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The hidden subgroup problem~(HSP) is one of the most important problems inquantum computation. Many problems for which quantum algorithm achievesexponential speedup over its classical counterparts can be reduced to theAbelian HSP. However, there is no efficient quantum algorithm for solving thenon-Abelian HSP. We find that the HSP can be reduced to a nested structuredsearch problem that is solved efficiently by using a quantum algorithm viamultistep quantum computation. Then we solve the HSP and problems that can bereduced to both the Abelian and the non-Abelian HSP in polynomial time by usingthis algorithm.

Polynomial-time quantum algorithm for solving the hidden subgroup problem