Introduction to Recent Quantum Algorithms

We discuss some of the recent progress in quantum algorithmics. We review most of the primary techniques used in proving upper and lower bounds and illustrate how to apply the techniques to a variety of problems, including the threshold function, parity, searching and sorting. We also give a set of open questions and possible future research directions. Our aim is to give a basic overview and we include suggestions to further reading.

[1]  D. Deutsch Quantum computational networks , 1989, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[2]  D. Deutsch,et al.  Rapid solution of problems by quantum computation , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[3]  Edward M. Reingold,et al.  Determining the Majority , 1993, Inf. Process. Lett..

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

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

[6]  Christoph Dürr,et al.  A Quantum Algorithm for Finding the Minimum , 1996, ArXiv.

[7]  Lov K. Grover A fast quantum mechanical algorithm for estimating the median , 1996, quant-ph/9607024.

[8]  Gilles Brassard,et al.  Tight bounds on quantum searching , 1996, quant-ph/9605034.

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

[10]  Lov K. Grover Quantum Mechanics Helps in Searching for a Needle in a Haystack , 1997, quant-ph/9706033.

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

[12]  Gilles Brassard,et al.  Quantum Algorithm for the Collision Problem , 1997 .

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

[14]  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.

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

[16]  M. Sipser,et al.  A Limit on the Speed of Quantum Computation for Insertion into an Ordered List , 1998 .

[17]  A. Selman,et al.  Complexity theory retrospective II , 1998 .

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

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

[20]  Thomas P. Hayes,et al.  On the Quantum Complexity of Majority , 1998 .

[21]  A. Berthiaume Quantum computation , 1998 .

[22]  R. Cleve,et al.  Quantum algorithms revisited , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[23]  Gilles Brassard,et al.  Quantum Cryptanalysis of Hash and Claw-Free Functions , 1998, LATIN.

[24]  Lov K. Grover,et al.  Quantum computation , 1999, Proceedings Twelfth International Conference on VLSI Design. (Cat. No.PR00013).

[25]  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).

[26]  Felix Wu,et al.  The quantum query complexity of approximating the median and related statistics , 1998, STOC '99.

[27]  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).

[28]  Christof Zalka GROVER'S QUANTUM SEARCHING ALGORITHM IS OPTIMAL , 1997, quant-ph/9711070.

[29]  M. Sipser,et al.  Invariant quantum algorithms for insertion into an ordered list , 1999, quant-ph/9901059.

[30]  Ronald de Wolf,et al.  A Lower Bound for Quantum Search of an Ordered List , 1999, Inf. Process. Lett..

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

[32]  Lance Fortnow,et al.  Complexity limitations on quantum computation , 1999, J. Comput. Syst. Sci..

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

[34]  V. Vinay,et al.  String Matching in ${\tilde O}(\sqrt{n}+\sqrt{m})$ Quantum Time , 2000, quant-ph/0011049.

[35]  Eleanor G. Rieffel,et al.  J an 2 00 0 An Introduction to Quantum Computing for Non-Physicists , 2002 .

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

[37]  R. Cleve An Introduction to Quantum Complexity Theory , 1999, quant-ph/9906111.

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

[39]  Frédéric Magniez,et al.  Quantum algorithms for element distinctness , 2000, Proceedings 16th Annual IEEE Conference on Computational Complexity.

[40]  Erich Novak Quantum Complexity of Integration , 2001, J. Complex..

[41]  Jan Neerbek,et al.  Quantum Complexities of Ordered Searching, Sorting, and Element Distinctness , 2002, Algorithmica.

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

[43]  Frédéric Magniez,et al.  Efficient Quantum Algorithms For Some Instances Of The Non-Abelian Hidden Subgroup Problem , 2003, Int. J. Found. Comput. Sci..