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On the Role of Hadamard Gates in Quantum Circuits

Dan Shepherd
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Abstract

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.

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Dan Shepherd. 2006-03-30. On the Role of Hadamard Gates in Quantum Circuits. https://arxiv.org/abs/quant-ph/0508153

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