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Quantum Mechanical Algorithms for the Nonabelian Hidden Subgroup Problem

2004
*
Combinatorica
*

We give a short exposition of new and known results on

doi:10.1007/s00493-004-0009-8
fatcat:7zmhojofbndkfl77yqbqlb7afq
*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*. ...##
###
Quantum mechanical algorithms for the nonabelian hidden subgroup problem

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

doi:10.1145/380752.380769
dblp:conf/stoc/GrigniSVV01
fatcat:6ah7e2jrtncnjjqs6ditv7uxua
*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*. ...##
###
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]

2004
*
arXiv
*
pre-print

An overview of

arXiv:quant-ph/0411037v1
fatcat:3fu3ehmejzbm3n43ihkig4bbha
*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? ...##
###
Quantum Algorithms for Simon's Problem Over General Groups

2007
*
arXiv
*
pre-print

Daniel Simon's 1994 discovery of an efficient

arXiv:quant-ph/0603251v2
fatcat:pdrmcqlmffgpvbxmyiag7eoyam
*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] . ...##
###
Quantum Algorithms for Hidden Nonlinear Structures

2007
*
48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)
*

Attempts to find new

doi:10.1109/focs.2007.18
dblp:conf/focs/Marx07
fatcat:jnudnl5e6vc5vmxuoyvcmzt45i
*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 ...##
###
Quantum Algorithms for Hidden Nonlinear Structures

2007
*
Foundations of Computer Science (FOCS), IEEE Annual Symposium on
*

Attempts to find new

doi:10.1109/focs.2007.4389510
fatcat:new2tatdqzguneb4ti2evxh55u
*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 ...##
###
Quantum Algorithms for Some Hidden Shift Problems

2006
*
SIAM journal on computing (Print)
*

We also define

doi:10.1137/s009753970343141x
fatcat:ad2sui2s7rbd3bs3r4hywbwy5e
*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 ...##
###
Page 3696 of Mathematical Reviews Vol. , Issue 2000e
[page]

2000
*
Mathematical Reviews
*

*The*first

*problem*involves determining

*the*“

*hidden*”

*subgroup*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]

2002
*
arXiv
*
pre-print

We also define

arXiv:quant-ph/0211140v1
fatcat:vt4kl5vvijdi5kni466g4zyche
*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 ...##
###
Normalizer Circuits and Quantum Computation
[article]

2016
*
arXiv
*
pre-print

Finally, we devise new

arXiv:1611.09274v1
fatcat:y6tw3ta2mfb5fkpzdyrbrmuhwu
*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). ...##
###
The Symmetric Group Defies Strong Fourier Sampling

2008
*
SIAM journal on computing (Print)
*

Introduction:

doi:10.1137/050644896
fatcat:yjeyr4lmizbchpx52aaileujxa
*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. ...##
###
Fully graphical treatment of the quantum algorithm for the Hidden Subgroup Problem
[article]

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. ...

##
###
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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