is percolation to distance n but not to infinity is not exponentially small in n. This narrows the range of methods available for proving the conjecture

Asymptotic Moments for Interference Mitigation in Correlated Fading Channels

We consider a certain class of large random matrices, composed of independent column vectors with zero mean and different covariance matrices, and derive asymptotically tight deterministic approximations of their moments. This random matrix model arises in several wireless communication systems of recent interest, such as distributed antenna systems or large antenna arrays. Computing the linear minimum mean square error (LMMSE) detector in such systems requires the inversion of a large covariance matrix which becomes prohibitively complex as the number of antennas and users grows. We apply the derived moment results to the design of a low-complexity polynomial expansion detector which approximates the matrix inverse by a matrix polynomial and study its asymptotic performance. Simulation results corroborate the analysis and evaluate the performance for finite system dimensions.Comment: 7 pages, 2 figures, to be presented at IEEE International Symposium on Information Theory (ISIT), Saint Petersburg, Russia, July 31 - August 5, 201

High threshold distributed quantum computing with three-qubit nodes

In the distributed quantum computing paradigm, well-controlled few-qubit `nodes' are networked together by connections which are relatively noisy and failure prone. A practical scheme must offer high tolerance to errors while requiring only simple (i.e. few-qubit) nodes. Here we show that relatively modest, three-qubit nodes can support advanced purification techniques and so offer robust scalability: the infidelity in the entanglement channel may be permitted to approach 10% if the infidelity in local operations is of order 0.1%. Our tolerance of network noise is therefore a order of magnitude beyond prior schemes, and our architecture remains robust even in the presence of considerable decoherence rates (memory errors). We compare the performance with that of schemes involving nodes of lower and higher complexity. Ion traps, and NV- centres in diamond, are two highly relevant emerging technologies.Comment: 5 figures, 12 pages in single column format. Revision has more detailed comparison with prior scheme