  We study a reduced quantum circuit computation paradigm in which the only
allowable gates either permute the computational basis states or else apply a
"global Hadamard operation", i.e. apply a Hadamard operation to every qubit
simultaneously. In this model, we discuss complexity bounds (lower-bounding the
number of global Hadamard operations) for common quantum algorithms : we
illustrate upper bounds for Shor's Algorithm, and prove lower bounds for
Grover's Algorithm. We also use our formalism to display a gate that is neither
quantum-universal nor classically simulable, on the assumption that Integer
Factoring is not in BPP.
