An initialization-free quantum algorithm solves the hidden subgroup problem over all finite abelian groups, reusing an arbitrary mixed auxiliary state and restoring it, with O(log|G|) queries and O(log^3|G|) operations.
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An Initialization-free Quantum Algorithm for General Abelian Hidden Subgroup Problem
An initialization-free quantum algorithm solves the hidden subgroup problem over all finite abelian groups, reusing an arbitrary mixed auxiliary state and restoring it, with O(log|G|) queries and O(log^3|G|) operations.