Adversary lower bounds for the collision and the set equality problems

We prove tight $\Omega(n^{1/3})$ lower bounds on the quantum query complexity of the Collision and the Set Equality problems, provided that the size of the alphabet is large enough. We do this using the negative-weight adversary method. Thus, we reprove the result by Aaronson and Shi, as well as a more recent development by Zhandry.

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

[2]  Mark Zhandry,et al.  A note on the quantum collision and set equality problems , 2013, Quantum Inf. Comput..

[3]  Andris Ambainis,et al.  Symmetry-Assisted Adversaries for Quantum State Generation , 2011, 2011 IEEE 26th Annual Conference on Computational Complexity.

[4]  Aleksandrs Belovs,et al.  Span programs for functions with constant-sized 1-certificates: extended abstract , 2011, STOC '12.

[5]  Gilles Brassard,et al.  Quantum cryptanalysis of hash and claw-free functions , 1997, SIGA.

[6]  Mario Szegedy,et al.  All Quantum Adversary Methods Are Equivalent , 2005, ICALP.

[7]  G. James,et al.  The Representation Theory of the Symmetric Group , 2009 .

[8]  Andris Ambainis,et al.  Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range , 2003, Theory Comput..

[9]  Bruce E. Sagan,et al.  The symmetric group - representations, combinatorial algorithms, and symmetric functions , 2001, Wadsworth & Brooks / Cole mathematics series.

[10]  Samuel Kutin,et al.  Quantum Lower Bound for the Collision Problem with Small Range , 2005, Theory Comput..

[11]  Robert Spalek,et al.  Adversary Lower Bound for the Orthogonal Array Problem , 2013, ArXiv.

[12]  Troy Lee,et al.  A strong direct product theorem for quantum query complexity , 2012, 2012 IEEE 27th Conference on Computational Complexity.

[13]  Randall R. Holmes.  Linear Representations of Finite Groups , 2008 .

[14]  Jérémie Roland,et al.  Explicit relation between all lower bound techniques for quantum query complexity , 2012, STACS.

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

[16]  Aleksandrs Belovs,et al.  Learning-Graph-Based Quantum Algorithm for k-Distinctness , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[17]  Scott Aaronson,et al.  Quantum lower bounds for the collision and the element distinctness problems , 2004, JACM.

[18]  Troy Lee,et al.  Quantum Query Complexity of State Conversion , 2010, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[19]  C. Curtis,et al.  Representation theory of finite groups and associated algebras , 1962 .

[20]  Gilles Brassard,et al.  Quantum Algorithm for the Collision Problem , 2016, Encyclopedia of Algorithms.

[21]  Aleksandrs Belovs,et al.  Adversary lower bound for the k-sum problem , 2012, ITCS '13.

[22]  W. Haemers,et al.  Association schemes , 1996 .

[23]  Gatis Midrijanis.  A Polynomial Quantum Query Lower Bound for the Set Equality Problem , 2004, ICALP.

[24]  Ansis Rosmanis,et al.  On the Power of Non-adaptive Learning Graphs , 2012, 2013 IEEE Conference on Computational Complexity.

[25]  Ronald de Wolf,et al.  Quantum lower bounds by polynomials , 2001, JACM.

[26]  Shengyu Zhang,et al.  On the power of Ambainis lower bounds , 2005, Theor. Comput. Sci..

[27]  Troy Lee,et al.  Negative weights make adversaries stronger , 2007, STOC '07.

[28]  Ben Reichardt,et al.  Reflections for quantum query algorithms , 2010, SODA '11.

[29]  Andris Ambainis,et al.  The Need for Structure in Quantum Speedups , 2009, Theory Comput..

[30]  Robert Spalek.  The Multiplicative Quantum Adversary , 2008, 2008 23rd Annual IEEE Conference on Computational Complexity.

[31]  Andris Ambainis,et al.  Polynomial degree vs. quantum query complexity , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[32]  Aleksandrs Belovs.  Adversary Lower Bound for Element Distinctness , 2012, ArXiv.

[33]  Shengyu Zhang.  On the Power of Ambainis's Lower Bounds , 2004, ICALP.

[34]  Mark Zhandry,et al.  How to Construct Quantum Random Functions , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[35]  Ben Reichardt,et al.  Span-program-based quantum algorithm for evaluating formulas , 2007, Theory Comput..