  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
Deutsch-Jozsa, 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.
