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Jisuan wuli

General Multiphase Matching for Grover Quantum Search Algorithm

Chen Xiangdong

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

Since a quantum system is inevitably influenced by some unpredictable perturbations,we thereby conclude that all the experimental realizations of Grover quantum search algorithm reported were,in fact,achieved in a three-dimensional complex subspace.We also prove that in a two-dimensional complex subspace,for any given initial superposition of basis states |γ0〉=cos β0|α〉+sin β0eiζ|β〉(β0 is a small positive real number,ζ is an arbitrary real number),there exists a set of solutions Fj={(θj,θj-1,…,θ1),(φj,φj-1,…,φ1)} such that a desired state can be found with certainty for some positive integer j≥2,where the phase rotation angles θl and φl are real numbers but not equal to 2k′π,1≤l≤j,k′ is an arbitrary integer.If it is only required that a desired state can be found with high success probability,then as the total number of the desired and undesired states in an unsorted database is sufficiently large the above set of solutions Fj can be written in the form Σji=1θl=Σji=1φl.for a relatively small positive integer j.

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Multiphase matching in the Grover algorithm

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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 $25/27$ for finding the equal-amplitude superposition of marked states when the fraction of the marked states stored in a database state is greater than $1/3$. 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 $100\text{ }%$ for a fraction of marked states of $1/10$ or larger.

arXiv:0801.2956•Physical Review A•Apr 30, 2008•26 citations
Since a quantum system is inevitably influenced by some unpredictable perturbations,we thereby conclude that all the experimental realizations of Grover quantum search algorithm reported were,in fact,achieved in a three-dimensional complex subspace.We also prove that in a two-dimensional complex subspace,for any given initial superposition of basis states |γ0〉=cos β0|α〉+sin β0eiζ|β〉(β0 is a small positive real number,ζ is an arbitrary real number),there exists a set of solutions Fj={(θj,θj-1,…,θ1),(φj,φj-1,…,φ1)} such that a desired state can be found with certainty for some positive integer j≥2,where the phase rotation angles θl and φl are real numbers but not equal to 2k′π,1≤l≤j,k′ is an arbitrary integer.If it is only required that a desired state can be found with high success probability,then as the total number of the desired and undesired states in an unsorted database is sufficiently large the above set of solutions Fj can be written in the form Σji=1θl=Σji=1φl.for a relatively small positive integer j.
For the case when the search space is confined to a two-dimensional complex subspace,if the total number of an unsorted database is sufficiently large,then a unique desired state can still be found with high success probability provided any two sets of phase rotation angles,which are completely independent(unrelated to each other) but the number of elements in one of the sets is relatively small compared to the total number of an unsorted database,satisfy the multiphase matching equation.In this paper we showed that for a quantum search problem with some specific requirements,one may combine the multiphase matching equation and the classical exhaustive algorithm such that a unique desired state can be found with success probability almost close to unity.Finally,we gave a numerical example.
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.
Abstract We systematically generalize the Grover algorithm in a density matrix formalism by exploiting the underlying two-dimensional subspace of the problem. Using this, we derive analytic expressions for the success probability after arbitrary iterations of the generalized Grover operator with two generic phase angles . We show for the phase matching condition with three iterations, success probability can be achieved only with knowledge about the lower bound , where λ is the ratio of marked to total number of states in the database. This result will improve the quantum search algorithm when applied to databases with unknown number of marked states in the specified regime of λ , at the cost of decreased efficiency in the smaller λ region.
We present a general phase matching condition for the quantum search algorithm with arbitrary unitary transformation and arbitrary phase rotations. We show by an explicit expression that the phase matching condition depends both on the unitary transformation U and the initial state. Assuming that the initial amplitude distribution is an arbitrary superposition sinθ_0 |1> + cosθ_0 e^{iδ} |2> with |1> = {1 / sinβ} \sum_k |τ_k> and |2> = {1 / cosβ} \sum_{i \ne τ}|i> , where |τ_k> is a marked state and \sinβ= \sqrt{\sum_k|U_{τ_k 0}|^2} is determined by the matrix elements of unitary transformation U between |τ_k> and the |0> state, then the general phase matching condition is tan{θ/ 2} [cos 2β+ tanθ_0 cosδsin 2β]= tan{ϕ/ 2} [1-tanθ_0 sinδsin 2βtan{θ/ 2}], where θand ϕare the phase rotation angles for |0> and |τ_k>, respectively. This generalizes previous conclusions in which the dependence of phase matching condition on UUU and the initial state has been disguised. We show that several phase conditions previously discussed in the literature are special cases of this general one, which clarifies the question of which condition should be regarded as exact.
We generalize Grover algorithm with two arbitrary phases in a density matrix set up. We give exact analytic expressions for the success probability after arbitrary number of iteration of the generalized Grover operator as a function of number of iterations, two phase angles (α, \{beta}) and parameter ξ introduced in the off diagonal terms of the density matrix in a sense to capture the coherence present in the initial quantum register. We extend Li and Li's idea and show for the phase matching condition α = -\{beta} = 0.35π with two iterations and ξ = 1, we can achieve success probability >= 0.8 only with a knowledge about the lower bound of λ = 0.166 where λ is the ratio of marked to total number states in the database. Finally we quantify success probability of the algorithm with decrease in coherence of the initial quantum state against modest noise in this simple model.
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.