Natural majorization of the Quantum Fourier Transformation in phase-estimation algorithms

Abstract

We prove that majorization relations hold step by step in the Quantum Fourier Transformation (QFT) for phase-estimation algorithms considered in the canonical decomposition. Our result relies on the fact that states which are mixed by Hadamard operators at any stage of the computation only differ by a phase. This property is a consequence of the structure of the initial state and of the QFT, based on controlled-phase operators and a single action of a Hadamard gate per qubit. As a consequence, Hadamard gates order the probability distribution associated to the quantum state, whereas controlled-phase operators carry all the entanglement but are immaterial to majorization. We also prove that majorization in phase-estimation algorithms follows in a most natural way from unitary evolution, unlike its counterpart in Grover's algorithm.