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Journal of the Physical Society of Japan

Different Fixed-Phases for Quantum Search Operators

Sahar Q. Saleh, Ahmed Younes

Published Nov 12, 2019• 2 citations
Publisher

Abstract

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.

DOI
10.7566/jpsj.88.124002

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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.
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.
Grover's algorithm provides a quadratic speedup for searching unstructured databases and is traditionally implemented with qubits in Hilbert spaces whose dimensions are powers of two. With the advent of quantum platforms utilizing qudits---quantum systems with more than two levels---there is a need to generalize Grover search to these architectures, including heterogeneous systems with qudits of varying dimensions. Here, we present a unified framework for qudit-based Grover search, detailing the construction of oracles and diffusion operators with and without ancilla qubits and generalizing deterministic and fixed-point search variants that ensure exact or bounded success probabilities. We analyze phase-matching techniques and provide explicit circuit decompositions suitable for diverse hardware platforms. We also compare the corresponding trajectories on the Bloch sphere to provide an intuitive visualization of how the different phase choices amplify the target state. These results facilitate flexible, hardware-oriented protocols for implementing Grover search on qudit processors, potentially reducing circuit depth and enhancing success probabilities, thereby offering a practical toolkit for quantum computation and sensing applications leveraging multilevel quantum systems.
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.
When the Grover’s algorithm is applied to search an unordered database, the probability of success usually decreases with the increase of marked items. To address this phenomenon, a fixed-phase quantum search algorithm with more flexible behavior is proposed. In proposed algorithm, the phase shifts can be fixed at the different values to meet the needs of different practical problems. If research requires a relatively rapid speed, the value of the phase shifts should be appropriately increased, if search requires a higher success probability, the value of the phase shifts should be appropriately decreased. When the phase shifts are fixed at , the success probability of at least 99.38% can be obtained in iterations.
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 and its generalization, quantum amplitude amplification, provide a quadratic advantage over classical algorithms for a diverse set of tasks but are tricky to use without knowing beforehand what fraction λ of the initial state is comprised of the target states. In contrast, fixed-point search algorithms need only a reliable lower bound on this fraction but, as a consequence, lose the very quadratic advantage that makes Grover's algorithm so appealing. Here we provide the first version of amplitude amplification that achieves fixed-point behavior without sacrificing the quantum speedup. Our result incorporates an adjustable bound on the failure probability and, for a given number of oracle queries, guarantees that this bound is satisfied over the broadest possible range of λ.