On Quantum Versions of the Yao Principle

The classical Yao principle states that the complexity R?(f) of an optimal randomized algorithm for a function f with success probability 1 - ? equals the complexity maxµ D?µ (f) of an optimal deterministic algorithm for f that is correct on a fraction 1 - ? of the inputs, weighed according to the hardest distribution µ over the inputs. In this paper we investigate to what extent such a principle holds for quantum algorithms. We propose two natural candidate quantum Yao principles, a "weak" and a "strong" one. For both principles, we prove that the quantum bounded-error complexityis a lower bound on the quantum analogues of maxµ D?µ (f). We then prove that equality cannot be obtained for the "strong" version, by exhibiting an exponential gap. On the other hand, as a positive result we prove that the "weak" version holds up to a constant factor for the query complexity of all symmetric Boolean functions.

[1]  Alain Tapp,et al.  Quantum Entanglement and the Communication Complexity of the Inner Product Function , 1998, QCQC.

[2]  Richard Cleve,et al.  The query complexity of order-finding , 1999, Proceedings 15th Annual IEEE Conference on Computational Complexity.

[3]  Daniel R. Simon,et al.  On the power of quantum computation , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[4]  Rajeev Motwani,et al.  Randomized Algorithms , 1995, SIGA.

[5]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[6]  Andris Ambainis,et al.  Quantum lower bounds by quantum arguments , 2000, STOC '00.

[7]  Avi Wigderson,et al.  Quantum vs. classical communication and computation , 1998, STOC '98.

[8]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[9]  Andris Ambainis,et al.  A Note on Quantum Black-Box Complexity of Almost all Boolean Functions , 1998, Inf. Process. Lett..

[10]  Ronald de Wolf,et al.  Bounds for small-error and zero-error quantum algorithms , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[11]  W. L. Nicholson,et al.  On the Normal Approximation to the Hypergeometric Distribution , 1956 .

[12]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

[13]  Ronald de Wolf,et al.  Quantum lower bounds by polynomials , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[14]  Umesh V. Vazirani,et al.  Quantum Complexity Theory , 1997, SIAM J. Comput..

[15]  G. Brassard,et al.  Quantum Amplitude Amplification and Estimation , 2000, quant-ph/0005055.

[16]  Andrew Chi-Chih Yao,et al.  Probabilistic computations: Toward a unified measure of complexity , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[17]  Ramamohan Paturi,et al.  On the degree of polynomials that approximate symmetric Boolean functions (preliminary version) , 1992, STOC '92.

[18]  David A. Meyer,et al.  Quantum Games and Quantum Algorithms Quantum Games & Quantum Algorithms , 2000 .