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Abstract: We compare classical and quantum query complexities of total Boolean functions. It is known that for worst-case complexity, the gap between quantum and classical can be at most polynomial [BBC + 98]. We show that for average-case complexity under the uniform distribution, quantum algorithms can be exponentially faster than classical algorithms. Under non-uniform distributions the gap can even be super-exponential. We also prove some general bounds for average-case complexity and show that the ... (Update)
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...1 2 log log n 1 2 . 2 There has also been some work on average (block) sensitivity [Ber96] and its applications [Bop97, Shi99, AW00] 6 Wegener [Weg85] proved that this theorem is tight up to the O(log log n) term by means of the monotone address function. We now prove...
.... It may well be true that bs(f) 2 O(s(f) 2 ) 3 There has also been some work on average (block) sensitivity [5] and its applications [7,40,2]. In particular, Shi [40] has shown that the average sensitivity of a total function f is a lower bound on its approximate degree g deg(f...
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BibTeX entry: (Update)
A. Ambainis and R. de Wolf. Average-case quantum query complexity. In Proceedings of 17th Annual Symposium on Theoretical Aspects of Computer Science (STACS'2000), Lecture Notes in Computer Science. Springer, 2000. To appear. Also quant-ph/9904079. http://citeseer.ist.psu.edu/article/ambainis00averagecase.html More
@article{ ambainis00averagecase,
author = "Andris Ambainis and Ronald de Wolf",
title = "Average-Case Quantum Query Complexity",
journal = "Lecture Notes in Computer Science",
volume = "1770",
pages = "133--??",
year = "2000",
url = "citeseer.ist.psu.edu/article/ambainis00averagecase.html" }
Citations (may not include all citations):957 The Probabilistic Method (context) - Alon, Spencer - 1992 ACM DBLP
210 A fast quantum mechanical algorithm for database search - Grover - 1996 ACM DBLP
164 Polynomial-time algorithms for prime factorization and discr.. - Shor - 1997 ACM DBLP
144 the power of quantum computation - Simon - 1997
126 Rapid solution of problems by quantum computation (context) - Deutsch, Jozsa - 1992 ACM
112 Average case complete problems (context) - Levin - 1986 ACM DBLP
102 Strengths and weaknesses of quantum computing - Bennett, Bernstein et al. - 1997 ACM DBLP
73 Tight bounds on quantum searching - Boyer, Brassard et al. - 1998 ACM
70 Average-case analysis of algorithms and data structures - Vitter, Flajolet - 1990 ACM DBLP
30 Quantum counting - Brassard, Hoyer et al. - 1998 ACM DBLP
15 Quantum algorithm for the collision problem - Brassard, Hoyer et al. - 1997
15 The quantum query complexity of approximating the median and.. - Nayak, Wu - 1999 ACM DBLP
11 Earlier version in STOC (context) - Nisan, decision et al. - 1991
11 Quantum amplitude amplification and estimation - Brassard, Hoyer et al.
10 counting and amplitude amplification by eigenvector analysis (context) - Mosca - 1998
7 The average-case complexity of determining the majority (context) - Alonso, Reingold et al. - 1997 ACM DBLP
3 the normal approximation to the hypergeometric distribution (context) - Nicholson - 1956
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