44,657 research outputs found

    Simulating quantum computation by contracting tensor networks

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    The treewidth of a graph is a useful combinatorial measure of how close the graph is to a tree. We prove that a quantum circuit with TT gates whose underlying graph has treewidth dd can be simulated deterministically in TO(1)exp[O(d)]T^{O(1)}\exp[O(d)] time, which, in particular, is polynomial in TT if d=O(logT)d=O(\log T). Among many implications, we show efficient simulations for log-depth circuits whose gates apply to nearby qubits only, a natural constraint satisfied by most physical implementations. We also show that one-way quantum computation of Raussendorf and Briegel (Physical Review Letters, 86:5188--5191, 2001), a universal quantum computation scheme with promising physical implementations, can be efficiently simulated by a randomized algorithm if its quantum resource is derived from a small-treewidth graph.Comment: 7 figure

    Universal spectral statistics in Wigner-Dyson, chiral and Andreev star graphs II: semiclassical approach

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    A semiclassical approach to the universal ergodic spectral statistics in quantum star graphs is presented for all known ten symmetry classes of quantum systems. The approach is based on periodic orbit theory, the exact semiclassical trace formula for star graphs and on diagrammatic techniques. The appropriate spectral form factors are calculated upto one order beyond the diagonal and self-dual approximations. The results are in accordance with the corresponding random-matrix theories which supports a properly generalized Bohigas-Giannoni-Schmit conjecture.Comment: 15 Page

    Universal spectral form factor for chaotic dynamics

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    We consider the semiclassical limit of the spectral form factor K(τ)K(\tau) of fully chaotic dynamics. Starting from the Gutzwiller type double sum over classical periodic orbits we set out to recover the universal behavior predicted by random-matrix theory, both for dynamics with and without time reversal invariance. For times smaller than half the Heisenberg time THf+1T_H\propto \hbar^{-f+1}, we extend the previously known τ\tau-expansion to include the cubic term. Beyond confirming random-matrix behavior of individual spectra, the virtue of that extension is that the ``diagrammatic rules'' come in sight which determine the families of orbit pairs responsible for all orders of the τ\tau-expansion.Comment: 4 pages, 1 figur

    Compressed Sensing Using Binary Matrices of Nearly Optimal Dimensions

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    In this paper, we study the problem of compressed sensing using binary measurement matrices and 1\ell_1-norm minimization (basis pursuit) as the recovery algorithm. We derive new upper and lower bounds on the number of measurements to achieve robust sparse recovery with binary matrices. We establish sufficient conditions for a column-regular binary matrix to satisfy the robust null space property (RNSP) and show that the associated sufficient conditions % sparsity bounds for robust sparse recovery obtained using the RNSP are better by a factor of (33)/22.6(3 \sqrt{3})/2 \approx 2.6 compared to the sufficient conditions obtained using the restricted isometry property (RIP). Next we derive universal \textit{lower} bounds on the number of measurements that any binary matrix needs to have in order to satisfy the weaker sufficient condition based on the RNSP and show that bipartite graphs of girth six are optimal. Then we display two classes of binary matrices, namely parity check matrices of array codes and Euler squares, which have girth six and are nearly optimal in the sense of almost satisfying the lower bound. In principle, randomly generated Gaussian measurement matrices are "order-optimal". So we compare the phase transition behavior of the basis pursuit formulation using binary array codes and Gaussian matrices and show that (i) there is essentially no difference between the phase transition boundaries in the two cases and (ii) the CPU time of basis pursuit with binary matrices is hundreds of times faster than with Gaussian matrices and the storage requirements are less. Therefore it is suggested that binary matrices are a viable alternative to Gaussian matrices for compressed sensing using basis pursuit. \end{abstract}Comment: 28 pages, 3 figures, 5 table
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