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Applied Mathematics & Information Sciences

Towards More Reliable Fixed Phase Quantum Search Algorithm,

Ahmed Younes

Published Jan 1, 2013• 19 citations
Publisher

Abstract

Building quantum devices using fixed operators is a must to simplify hardware construction of a quantum computer. Quan- tum search engine is not an exception. In this paper, a fixed phase quantum search algorithm that searches for M matches in an unstructured list of size N will be proposed. Fixing phase shifts to 1.91684π in the standard amplitude amplification will make the minimum probability of success is 99.58% in O N/M for 0 <M ≤ N better than any know fixed operator quantum search algorithm. The algorithm will be able to handle either a single match or multiple matches in the search space. The algorithm will find a match in O N/M whether the number of matches is known or not in advance.

DOI
10.12785/amis/070110

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Building quantum devices using fixed operators is a must to simplify hardware construction of a quantum computer. Quan- tum search engine is not an exception. In this paper, a fixed phase quantum search algorithm that searches for M matches in an unstructured list of size N will be proposed. Fixing phase shifts to 1.91684π in the standard amplitude amplification will make the minimum probability of success is 99.58% in O N/M for 0 <M ≤ N better than any know fixed operator quantum search algorithm. The algorithm will be able to handle either a single match or multiple matches in the search space. The algorithm will find a match in O N/M whether the number of matches is known or not in advance.
Building quantum devices using fixed operators is a must to simplify the hardware construction. Quantum search engine is not an exception. In this paper, a fixed phase quantum search algorithm that searches for M matches in an unstructured search space of size N will be presented. Selecting phase shifts of 1.91684πin the standard amplitude amplification will make the technique perform better so as to get probability of success at least 99.58% in O(sqrt(N/M)) better than any know fixed operator quantum search algorithms. The algorithm will be able to handle either a single match or multiple matches in the search space. The algorithm will find a match in O(sqrt(N/M)) whether the number of matches is known or not in advance.
Amplitude amplification techniques are the basic building blocks in many quantum algorithms. Amplifying the probability of success to the highest possible probability to find solutions in the search space is an essential aim. To simplify the construction of the hardware, the search operators with fixed phases must be used to handle the whole search space. This paper presents more reliable fixed-phase quantum search algorithm, where the phase shifts that used in the oracle and the diffusion operator are not matched, in contrary to the wide belief in literature that the matching condition between the phase shifts is the best choice. This paper shows that there are many phase shifts which can be used for the search operators to give better results when the number of solutions is known in advance. The minimum probability of success of the proposed algorithm is 99.61% in quadratic speed-up which is better than any other known fixed operator quantum search algorithms.
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%.
Grover's quantum search algorithm provides a fundamental quadratic speedup for unstructured datasets, reducing query complexity from O(N)\mathcal{O}(N)O(N) to O(N)\mathcal{O}(\sqrt{N})O(N​). However, the algorithm's reliance on precise iteration counts leads to the ``soufflé problem,'' where over-rotation results in a sharp decline in success probability. This limitation is particularly restrictive when the number of solution states, MMM, is unknown. In this work, we present an ancilla-mediated fixed-point quantum search algorithm that achieves robust convergence by mapping the solution amplitude to a dedicated ancilla qubit. Unlike existing phase-matching fixed-point methods, our approach utilizes Grover's real-plane reflections, thereby maintaining the intuitive geometric architecture of the original algorithm. We demonstrate that this method achieves a success probability of at least 92.6%92.6\%92.6% with a query complexity of approximately O(N/M)\mathcal{O}(\sqrt{N/M})O(N/M​), effectively bridging the gap between standard amplitude amplification and robust fixed-point convergence.
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.
This paper presents the Single-Rotation Quantum Search Algorithm, which achieves amplitude-amplification with exactly one oracle call and one rotation operation. We prove that for unstructured search over N items, a single rotation by π/4 radians is both necessary and sufficient to amplify the marked state probability from 1/N to 1/2. This represents the theoretical minimum: no algorithm can succeed with fewer than one oracle call, and no additional rotations are needed beyond the single π/4 rotation. We provide explicit closed forms for the rotation operator with numerical values, demonstrating complete physical realizability. The algorithm is implemented by one oracle call, one diffusion call, and log(N) reflections. For N = 2n search spaces, the algorithm achieves success probability exactly 1/2 with one iteration, compared to O( N ) iterations for Grover’s algorithm.