The hidden subgroup problem (HSP) plays an important role in quantum computation, because many quantum algorithms that are exponentially faster than classical algorithms can be casted in the HSP structure. In this paper, we present a new polynomial-time quantum algorithm that solves the HSP over the group Zpr ⋊ Zqs , when p r /q = poly(log p r ), where p, q are any odd prime numbers and r, s are any positive integers. To find the hidden subgroup, our algorithm uses the abelian quantum Fourier transform and a reduction procedure that simplifies the problem to find cyclic subgroups.2 The Structure of the Group Z p r ⋊ Z q s Let p, q be prime numbers and r, s positive integers. The semidirect product Z p r ⋊ φ Z q s , where Z p r and Z q s are cyclic groups and φ :
ABSTRACT. The hidden subgroup problem (HSP) plays an important role in quantum computing because many quantum algorithms that are exponentially faster than classical algorithms are special cases of the HSP. In this paper we show that there exists a new efficient quantum algorithm for the HSP on groups Z N Z q s where N is an integer with a special prime factorization, q prime number and s any positive integer.
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