The quantum query complexity of the abelian hidden subgroup problem

Simon, in his FOCS'94 paper, was the first to show an exponential gap between classical and quantum computation. The problem he dealt with is now part of a well-studied class of problems, the hidden subgroup problems. We study Simon's problem from the point of view of quantum query complexity and give here a first non-trivial lower bound on the query complexity of a hidden subgroup problem, namely Simon's problem. More generally, we give a lower bound which is optimal up to a constant factor for any abelian group.

[1]  H. Kurzweil,et al.  The theory of finite groups , 2003 .

[2]  Emanuel Knill,et al.  The quantum query complexity of the hidden subgroup problem is polynomial , 2004, Inf. Process. Lett..

[3]  Lisa R. Hales.  The quantum fourier transform and extensions of the abelian hidden subgroup problem , 2002 .

[4]  P. Høyer.  Conjugated operators in quantum algorithms , 1999 .

[5]  Andris Ambainis,et al.  Quantum walk algorithm for element distinctness , 2003, 45th Annual IEEE Symposium on Foundations of Computer Science.

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

[7]  今井 浩.  20世紀の名著名論：Peter Shor : Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 2004 .

[8]  Vincent Nesme,et al.  Adversary Lower Bounds for Nonadaptive Quantum Algorithms , 2008 .

[9]  Mika Hirvensalo,et al.  Introduction to Evolutionary Computing , 2002, Natural Computing Series.

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

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

[12]  Greg Kuperberg.  A Subexponential-Time Quantum Algorithm for the Dihedral Hidden Subgroup Problem , 2005, SIAM J. Comput..

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

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

[15]  H. Kurzweil,et al.  The theory of finite groups : an introduction , 2004 .

[16]  M. Hirvensalo.  Quantum Computing (Natural Computing Series) , 2004 .

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

[18]  R. Jozsa.  Quantum algorithms and the Fourier transform , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[19]  Daniel R. Simon.  On the Power of Quantum Computation , 1997, SIAM J. Comput..

[20]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[21]  Scott Aaronson,et al.  Quantum lower bound for the collision problem , 2001, STOC '02.

[22]  Samuel Kutin.  A quantum lower bound for the collision problem , 2003 .

[23]  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.

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

[25]  Robert Beals,et al.  Quantum computation of Fourier transforms over symmetric groups , 1997, STOC '97.

[26]  Andris Ambainis,et al.  A better lower bound for quantum algorithms searching an ordered list , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

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