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International Journal of Quantum Information

MÖBIUS TRANSFORMATIONS IN QUANTUM AMPLITUDE AMPLIFICATION WITH GENERALIZED PHASES

César Bautista-Ramos, Nora Castillo-Tépox

Published Sep 1, 2010• 1 citation
Publisher

Abstract

The iteration of the operators employed in quantum amplitude amplification with generalized phases is analyzed by using elementary properties (geometric and algebraic) of the Möbius transformations (fractional linear transformations). It is shown that, for a given quantum algorithm without measurement, which produces a good state with probability a of success, if the phase angles φ and ϕ which mark the good and initial states respectively satisfy φ = ϕ with a small enough, then, for a number n of iterations with [Formula: see text] we get an error probability that is at most O(aϕ2).

DOI
10.1142/s0219749910005764

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The iteration of the operators employed in quantum amplitude amplification with generalized phases is analyzed by using elementary properties (geometric and algebraic) of the Möbius transformations (fractional linear transformations). It is shown that, for a given quantum algorithm without measurement, which produces a good state with probability a of success, if the phase angles φ and ϕ which mark the good and initial states respectively satisfy φ = ϕ with a small enough, then, for a number n of iterations with [Formula: see text] we get an error probability that is at most O(aϕ2).
We consider the use of arbitrary phases in quantum amplitude amplification, which is a generalization of quantum searching. We prove that the phase condition in amplitude amplification is given by $\mathrm{tan}(\ensuremath{\varphi}/2)=\mathrm{tan}(\ensuremath{\varphi}/2)(1\ensuremath{-}2a),$ where \ensuremath{\varphi} and \ensuremath{\varphi} are the phases used and where a is the success probability of the given algorithm. Thus the choice of phases depends nontrivially and nonlinearly on the success probability. Utilizing this condition, we give methods for constructing quantum algorithms that succeed with certainty and for implementing arbitrary rotations. We also conclude that phase errors of order up to 1/a1/\sqrt{a}1/a​ can be tolerated in amplitude amplification.
One iterative in Grover's original quantum search algorithm consists of two Hadamard–Walsh transformations, a selective amplitude inversion and a diffusion amplitude inversion. We concentrate on the relation among the probability of success of the algorithm, the phase shifts, the number of target items and the number of iterations via replacing the two amplitude inversions by phase shifts of an arbitrary ϕ = ϕ(0 ⩽ ϕ, ϕ ⩽ 2π). Then, according to the relation we find out the optimal phase shifts when the number of iterations is given. We present a new quantum search algorithm based on the optimal phase shifts of 1.018 after 0.5π/ iterations. The new algorithm can obtain either a single target item or multiple target items in the search space with the probability of success at least 93.43%.
The recursion equation analysis of Grover's quantum search algorithm presented by Biham et al. [Phys. Rev. A 60, 2742 (1999)] is generalized. It is applied to the large class of Grover-type algorithms in which the Hadamard transform is replaced by any other unitary transformation and the phase inversion is replaced by a rotation by an arbitrary angle. The time evolution of the amplitudes of the marked and unmarked states, for any initial complex amplitude distribution, is expressed using first-order linear difference equations. These equations are solved exactly. The solution provides the number of iterations T after which the probability of finding a marked state upon measurement is the highest, as well as the value of this probability, Pmax.{P}_{\mathrm{max}}.Pmax​. Both T and Pmax{P}_{\mathrm{max}}Pmax​ are found to depend on the averages and variances of the initial amplitude distributions of the marked and unmarked states, but not on higher moments.
Each iteration in Grover’s original quantum search algorithm contains 4 steps: two Hadamard-Walsh transformations and two amplitudes inversions. When the inversion of the marked state is replaced by arbitrary phase rotation θ and the inversion for the prepared state |γ 〉 is replaced by rotation through φ, we found that these phase rotations must satisfy a matching condition θ = φ. Approximate formula for the amplitude of the marked state after an arbitrary number of iterations are also derived. When phase matching is obtained, we found that the generalized quantum search is still a rotation in 2-dimensional space, but with a small angle β ′ ≈ (2sin θ 2)β, where θ is the angle of phase rotation of the marked state and 2β is the angle of U(2) rotation when the phase rotations are the inversions in the original version. We give also a simple explanation of the phase matching requirement.
Quantum search/amplitude amplification algorithms are designed to be able to amplify the amplitude in the target state linearly with the number of operations. Since the probability is the square of the amplitude, this results in the success probability rising quadratically with the number of operations. This paper presents a new kind of quantum search algorithm in which the amplitude of the target state, itself increases quadratically with the number of operations. However, the domain of applications of this is much more limited than standard amplitude amplification.
Phase matching has been studied for the Grover algorithm as a way of enhancing the efficiency of the quantum search. Recently Li and Li found that a particular form of phase matching yields, with a single Grover operation, a success probability greater than 25/2725/2725/27 for finding the equal-amplitude superposition of marked states when the fraction of the marked states stored in a database state is greater than 1/31/31/3. Although this single operation eliminates the oscillations of the success probability that occur with multiple Grover operations, the latter oscillations reappear with multiple iterations of Li and Li's phase matching. In this paper we introduce a multiphase matching subject to a certain matching rule by which we can obtain a multiple Grover operation that with only a few iterations yields a success probability that is almost constant and unity over a wide range of the fraction of marked items. As an example we show that a multiphase operation with six iterations yields a success probability between 99.8 and 100 100\text{ }%100  for a fraction of marked states of 1/101/101/10 or larger.