Since Grover's seminal work, quantum search has been studied in great detail. In the usual search problem, we have a collection of $n$ items $x_1, ldots, x_n$ and we would like to find $i: x_i=1$. We consider a new variant of this problem in which evaluating $x_i$ for different $i$ may take a different number of time steps. Let $t_i$ be the number of time steps required to evaluate $x_i$. If the numbers $t_i$ are known in advance, we give an algorithm that solves the problem in $O(sqrt{t_1^2+t_2^2+ldots+t_n^2)$ steps. This is optimal, as we also show a matching lower bound. The case, when $t_i$ are not known in advance, can be solved with a polylogarithmic overhead. We also give an application of our new search algorithm to computing read-once functions.
@InProceedings{ambainis:LIPIcs.STACS.2008.1333, author = {Ambainis, Andris}, title = {{Quantum search with variable times}}, booktitle = {25th International Symposium on Theoretical Aspects of Computer Science}, pages = {49--60}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-06-4}, ISSN = {1868-8969}, year = {2008}, volume = {1}, editor = {Albers, Susanne and Weil, Pascal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1333}, URN = {urn:nbn:de:0030-drops-13333}, doi = {10.4230/LIPIcs.STACS.2008.1333}, annote = {Keywords: } }
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