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arXiv:2207.09762

Exact results on Quantum search algorithm

Saptarshi Roy Chowdhury, Swarupananda Pradhan

Published Jul 20, 2022• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2207.09762

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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.
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.
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.
We investigate the phase matching problem in the fixed-point quantum search algorithm proposed by Grover [Phys. Rev. Lett.95 (2005) 150501]. We show that the optimal phase shift is π/3.61, which replaces the π/3 phase shift in fixed-point quantum search algorithm. The π/3.61-phase algorithm can be achieved in [Formula: see text] with the success probability of at least 94.11%, which offsets disadvantage that the success probability of getting correct results usually decreases with the increase of marked items when original Grover quantum search algorithm is applied to search an unordered database. In the meantime, this work also indicates that Grover quantum search algorithm is considerably robust to certain kinds of perturbations and is robust against modest noise in the initialization procedure.
Grover proposed how a quantum computer can find a single marked object in a database of size N by using only O(N) queries of the oracle that identifies the object with the probability of finding the marked object being not exactly 1.The Grover quantum search algorithm is generalized to deal with an arbitrary mixed initial state.The probability of finding the marked object is calculated and it is found that it depends strongly on the specific initial state.Our results contribute to analyze the influence of quantum noise on Grover algorithm based on pure state.Finally,we give few examples to show that for some extremely mixed initial states.
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.
The success probability of a search of MMM targets from a database of size NNN, using Grover's search algorithm depends critically on the number of iterations of the composite operation of the oracle followed by Grover's diffusion operation. Although the required number of iterations scales as O(N)\mathcal{O}(\sqrt{N})O(N​) for large NNN, the asymptote is not a good indicator of the optimal number of iterations when M/N\sqrt{M/N}M/N​ is not small. A scheme for the determination of the exact number of iterations, subject to a threshold set for the success probability of the search (probability of detecting the target state(s)), is crucial for the efficacy of the algorithm. In this work, a general scheme for the construction of nnn-qubit Grover's search algorithm with 1≤M≤N1 \leq M \leq N1≤M≤N target states is presented, along with the procedure to find the optimal number of iterations for a successful search. It is also shown that for given NNN and MMM, there is an upper-bound on the success probability of the algorithm.