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Quantum Mechanical Algorithms for the Nonabelian Hidden Subgroup Problem

Michelangelo Grigni, Leonard J. Schulman, Monica Vazirani, Umesh Vazirani
2004 Combinatorica  
We give a short exposition of new and known results on the "standard method" of identifying a hidden subgroup of a nonabelian group using a quantum computer.  ...  Abstract We give a short exposition of new and known results on the "standard method" of identifying a hidden subgroup of a nonabelian group using a quantum computer.  ...  Normal Recall that in the "standard method" for the hidden subgroup problem we sample from the Fourier transform of the uniform superposition over a random coset of the hidden subgroup .  ... 
doi:10.1007/s00493-004-0009-8 fatcat:7zmhojofbndkfl77yqbqlb7afq

Quantum mechanical algorithms for the nonabelian hidden subgroup problem

Michelangelo Grigni, Leonard Schulman, Monica Vazirani, Umesh Vazirani
2001 Proceedings of the thirty-third annual ACM symposium on Theory of computing - STOC '01  
We give a short exposition of new and known results on the "standard method" of identifying a hidden subgroup of a nonabelian group using a quantum computer.  ...  Abstract We give a short exposition of new and known results on the "standard method" of identifying a hidden subgroup of a nonabelian group using a quantum computer.  ...  Normal Recall that in the "standard method" for the hidden subgroup problem we sample from the Fourier transform of the uniform superposition over a random coset of the hidden subgroup .  ... 
doi:10.1145/380752.380769 dblp:conf/stoc/GrigniSVV01 fatcat:6ah7e2jrtncnjjqs6ditv7uxua

Page 8261 of Mathematical Reviews Vol. , Issue 2004j [page]

2004 Mathematical Reviews  
The hidden subgroup problem (HSP) plays a central role in most of the practically relevant quantum algorithms.  ...  It is also known that an efficient solution to the HSP for S,, gives an efficient algorithm for the graph isomorphism problem.  ... 

The Hidden Subgroup Problem - Review and Open Problems [article]

Chris Lomont
2004 arXiv   pre-print
An overview of quantum computing and in particular the Hidden Subgroup Problem are presented from a mathematical viewpoint.  ...  Detailed proofs are supplied for many important results from the literature, and notation is unified, making it easier to absorb the background necessary to begin research on the Hidden Subgroup Problem  ...  With that said, let's begin analyzing the general (nonabelian case). The General Hidden Subgroup Problem Why do we want to find hidden subgroups of nonabelian groups?  ... 
arXiv:quant-ph/0411037v1 fatcat:3fu3ehmejzbm3n43ihkig4bbha

Quantum Algorithms for Simon's Problem Over General Groups

Gorjan Alagic and Cristopher Moore and Alexander Russell
2007 arXiv   pre-print
Daniel Simon's 1994 discovery of an efficient quantum algorithm for solving the hidden subgroup problem (HSP) over Z_2^n provided one of the first algebraic problems for which quantum computers are exponentially  ...  In the current parlance, this is the hidden subgroup problem (HSP) over groups of the form G^n, where G is a nonabelian group of constant size, and where the hidden subgroup is either trivial or has order  ...  This problem fits into the framework of the Hidden Subgroup Problem (HSP), which also underlies Shor's celebrated quantum algorithms for factoring and discrete logarithm [17] .  ... 
arXiv:quant-ph/0603251v2 fatcat:pdrmcqlmffgpvbxmyiag7eoyam

Quantum Algorithms for Hidden Nonlinear Structures

Andrew M. Childs, Leonard J. Schulman, Umesh V. Vazirani
2007 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)  
Attempts to find new quantum algorithms that outperform classical computation have focused primarily on the nonabelian hidden subgroup problem, which generalizes the central problem solved by Shor's factoring  ...  We also give some positive results on the quantum query complexity of finding hidden nonlinear structures.  ...  of small characteristic and for drawing our attention to Lemma 1 of [19] , and John Watrous for suggesting the problem of finding an efficient quantum procedure for moving amplitude from spheres to their  ... 
doi:10.1109/focs.2007.18 dblp:conf/focs/Marx07 fatcat:jnudnl5e6vc5vmxuoyvcmzt45i

Quantum Algorithms for Hidden Nonlinear Structures

Andrew M. Childs, Leonard J. Schulman, Umesh V. Vazirani
2007 Foundations of Computer Science (FOCS), IEEE Annual Symposium on  
Attempts to find new quantum algorithms that outperform classical computation have focused primarily on the nonabelian hidden subgroup problem, which generalizes the central problem solved by Shor's factoring  ...  We also give some positive results on the quantum query complexity of finding hidden nonlinear structures.  ...  of small characteristic and for drawing our attention to Lemma 1 of [19] , and John Watrous for suggesting the problem of finding an efficient quantum procedure for moving amplitude from spheres to their  ... 
doi:10.1109/focs.2007.4389510 fatcat:new2tatdqzguneb4ti2evxh55u

