Natural Majorization of the Quantum Fourier Transformation in Phase-Estimation Algorithms |
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Authors: | Orús Román Latorre José I. Martín-Delgado Miguel A. |
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Affiliation: | (1) Department d'Estructura i Constituents de la Matèria, Univ. Barcelona, 08028 Barcelona, Spain;(2) Departamento de Física Teórica I, Universidad Complutense, 28040 Madrid, Spain |
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Abstract: | 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.PACS: 03.67.-a, 03.67.Lx |
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Keywords: | majorization quantum Fourier transformation quantum phase-estimation quantum algorithms |
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