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Journal of Physics A Mathematical and General

AnSO(3) picture for quantum searching

Gui Lu Long, Chang Cun Tu, Yan Li, Wei Lin Zhang, Hai Yan

Published Jan 18, 2001• 21 citations
Publisher

Abstract

An SO (3) picture of the generalized Grover quantum searching algorithm is constructed. In this picture, various aspects of quantum searching, the phase matching requirement in particular, are easily understood. It also offers a simple quantitative description of quantum searching. Exact formulae for the rotation angle and rotational axis are given. The probability of finding the marked state is just the projection of the state vector onto the z -axis in this picture. Applications of the picture to the standard Grover algorithm and to a generalized quantum search algorithm with arbitrary phase rotations are given.

DOI
10.1088/0305-4470/34/4/312

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An SO (3) picture of the generalized Grover quantum searching algorithm is constructed. In this picture, various aspects of quantum searching, the phase matching requirement in particular, are easily understood. It also offers a simple quantitative description of quantum searching. Exact formulae for the rotation angle and rotational axis are given. The probability of finding the marked state is just the projection of the state vector onto the z -axis in this picture. Applications of the picture to the standard Grover algorithm and to a generalized quantum search algorithm with arbitrary phase rotations are given.
An SO(3) picture of the generalized Grover's quantum searching algorithm,with arbitrary unitary transformation and with arbitrary phase rotations, is constructed. In this picture, any quantum search operation is a rotation in a 3 dimensional space. Exact formulas for the rotation angle and rotational axis are given. The probability of finding the marked state is just (z+1)/2(z+1)/2(z+1)/2, where z is the z-component of the state vector. Exact formulas for this probability is easily obtained. The phase matching requirement and the failure of algorithm when phase mismatches are clearly explained.
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.
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.
A general quantum search algorithm with arbitrary unitary transformations and an arbitrary initial state is considered in this work. To search a marked state with certainty, we have derived, using an SU(2) representation: (1) the matching condition relating the phase rotations in the algorithm, (2) a concise formula for evaluating the required number of iterations for the search, and (3) the final state after the search, with a complex phase in its amplitude. Moreover, the optimal choices and modifications of the phase angles in the Grover kernel are also studied.
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.