We use a categorical topological semantics to examine the Deutsch-Jozsa,
hidden subgroup and single-shot Grover algorithms. This reveals important
structures hidden by conventional algebraic presentations, and allows novel
proofs of correctness via local topological operations, giving for the first
time a satisfying high-level explanation for why these procedures work. We also
investigate generalizations of these algorithms, providing improved analyses of
those already in the literature, and a new generalization of the single-shot
Grover algorithm.Comment: 33 pages. Updated to match the final published articl