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Journal of Physics A Mathematical and General

Gate imperfection in the quantum random-walk search algorithm

Yun Li, Lei Ma, Jie Zhou

Published Jul 5, 2006• 7 citations
Publisher

Abstract

We study how imperfect quantum gates affect the quantum random-walk search algorithm. We find that systematic errors in phase inversions result in the reduction of the maximum probability of the marked state and lower the algorithm efficiency with an increasing degree of inaccuracy. The size of the database should be limited due to the inevitable errors. Finally, we compare the phase noise caused by such errors in the random-walk search algorithm with that in the Grover search algorithm.

DOI
10.1088/0305-4470/39/29/021

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We study how imperfect quantum gates affect the quantum random-walk search algorithm. We find that systematic errors in phase inversions result in the reduction of the maximum probability of the marked state and lower the algorithm efficiency with an increasing degree of inaccuracy. The size of the database should be limited due to the inevitable errors. Finally, we compare the phase noise caused by such errors in the random-walk search algorithm with that in the Grover search algorithm.
It is found that systematic errors in phase inversions and random errors in Hadmard-Walsh transformations are the dominant gate imperfection in Grover's quantum search algorithm. They lead to reductions in the maximum probability of the marked state and affect the efficiency of the algorithm. Given the degree of inaccuracy, we find that to guarantee a half-rate of success, the size of the database should be on the order of $O(1/{\ensuremath{\delta}}^{2}),$ where $\ensuremath{\delta}$ is the uncertainty.
This study investigates the effects of systematic errors in phase inversions on the success rate and number of iterations in the optimized quantum random-walk search algorithm. Using the geometric description of this algorithm, a model of the algorithm with phase errors is established, and the relationship between the success rate of the algorithm, the database size, the number of iterations, and the phase error is determined. For a given database size, we obtain both the maximum success rate of the algorithm and the required number of iterations when phase errors are present in the algorithm. Analyses and numerical simulations show that the optimized quantum random-walk search algorithm is more robust against phase errors than Grover’s algorithm.
It is found that Grover's quantum search algorithm is not robust against phase inversion and Hadmard transformation inaccuracies. Imperfect phase inversions and Hadmard-Walsh transformations in Grover's quantum search algorithm lead to reductions in the maximum probability of the marked state and affect the efficiency of the algorithm. even in the absence of decoherence. Given the degrees of inaccuracies, we find that to guarantee half rate of success, the size of the database should be in the order of O(1δ2)O({1 \over δ^2})O(δ21​), where δδδ is the uncertainty.
As the matching condition in Grover search algorithm is transgressed due to inevitable errors in phase inversions, it gives a reduction in maximum probability of success. With a given degree of maximum success, we have derive the generalized and imroved criterion for tolerated error and corresponding size of quantum database under the inevitable gate imperfections. The vanished inaccurancy to this condition has also been shown. Besides, a concise formula for evaluating minimum number of iterations is also presented in this work.
The quantum random walk is a possible approach to construct new quantum search algorithms. It has been shown by Shenvi et al. [Phys. Rev. A 67 (2003) 52307 that a kind of algorithm can perform an oracle search on a database of N items with O ( N 1/2 ) calling to the oracle, yielding a speedup similar to other quantum search algorithms. We study the effect of white or Gaussian noise on this algorithm. The algorithm loses efficiency when noise is added. We also show that noise on the target state plays a more important role than that on other states. Finally we compare the effects of similar types of noise in the quantum random walk search algorithm and Grover's search algorithm.
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.