This paper describes a quantum algorithm for efficiently decomposing finite Abelian groups into a product of cyclic groups. Such a decomposition is needed in order to apply the Abelian hidden subgroup algorithm. Such a decomposition (assuming the Generalized Riemann Hypothesis) also leads to an efficient algorithm for computing class numbers (known to be at least as difficult as factoring).

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Over the years, various lift-and-project methods have been proposed to construct hierarchies of successive linear or semidefinite relaxations of a 0–1 polytope P ⫅ ℝn that converge to P in n steps. Many such methods have been shown to require n steps in the worst case. In this paper, we show that the method of Lasserre also requires n steps in the worst case.

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