Abstract
Suppose we have n algorithms, quantum or classical, each computing some bit-value with bounded error probability. We describe a quantum algorithm that uses O(\( \sqrt n \)) repetitions of the base algorithms and with high probability finds the index of a 1-bit among these n bits (if there is such an index). This shows that it is not necessary to first significantly reduce the error probability in the base algorithms to O(1/poly(n)) (which would require O(\( \sqrt n \log n \) log n) repetitions in total). Our technique is a recursive interleaving of amplitude amplification and error-reduction, and may be of more general interest. Essentially, it shows that quantum amplitude amplification can be made to work also with a bounded-error verifier. As a corollary we obtain optimal quantum upper bounds of O(\( \sqrt N \)) queries for all constant-depth AND-OR trees on N variables, improving upon earlier upper bounds of O(\( \sqrt N \) polylog(N)).
Supported in part by the Alberta Ingenuity Fund and the Pacific Institute for the Mathematical Sciences.
This research was (partially) funded by projects QAIP (IST-1999-11234) and RESQ (IST-2001-37559) of the IST-FET programme of the EC.
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Høyer, P., Mosca, M., de Wolf, R. (2003). Quantum Search on Bounded-Error Inputs. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds) Automata, Languages and Programming. ICALP 2003. Lecture Notes in Computer Science, vol 2719. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45061-0_25
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DOI: https://doi.org/10.1007/3-540-45061-0_25
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