Abstract
We describe a new polynomial-time quantum algorithm that solves the hidden subgroup problem (HSP) for a special class of metacyclic groups, namely \(\mathbb{Z}_{p} \rtimes \mathbb{Z}_{q^s}\), with q |(p − 1) and p/q = poly(log p), where p, q are any odd prime numbers and s is any positive integer. This solution generalizes previous algorithms presented in the literature. In a more general setting, without imposing a relation between p and q, we obtain a quantum algorithm with time and query complexity \(2^{O(\sqrt{\log p})}\). In any case, those results improve the classical algorithm, which needs \({\Omega}(\sqrt{p})\) queries.
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Gonçalves, D.N., Portugal, R., Cosme, C.M.M. (2009). Solutions to the Hidden Subgroup Problem on Some Metacyclic Groups. In: Childs, A., Mosca, M. (eds) Theory of Quantum Computation, Communication, and Cryptography. TQC 2009. Lecture Notes in Computer Science, vol 5906. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10698-9_1
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DOI: https://doi.org/10.1007/978-3-642-10698-9_1
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