Abstract
Grover’s algorithm is a quantum search algorithm solving the unstructured search problem of size n in \(O(\sqrt{n})\) queries, while any classical algorithm needs O(n) queries [3].
However, if query has some small probability of failing (reporting that none of the elements are marked), then quantum speed-up disappears: no quantum algorithm can be faster than a classical exhaustive search by more than a constant factor [8].
We study the behaviour of Grover’s algorithm in the model there query may report some marked elements as unmarked (each marked element has its own error probability, independent of other marked elements).
We analyse the limiting behaviour of Grover’s algorithm for a large number of steps and prove the existence of limiting state ρ lim . Interestingly, the limiting state is independent of error probabilities of individual marked elements. If we measure ρ lim , the probability of getting one of the marked states i 1, …, i k is \(\frac{k}{k+1}\). We show that convergence time is O(n).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Ambainis, A.: Quantum search algorithms. SIGACT News 35(2), 22–35 (2004)
Buhrman, H., Newman, I., Röhrig, H., de Wolf, R.: Robust Polynomials and Quantum Algorithms. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 593–604. Springer, Heidelberg (2005)
Grover, L.: A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th ACM STOC, Philadelphia, Pennsylvania, pp. 212–219. ACM Press (1996)
Horn, R., Johnson, C.: Matrix Analysis. Cambridge University Press (2006)
Long, G.L., Li, Y.S., Zhang, W.L., Tu, C.C.: An intrinsic limitation on the size of quantum database. Physical Review A 61, 042305 (2000); Also arXiv:quant-ph/9910076
Kaye, P., Laflamme, R., Mosca, M.: An Introduction to Quantum Computing. Cambridge University Press (2007)
Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press (2000)
Regev, O., Schiff, L.: Impossibility of a Quantum Speed-Up with a Faulty Oracle. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 773–781. Springer, Heidelberg (2008)
Shapira, D., Mozes, S., Biham, O.: The effect of unitary noise on Grover’s quantum search algorithm. Physical Review A 67, 042301 (2003); Also arXiv:quant-ph/0307142
Shenvi, N., Brown, K.R., Whaley, K.B.: Effects of Noisy Oracle on Search Algorithm Complexity. Physical Review A 68, 052313 (2003); Also quant-ph/0304138
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ambainis, A., Bačkurs, A., Nahimovs, N., Rivosh, A. (2013). Grover’s Algorithm with Errors. In: Kučera, A., Henzinger, T.A., Nešetřil, J., Vojnar, T., Antoš, D. (eds) Mathematical and Engineering Methods in Computer Science. MEMICS 2012. Lecture Notes in Computer Science, vol 7721. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36046-6_17
Download citation
DOI: https://doi.org/10.1007/978-3-642-36046-6_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-36044-2
Online ISBN: 978-3-642-36046-6
eBook Packages: Computer ScienceComputer Science (R0)