Unbounded-Error Quantum Query Complexity

This work studies the quantum query complexity of Boolean functions in an unbounded-error scenario where it is only required that the query algorithm succeeds with a probability strictly greater than 1/2. We show that, just as in the communication complexity model, the unbounded-error quantum query complexity is exactly half of its classical counterpart for any (partial or total) Boolean function. Moreover, connecting the query and communication complexity results, we show that the ''black-box'' approach to convert quantum query algorithms into communication protocols by Buhrman-Cleve-Wigderson [STOC'98] is optimal even in the unbounded-error setting. We also study a related setting, called the weakly unbounded-error setting, where the cost of a query algorithm is given by q+log(1/2(p-1/2)), where q is the number of queries made and p>1/2 is the success probability of the algorithm. In contrast to the case of communication complexity, we show a tight multiplicative @Q(logn) separation between quantum and classical query complexity in this setting for a partial Boolean function. The asymptotic equivalence between them is also shown for some well-studied total Boolean functions.

[1]  James Aspnes,et al.  The expressive power of voting polynomials , 1994, Comb..

[2]  Jürgen Forster.  A linear lower bound on the unbounded error probabilistic communication complexity , 2002, J. Comput. Syst. Sci..

[3]  A. Razborov.  Quantum communication complexity of symmetric predicates , 2002, quant-ph/0204025.

[4]  Wim van Dam,et al.  Quantum Oracle Interrogation: Getting All Information for Almost Half the Price , 1999 .

[5]  Hartmut Klauck.  Lower Bounds for Quantum Communication Complexity , 2007, SIAM J. Comput..

[6]  Raymond H. Putra,et al.  Unbounded-Error One-Way Classical and Quantum Communication Complexity , 2007, ICALP.

[7]  Yaoyun Shi.  Lower bounds of quantum black-box complexity and degree of approximating polynomials by influence of Boolean variables , 2000, Inf. Process. Lett..

[8]  D. Richard Kuhn,et al.  Quantum computing and communication , 2002, Adv. Comput..

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

[10]  Janos Simon,et al.  Probabilistic Communication Complexity , 1986, J. Comput. Syst. Sci..

[11]  M. Sipser,et al.  Limit on the Speed of Quantum Computation in Determining Parity , 1998, quant-ph/9802045.

[12]  Richard Beigel,et al.  Perceptrons, PP, and the polynomial hierarchy , 1992, [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference.

[13]  Peter Frankl,et al.  Complexity classes in communication complexity theory , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[14]  N. Alon,et al.  Wiley‐Interscience Series in Discrete Mathematics and Optimization , 2004 .

[15]  Raymond H. Putra,et al.  Unbounded-Error Classical and Quantum Communication Complexity , 2007, ISAAC.

[16]  Noga Alon,et al.  The Probabilistic Method, Second Edition , 2004 .

[17]  Ashwin Nayak,et al.  Limits on the ability of quantum states to convey classical messages , 2006, JACM.

[18]  Scott Aaronson,et al.  BQP and the polynomial hierarchy , 2009, STOC '10.

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

[20]  Ronald de Wolf,et al.  Improved Quantum Communication Complexity Bounds for Disjointness and Equality , 2001, STACS.

[21]  Robert Spalek,et al.  Lower Bounds on Quantum Query Complexity , 2005, Bull. EATCS.

[22]  Shengyu Zhang.  On the Tightness of the Buhrman-Cleve-Wigderson Simulation , 2009, ISAAC.

[23]  Alexander A. Sherstov.  Halfspace Matrices , 2007, Computational Complexity Conference.

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

[25]  Kathie Cameron,et al.  Monotone path systems in simple regions , 1994, Comb..

[26]  Karl Zeller,et al.  Schwankung von Polynomen zwischen Gitterpunkten , 1964 .

[27]  Elliott Ward Cheney,et al.  A Comparison of Uniform Approximations on an Interval and a Finite Subset Thereof , 1966 .

[28]  François Le Gall,et al.  Quantum weakly nondeterministic communication complexity , 2005, Theor. Comput. Sci..

[29]  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).

[30]  Andris Ambainis,et al.  Quantum search of spatial regions , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[31]  Ryan O'Donnell,et al.  Extremal properties of polynomial threshold functions , 2003, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..

[32]  Harry Buhrman,et al.  Quantum Computing and Communication Complexity , 2001, Bull. EATCS.

[33]  H. Buhrman,et al.  Complexity measures and decision tree complexity: a survey , 2002, Theor. Comput. Sci..

[34]  Noam Nisan,et al.  On the degree of boolean functions as real polynomials , 1992, STOC '92.

[35]  Nikolai K. Vereshchagin,et al.  On Computation and Communication with Small Bias , 2007, Computational Complexity Conference.

[36]  Ronald de Wolf,et al.  Nondeterministic Quantum Query and Communication Complexities , 2003, SIAM J. Comput..

[37]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[38]  Pr.  The Goldreich-levin Theorem , .

[39]  Rüdiger Reischuk,et al.  Relations between Communication Complexity Classes , 1990, J. Comput. Syst. Sci..

[40]  Dave Bacon,et al.  Classical simulation of quantum entanglement without local hidden variables , 2001 .

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