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Quantum hidden subgroup algorithms on free groups, (in preparation
"... Abstract. One of the most promising and versatile approaches to creating new quantum algorithms is based on the quantum hidden subgroup (QHS) paradigm, originally suggested by Alexei Kitaev. This class of quantum algorithms encompasses the DeutschJozsa, Simon, Shor algorithms, and many more. In thi ..."
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Abstract. One of the most promising and versatile approaches to creating new quantum algorithms is based on the quantum hidden subgroup (QHS) paradigm, originally suggested by Alexei Kitaev. This class of quantum algorithms encompasses the DeutschJozsa, Simon, Shor algorithms, and many more. In this paper, our strategy for finding new quantum algorithms is to decompose Shor’s quantum factoring algorithm into its basic primitives, then to generalize these primitives, and finally to show how to reassemble them into new QHS algorithms. Taking an ”alphabetic building blocks approach, ” we use these primitives to form an ”algorithmic toolkit ” for the creation of new quantum algorithms, such as wandering Shor algorithms, continuous Shor algorithms, the quantum circle algorithm, the dual Shor algorithm, a QHS algorithm for Feynman integrals, free QHS algorithms, and more. Toward the end of this paper, we show how Grover’s algorithm is most surprisingly “almost ” a QHS algorithm, and how this result suggests the possibility of an even more complete ”algorithmic tookit ” beyond the QHS algorithms. Contents
Ramanujan Primes and Bertrand's Postulate
"... 8, pp. 208209] published one year before his death in 1920 at the age of 32, the Indian mathematical genius Srinivasa Ramanujan wrote: Landau in his Handbuch [4], pp. 8992, gives a proof of a theorem the truth of which was conjectured by Bertrand: namely that there is at least one prime p such tha ..."
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8, pp. 208209] published one year before his death in 1920 at the age of 32, the Indian mathematical genius Srinivasa Ramanujan wrote: Landau in his Handbuch [4], pp. 8992, gives a proof of a theorem the truth of which was conjectured by Bertrand: namely that there is at least one prime p such that