 Publication Date:
 NSFPAR ID:
 10176075
 Journal Name:
 2019 IEEE 58th Conference on Decision and Control (CDC)
 Page Range or eLocationID:
 4916 to 4921
 Sponsoring Org:
 National Science Foundation
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We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speedups. We consider SDP instances with m constraint matrices, each of dimension n, rank at most r, and sparsity s. The first algorithm assumes an input model where one is given access to an oracle to the entries of the matrices at unit cost. We show that it has run time O~(s^2 (sqrt{m} epsilon^{10} + sqrt{n} epsilon^{12})), with epsilon the error of the solution. This gives an optimal dependence in terms of m, n and quadratic improvement over previous quantum algorithms (when m ~~ n). The second algorithm assumes a fully quantum input model in which the input matrices are given as quantum states. We show that its run time is O~(sqrt{m}+poly(r))*poly(log m,log n,B,epsilon^{1}), with B an upper bound on the tracenorm of all input matrices. In particular the complexity depends only polylogarithmically in n and polynomially in r. We apply the second SDP solver to learn a good description of a quantum state with respect to a set of measurements: Given m measurements and a supply of copies of an unknown state rho with rank at most r, we show we can find in time sqrt{m}*poly(logmore »

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Semidefinite programs (SDP) are important in learning and combinatorial optimization with numerous applications. In pursuit of lowrank solutions and low complexity algorithms, we consider the Burer–Monteiro factorization approach for solving SDPs. For a large class of SDPs, upon random perturbation of the cost matrix, with high probability, we show that all approximate secondorder stationary points are approximate global optima for the penalty formulation of appropriately rankconstrained SDPs, as long as the number of constraints scales subquadratically with the desired rank. Our result is based on a simple penalty function formulation of the rankconstrained SDP along with a smoothed analysis to avoid worstcase cost matrices. We particularize our results to two applications, namely, MaxCut and matrix completion.

Abstract We study the sparsity of the solutions to systems of linear Diophantine equations with and without nonnegativity constraints. The sparsity of a solution vector is the number of its nonzero entries, which is referred to as the
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