Natural Majorization of the Quantum Fourier
Transformation in Phase-Estimation Algorithms
and Miguel A. Martı
Received July 4, 2002; accepted October 18, 2002
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.
KEY WORDS: majorization; quantum Fourier transformation; quantum phase-
estimation; quantum algorithms.
PACS: 03.67.-a, 03.67.Lx
Majorization theory emerges as the natural framework to analyze and
quantify the measure of disorder for classical probability distributions.
Majorization ordering is far more severe than the one proposed by standard
Shannon entropy. If one probability distribution majorizes another, a set of
inequalities must hold that constrain the former probabilities with respect to
the latter. These inequalities entail entropy ordering, but the converse is not
necessarily true. Quantum mechanically, majorization has proven to be at
1570-0755/02/0800-0283/0 # 2003 Plenum Publishing Corporation
Department d’Estructura i Constituents de la Mate
ria, Univ. Barcelona, 08028. Barcelona,
Departamento de Fı
rica I, Universidad Complutense, 28040. Madrid, Spain.
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Quantum Information Processing, Vol. 1, No. 4, August 2002 (# 2003)