Quantum property testing

A language <i>L</i> has a property tester if there exists a probabilistic algorithm that given an input <i>x</i> only asks a small number of bits of <i>x</i> and distinguishes the cases as to whether <i>x</i> is in <i>L</i> and <i>x</i> has large Hamming distance from all <i>y</i> in <i>L</i>. We define a similar notion of quantum property testing and show that there exist languages with quantum property testers but no good classical testers. We also show there exist languages which require a large number of queries even for quantumly testing.

[1]  N. S. Barnett,et al.  Private communication , 1969 .

[2]  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).

[3]  Noga Alon,et al.  A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem , 1985, J. Algorithms.

[4]  R. Rubinfeld.  A mathematical theory of self-checking, self-testing and self-correcting programs , 1991 .

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

[6]  Umesh V. Vazirani,et al.  Quantum complexity theory , 1993, STOC.

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

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

[9]  Ronitt Rubinfeld,et al.  Robust Characterizations of Polynomials with Applications to Program Testing , 1996, SIAM J. Comput..

[10]  Dana Ron,et al.  Property Testing in Bounded Degree Graphs , 2002, STOC '97.

[11]  Oded Goldreich,et al.  Combinatorial property testing (a survey) , 1997, Randomization Methods in Algorithm Design.

[12]  Gilles Brassard,et al.  Strengths and Weaknesses of Quantum Computing , 1997, SIAM J. Comput..

[13]  Gilles Brassard,et al.  An exact quantum polynomial-time algorithm for Simon's problem , 1997, Proceedings of the Fifth Israeli Symposium on Theory of Computing and Systems.

[14]  Ronitt Rubinfeld,et al.  Spot-checkers , 1998, STOC '98.

[15]  Andrew Chi-Chih Yao,et al.  Quantum cryptography with imperfect apparatus , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[16]  Dana Ron,et al.  Property testing and its connection to learning and approximation , 1998, JACM.

[17]  Noga Alon,et al.  Efficient Testing of Large Graphs , 2000, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

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

[19]  Noga Alon,et al.  Regular languages are testable with a constant number of queries , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[20]  Sean Hallgren,et al.  An improved quantum Fourier transform algorithm and applications , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[21]  Eldar Fischer,et al.  Large K r -free subgraphs in K s -free graphs and some other Ramsey-type problems , 2005 .

[22]  Eldar Fischer,et al.  Testing of matrix properties , 2001, STOC '01.

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

[24]  Noga Alon,et al.  Testing subgraphs in large graphs , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[25]  Luca Trevisan,et al.  Three Theorems regarding Testing Graph Properties , 2001, Electron. Colloquium Comput. Complex..

[26]  Eldar Fischer,et al.  Testing graphs for colorable properties , 2001, SODA '01.

[27]  D. Ron,et al.  Testing the diameter of graphs , 2002 .

[28]  Frédéric Magniez,et al.  Quantum Testers for Hidden Group Properties , 2003, MFCS.

[29]  N. Alon,et al.  Testing of Clustering , 2003, SIAM J. Discret. Math..

[30]  E. Fischer.  THE ART OF UNINFORMED DECISIONS: A PRIMER TO PROPERTY TESTING , 2004 .

[31]  Guy Kindler,et al.  Testing juntas , 2002, J. Comput. Syst. Sci..

[32]  Noga Alon,et al.  Testing subgraphs in directed graphs , 2004, J. Comput. Syst. Sci..