  We construct abstract models of blackbox quantum algorithms using a model of
quantum computation in sets and relations, a setting that is usually considered
for nondeterministic classical computation. This alternative model of quantum
computation (QCRel), though unphysical, nevertheless faithfully models its
computational structure. Our main results are models of the Deutsch-Jozsa,
single-shot Grovers, and GroupHomID algorithms in QCRel. These results provide
new tools to analyze the semantics of quantum computation and improve our
understanding of the relationship between computational speedups and the
structure of physical theories. They also exemplify a method of extending
physical/computational intuition into new mathematical settings.
