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Experimental NMR Realization of A Generalized Quantum Search Algorithm

G. L. Long, H. Y. Yan, Y. S. Li, C. C. Tu, J. X. Tao, H. M. Chen, M. L. Liu, Xue Zhang, J. L. Luo, Xiao Li, X. Z. Zeng

Published Jan 1, 2008• 37 citations

Abstract

A generalized quantum search algorithm, where phase inversions for the marked state and the prepared state are replaced by π/2 phase rotations, is realized in a 2-qubit NMR heteronuclear system. The quantum algorithm searches a marked state with a smaller step compared to standard Grover algorithm. Phase matching requirement in quantum searching is demonstrated by comparing it with another generalized algorithm where the two phase rotations are π/2 and 3π/2 respectively. Pulse sequences which include non 90 1 degree pulses are given. Typeset using REVTEX 2 Grover’s quantum search algorithm is one of the most important development in quantum computation [1]. It achieves quadratic speedup in searching a marked state in an unordered list over classical searching algorithms. It has many potential applications in various fields of interests, for instance in deciphering the DES encryption code [2] and algorithms that need searching. Typical examples of the algorithms that need searching are the Simon

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A generalized quantum search algorithm, where phase inversions for the marked state and the prepared state are replaced by π/2 phase rotations, is realized in a 2-qubit NMR heteronuclear system. The quantum algorithm searches a marked state with a smaller step compared to standard Grover algorithm. Phase matching requirement in quantum searching is demonstrated by comparing it with another generalized algorithm where the two phase rotations are π/2 and 3π/2 respectively. Pulse sequences which include non 90 1 degree pulses are given. Typeset using REVTEX 2 Grover’s quantum search algorithm is one of the most important development in quantum computation [1]. It achieves quadratic speedup in searching a marked state in an unordered list over classical searching algorithms. It has many potential applications in various fields of interests, for instance in deciphering the DES encryption code [2] and algorithms that need searching. Typical examples of the algorithms that need searching are the Simon

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.

The improved quantum scheduling algorithm proposed by Grover has been generalized using the generalized quantum search algorithm, in which a unitary operator replaces the Walsh-Hadamard transform, and

π/2

phase rotations replace the selective inversions, in order to make the quantum scheduling algorithm suitable for more cases. Our scheme is realized on a nuclear magnetic resonance (NMR) quantum computer. Experimental results show a good agreement between theory and experiment.

We briefly introduced some of our recent work related to the phase matching condition in quantum searching algorithms and the improved Grover algorithm. When one replaces the two phase inversions in the Grover algorithm with arbitrary phase rotations, the modified algorithm usually fails in searching the marked state unless a phase matching condition is satisfied between the two phases. The Grover algorithm is not 100% in success rate, an improved Grover algorithm with zero-failure rate is given by replacing the phase inversions with angles that depends on the size of the database. Other aspects of the Grover algorithm such as the SO(3) picture of quantum searching, the dominant gate imperfections in the Grover algorithm are also mentioned.

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.