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.
Shepherd Daniel (Sun,) studied this question.