Abstract
Grover’s algorithm is a quantum query algorithm solving the unstructured search problem of size N using \(O(\sqrt{N})\) queries. It provides a significant speed-up over any classical algorithm [2].
The running time of the algorithm, however, is very sensitive to errors in queries. Multiple authors have analysed the algorithm using different models of query errors and showed the loss of quantum speed-up [1, 4].
We study the behavior of Grover’s algorithm in the model where the search space contains both faulty and non-faulty marked elements. We show that in this setting it is indeed possible to find one of marked elements in \(O(\sqrt{N})\) queries.
This research was supported by EU FP7 project QALGO (Dmitry Kravchenko, Nikolajs Nahimovs) and ERC project MQC (Alexander Rivosh).
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Notes
- 1.
The limitation of at most one non-faulty marked element comes from the probability independence assumption – two or more marked elements with zero error probability of failure would not be independent.
- 2.
It is usually considered that \(k \ll N\), as for \(\frac{k}{N}\ge \lambda \) with sufficiently large \(\lambda \) the search problem can be trivially solved by a probabilistic algorithm in time \(O\left( \lambda ^{-1}\right) \).
- 3.
Orthodrome, also known as a great circle, of a sphere is the intersection of the sphere with a plane which passes through the center point of the sphere.
- 4.
\(A \le 0.25\pi \) means that there is at least as many non-faulty marked items as faulty marked items. Since we limit our considerations with only one faulty marked item, it suffices with only one non-faulty marked item.
- 5.
For one faulty marked item, it means existence of at least \(\lceil {{\mathrm{arccot}}}{0.1953\ldots \pi } \rceil = \lceil 1.02047\rceil =2\) non-faulty marked items.
References
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Kravchenko, D., Nahimovs, N., Rivosh, A. (2016). Grover’s Search with Faults on Some Marked Elements. In: Freivalds, R., Engels, G., Catania, B. (eds) SOFSEM 2016: Theory and Practice of Computer Science. SOFSEM 2016. Lecture Notes in Computer Science(), vol 9587. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49192-8_28
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DOI: https://doi.org/10.1007/978-3-662-49192-8_28
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