It is well-known that Shor's factorization algorithm, Simon's period-finding
algorithm, and Deutsch's original XOR algorithm can all be formulated as
solutions to a hidden subgroup problem. Here the salient features of the
information-processing in the three algorithms are presented from a different
perspective, in terms of the way in which the algorithms exploit the
non-Boolean quantum logic represented by the projective geometry of Hilbert
space. From this quantum logical perspective, the XOR algorithm appears
directly as a special case of Simon's algorithm, and all three algorithms can
be seen as exploiting the non-Boolean logic represented by the subspace
structure of Hilbert space in a similar way. Essentially, a global property of
a function (such as a period, or a disjunctive property) is encoded as a
subspace in Hilbert space representing a quantum proposition, which can then be
efficiently distinguished from alternative propositions, corresponding to
alternative global properties, by a measurement (or sequence of measurements)
that identifies the target proposition as the proposition represented by the
subspace containing the final state produced by the algorithm.
