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Random measurement bases, quantum state distinction and applications to the hidden subgroup problem
[article]

2005
*
arXiv
*
pre-print

*The*standard

*quantum*approach

*to*solving

*the*

*hidden*

*subgroup*

*problem*(HSP) is a special case of

*the*

*state*identification

*problem*where

*the*ensemble consists of so-called coset

*states*of candidate

*hidden*... This result gives us

*the*first sufficient condition

*and*an information-theoretic solution for

*the*following

*quantum*

*state*

*distinction*

*problem*: given an a priori known ensemble of

*quantum*

*states*, is there ... Acknowledgements

*The*author wishes

*to*thank Andris Ambainis, Martin Rötteler, Debbie Leung, Joseph Emerson

*and*Christoph Dankert for useful discussions,

*and*Jaikumar Radhakrishnan for feedback on an earlier ...

##
###
The Power of Strong Fourier Sampling: Quantum Algorithms for Affine Groups and Hidden Shifts

2007
*
SIAM journal on computing (Print)
*

These

doi:10.1137/s0097539705447177
fatcat:47ryliu55fg33kgarhc6vxb3tq
*hidden**subgroup**problems*are typically solved by Fourier sampling:*the**quantum*Fourier transform of ψ is computed*and**measured*. ... Many*quantum*algorithms, including Shor's celebrated factoring*and*discrete log algorithms, proceed by reduction*to*a*hidden**subgroup**problem*, in which an unknown*subgroup*H of a group G must be determined ... Kuperberg refers*to*this as*the*coherent*hidden**subgroup**problem*[18] . Step 4. Carry out*the**quantum*Fourier transform on ψ 3 or ρ H*and**measure**the*result. ...##
###
Quantum Algorithms
[article]

2008
*
arXiv
*
pre-print

amplification,

arXiv:0808.0369v1
fatcat:gsiyvpw7mnd2hlmki5tvgjwgvu
*quantum*algorithms for simulating*quantum*mechanical systems, several non-trivial generalizations of*the*Abelian*Hidden**Subgroup**Problem*(*and*related techniques),*the**quantum*walk paradigm ... This includes a summary of*the*early*quantum*algorithms, a description of*the*Abelian*Hidden**Subgroup*algorithms (including Shor's factoring*and*discrete logarithm algorithms),*quantum*searching*and*amplitude ... "Pretty Good*Measurements*" A natural approach*to*solving*the*non-Abelian*hidden**subgroup**problem*is*to*prepare several instances of a*random*coset*state*for*the**hidden**subgroup*K,*and*then try*to*determine ...##
###
Quantum Measurements for Hidden Subgroup Problems with Optimal Sample Complexity
[article]

2007
*
arXiv
*
pre-print

One of

arXiv:quant-ph/0604174v3
fatcat:o5yz5vlv5newfpvszzvh4xrqdq
*the*central issues in*the**hidden**subgroup**problem*is*to*bound*the*sample complexity, i.e.,*the*number of identical samples of coset*states*sufficient*and*necessary*to*solve*the**problem*. ... In particular, it does so for*the*important instances such as*the*dihedral*and**the*symmetric*hidden**subgroup**problems*. ... Given f H : G → S*and*a generator set of G,*the**hidden**subgroup**problem*(HSP) is*the**problem*of finding a set of generators for*the**hidden**subgroup*H. ...##
###
On Quantum Algorithms for Noncommutative Hidden Subgroups

2000
*
Advances in Applied Mathematics
*

A general framework for

doi:10.1006/aama.2000.0699
fatcat:tz3gpujsknhmbnfcfwe6c36tzy
*the*noncommutative*hidden**subgroup**problem*is discussed*and*we indicate future research directions. ...*Quantum*algorithms for factoring*and*finding discrete logarithms have previously been generalized*to*finding*hidden**subgroups*of finite Abelian groups. ... ACKNOWLEDGMENTS We would like*to*thank Dan Rockmore, David Maslen,*and*Hans J. ...##
###
Quantum algorithms for algebraic problems

2010
*
Reviews of Modern Physics
*

This article reviews

doi:10.1103/revmodphys.82.1
fatcat:z6xqtmdvdffrdnyo26wsjp4vee
*the*current*state*of*quantum*algorithms, focusing on algorithms with superpolynomial speedup over classical computation,*and*in particular, on*problems*with an algebraic flavor. ... Understanding what other computational*problems*can be solved significantly faster using*quantum*algorithms is one of*the*major challenges in*the*theory of*quantum*computation,*and*such algorithms motivate ... This article was written in part while AMC was at*the*Institute for*Quantum*Information at Caltech, where he received support from*the*National Science Foundation under grant PHY-456720*and*from*the*Army ...##
###
On Quantum Algorithms for Noncommutative Hidden Subgroups
[chapter]

1999
*
Lecture Notes in Computer Science
*

A general framework for

doi:10.1007/3-540-49116-3_45
fatcat:nsuvwurevvh4bf6riimikqnwjq
*the*noncommutative*hidden**subgroup**problem*is discussed*and*we indicate future research directions. ...*Quantum*algorithms for factoring*and*discrete logarithm have previously been generalized*to*finding*hidden**subgroups*of finite Abelian groups. ... Acknowledgements We would like*to*thank Dan Rockmore, David Maslen*and*Hans J. ...##
###
Simon's Algorithm, Clebsch-Gordan Sieves, and Hidden Symmetries of Multiple Squares
[article]

2008
*
arXiv
*
pre-print

*The*first

*quantum*algorithm

*to*offer an exponential speedup (in

*the*query complexity setting) over classical algorithms was Simon's algorithm for identifying a

*hidden*exclusive-or mask. ... This

*problem*previously admitted an efficient

*quantum*algorithm but a connection

*to*Clebsch-Gordan transforms had not been made. ... This work was supported by ARO/NSA

*quantum*algorithms grant number W911NSF-06-1-0379

*and*NSF grant number 0523359

*and*NSF grant number 0621621. ...

