We prove that majorization relations hold step by step in the Quantum Fourier Transformation (QFT) for phase-estimation algorithms. 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. The detail of our proof shows that Hadamard gates sort 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.
Quantum Information Processing – Springer Journals
Published: Oct 13, 2004
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