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Europhysics Letters (EPL)

Phase matching in quantum search algorithm

Saptarshi Roy Chowdhury, Sudarshan Baruah, Biswaranjan Dikshit

Published Feb 8, 2023• 2 citations
Publisher

Abstract

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.

DOI
10.1209/0295-5075/acba41

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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 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.
As searching targets in an unordered database with Grover's algorithm,difference in marked items is not taken into consideration.If fraction of marked items is greater than 1/4,success probability of search decreases rapidly with increase of marked items.When the fraction of marked items is greater than 1/2,the algorithm is disabled.Aiming at above problems,an improved Grover's algorithm with weighted targets is proposed in which every target is assigned a weight coefficient according to its significance.With these weight coefficients,targets are represented as quantum superpositions which make probability getting target equal to its weight coefficient.An adaptive phase matching method is proposed based on weighted targets.The directions of phase rotations are contrary,and amplitudes of the two phase rotations are determined by inner-product of target quantum superposition and initial state of the system.As the inner-product is greater than(((3-5)/8)~(1/2),) success probability is equal to 1 with two steps of Grover iteration at most.The improved quantum searching algorithm and the new phase matching are verified by an example.
When the Grover's algorithm is applied to search an unordered database, the probability of getting correct results usually decreases with the increase of marked items. The reason for this phenomenon is analyzed in this Letter, the Grover iteration is studied, and a new phase matching is proposed. With application of the new phase matching, when the fraction of marked items is greater than 1/3, the probability of getting correct results is greater than 25/27 with only one Grover iteration. The validity of the new phase matching is verified by a search example.
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.