Quantum Search on Bounded-Error Inputs
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)).
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- 1.A. Ambainis. Quantum lower bounds by quantum arguments. In Proceedings of 32nd ACM STOC, pages 636–643, 2000.Google Scholar
- 2.H. Barnum and M. Saks. A lower bound on the quantum query complexity of read-once functions. quant-ph/0201007, 3 Jan 2002.Google Scholar
- 3.R. Beals, H. Buhrman, R. Cleve, M. Mosca, and R. de Wolf. Quantum lower bounds by polynomials. In Proceedings of 39th IEEE FOCS, pages 352–361, 1998.Google Scholar
- 6.G. Brassard and P. Høyer. An exact quantum polynomial-time algorithm for Simon’s problem. In Proceedings of Fifth Israeli Symposium on Theory of Computing and Systems (ISTCS’97), pages 12–23, 1997.Google Scholar
- 8.G. Brassard, P. Høyer, M. Mosca, and A. Tapp. Quantum amplitude amplification and estimation. In Lomonaco, S. J., Jr. and Brandt, H. E. (eds.): Quantum Computation and Quantum Information: A Millennium Volume. AMS Contemporary Mathematics Series, 305:53–74, 2002.Google Scholar
- 9.H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and computation. In Proceedings of 30th ACM STOC, pages 63–68, 1998.Google Scholar
- 10.L. K. Grover. A fast quantum mechanical algorithm for database search. In Proceedings of 28th ACM STOC, pages 212–219, 1996.Google Scholar
- 11.P. Høyer and R. de Wolf. Improved quantum communication complexity bounds for disjointness and equality. In Proceedings of 19th Annual Symposium on Theoretical Aspects of Computer Science (STACS’2002), Lecture Notes in Computer Science, Vol. 2285, pages 299–310. Springer-Verlag, 2002.Google Scholar
- 12.M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2000.Google Scholar
- 13.Y. Shi. Approximating linear restrictions of Boolean functions. Unpublished manuscript, 2002.Google Scholar