Quantum Algorithms for Some Hidden Shift Problems

Wim van Dam, Sean Hallgren, Lawrence Ip
2006 SIAM journal on computing (Print)  
We also define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem.  ...  Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structures of functions, especially periodicity.  ...  We would like to thank the anonymous referee who pointed out the application of the shifted Legendre symbol problem to algebraically homomorphic cryptosystems, and Umesh Vazirani, whose many suggestions  ... 
doi:10.1137/s009753970343141x fatcat:ad2sui2s7rbd3bs3r4hywbwy5e

Page 3696 of Mathematical Reviews Vol. , Issue 2000e [page]

2000 Mathematical Reviews  
The first problem involves determining thehiddensubgroup of a function that is constant on cosets of a subgroup. This problem has known, efficient solutions only for the case of abelian groups.  ...  For the case of nonabelian groups, this is one of the better-known open problems in quantum computing. The second problem is to learn which group homomorphism is implemented by a quantum black box.  ... 

Quantum Algorithms for some Hidden Shift Problems [article]

Wim van Dam, Lawrence Ip
2002 arXiv   pre-print
We also define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem.  ...  Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structure of functions, especially periodicity.  ...  Acknowledgments We would like to thank the anonymous referee who pointed out the application of shifted Legendre symbol problem to algebraically homomorphic cryptosystems and Umesh Vazirani, whose many  ... 
arXiv:quant-ph/0211140v1 fatcat:vt4kl5vvijdi5kni466g4zyche

Normalizer Circuits and Quantum Computation [article]

Juan Bermejo-Vega
2016 arXiv   pre-print
Finally, we devise new quantum algorithms for finding hidden hyperstructures. The results offer new insights into the source of quantum speed-ups for several algebraic problems.  ...  Though Clifford circuits are efficiently classically simulable, we show that normalizer circuit models encompass Shor's celebrated factoring algorithm and the quantum algorithms for abelian Hidden Subgroup  ...  exponential quantum speed-ups of Shor's algorithm and the quantum algorithms for solving abelian hidden subgroup problems (HSP).  ... 
arXiv:1611.09274v1 fatcat:y6tw3ta2mfb5fkpzdyrbrmuhwu

The Symmetric Group Defies Strong Fourier Sampling

Cristopher Moore, Alexander Russell, Leonard J. Schulman
2008 SIAM journal on computing (Print)  
Introduction: The hidden subgroup problem. Many problems of interest in quantum computing can be reduced to an instance of the hidden subgroup problem (HSP).  ...  by Simon and Shor into an approach to the hidden subgroup problem.  ...  M. also thanks Rosemary Moore for providing a larger perspective. Finally, we thank Gorjan Alagic for his comments on the structured involutions material.  ... 
doi:10.1137/050644896 fatcat:yjeyr4lmizbchpx52aaileujxa

Fully graphical treatment of the quantum algorithm for the Hidden Subgroup Problem [article]

Stefano Gogioso, Aleks Kissinger
2017 arXiv   pre-print
The abelian Hidden Subgroup Problem (HSP) is extremely general, and many problems with known quantum exponential speed-up (such as integers factorisation, the discrete logarithm and Simon's problem) can  ...  Being fully diagrammatic, our proof extends beyond the traditional case of finite-dimensional quantum theory: for example, we can use it to show that Simon's problem can be efficiently solved in real quantum  ...  The authors would like to thank Bob Coecke for comments and suggestions, as well as Sukrita Chatterji and Nicolò Chiappori for support.  ... 
arXiv:1701.08669v1 fatcat:x3chwcdpwnhtfdxcpg36eryf4y

Page 5546 of Mathematical Reviews Vol. , Issue 2003g [page]

2003 Mathematical Reviews  
Kauffman, Quantum hidden subgroup algorithms: a mathematical perspective (139- 202); Elitza N. Maneva and John A. Smolin, Improved two-party and multi-party purification protocols (203-212); David A.  ...  Meyer, Quantum games and quantum algorithms (213-220); John M. Myers and F.  ... 

Page 586 of Mathematical Reviews Vol. , Issue 2001A [page]

2001 Mathematical Reviews  
This mechanism then allows for universal quantum computation by composing a generic pair of such loops. They also briefly address the associated problem of computational complex- ity.  ...  Abrams and Seth Lloyd, Computational complexity and physical law (167- 173); Michele Mosca and Artur Ekert, The hidden subgroup prob- lem and eigenvalue estimation on a quantum computer (174-188); Giuseppe  ... 
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