##
###
Optimal measurements for the dihedral hidden subgroup problem
[article]

2005
*
arXiv
*
pre-print

We consider

arXiv:quant-ph/0501044v1
fatcat:cgvue7uxerbmnltvowgadqxv6u
*the*dihedral*hidden**subgroup**problem*as*the**problem*of distinguishing*hidden**subgroup**states*. ... Thus*the*dihedral group provides an example of a group G for which Omega(log|G|)*hidden**subgroup**states*are necessary*to*solve*the**hidden**subgroup**problem*. ... We thank Carlos Mochon*and*Frank Verstraete for helpful discussions of Theorem 1,*and*Abie Flaxman for correspondence regarding*the*algorithm in [16]*and*an earlier version thereof. ...##
###
Is Grover's Algorithm a Quantum Hidden Subgroup Algorithm?

2007
*
Quantum Information Processing
*

But we then go on

doi:10.1007/s11128-007-0066-1
fatcat:qn57oocbvjaxjjhhac462up34q
*to*show that*the*standard non-abelian*quantum**hidden**subgroup*(QHS) algorithm can not find a solution*to*this particular HSP. ... Specifically, we show that Grover's algorithm can be viewed as a*quantum*algorithm which solves a non-abelian*hidden**subgroup**problem*(HSP). ...*The*U.S. Government is authorized*to*reproduce*and*distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. ...##
###
Is Grover's Algorithm a Quantum Hidden Subgroup Algorithm ?
[article]

2006
*
arXiv
*
pre-print

But we then go on

arXiv:quant-ph/0603140v1
fatcat:djsdyy445bfqlhotv2ops377ha
*to*show that*the*standard non-abelian*quantum**hidden**subgroup*(QHS) algorithm can not find a solution*to*this particular HSP. ... Specifically, we show that Grover's algorithm can be viewed as a*quantum*algorithm which solves a non-abelian*hidden**subgroup**problem*(HSP). ...*The*corresponding*quantum*form of this HSP is*stated*as follows:*Problem*2 (*Hidden**Subgroup**Problem*:*Quantum*Version). Let ϕ : G −→ S be a map with*hidden**subgroup*structure. ...##
###
Hidden symmetry detection on a quantum computer
[article]

2006
*
arXiv
*
pre-print

*The*fastest

*quantum*algorithms (for

*the*solution of classical computational tasks) known so far are basically variations of

*the*

*hidden*

*subgroup*

*problem*with f(U[x])=f(x). ... Following a discussion regarding which tasks might be solved efficiently by

*quantum*computers, it will be demonstrated by means of a simple example, that

*the*detection of more general

*hidden*(two-point ... a contradiction f (x) = f (x) + 2q;

*and*

*the*second example can be reduced

*to*Shor's case f (x + 2p) = f (x). ...

##
###
The quantum query complexity of the abelian hidden subgroup problem

2007
*
Theoretical Computer Science
*

We study Simon's

doi:10.1016/j.tcs.2007.02.057
fatcat:bkzyhyd4czg6tdpwyy4jmf2fwm
*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*. ...*The**problem*he dealt with is now part of a well-studied class of*problems*,*the**hidden**subgroup**problems*. ... Acknowledgements Many thanks*to*Xavier Caruso, Yves de Cornulier*and*Joël Riou for useful help. Thanks also go*to*Frédéric Magniez*and**to**the*ICALP 2005 referees for bibliographical hints. ...##
###
On the Power of Random Bases in Fourier Sampling: Hidden Subgroup Problem in the Heisenberg Group
[chapter]

2005
*
Lecture Notes in Computer Science
*

*The*

*hidden*

*subgroup*

*problem*(HSP) provides a unified framework

*to*study

*problems*of grouptheoretical nature in

*quantum*computing such as order finding

*and*

*the*discrete logarithm

*problem*. ... We define a parameter r(G) for a group G

*and*show that O((log |G|/r(G)) 2 ) iterations of

*the*

*random*strong method give enough classical information

*to*identify a

*hidden*

*subgroup*in G. ... Acknowledgments We thank Frédéric Magniez, Leonard Schulman, Cris Moore, Alex Russell

*and*Avery Miller for useful discussions. ...

##
###
McEliece and Niederreiter Cryptosystems That Resist Quantum Fourier Sampling Attacks
[chapter]

2011
*
Lecture Notes in Computer Science
*

Specifically, we show that

doi:10.1007/978-3-642-22792-9_43
fatcat:5oyobilk5vb3lpqyy5v3t5gufu
*the*natural case of*the**Hidden**Subgroup**Problem**to*which McEliece-type cryptosystems reduce cannot be solved by strong Fourier sampling, or by any*measurement*of a coset*state*... cryptosystems are vulnerablenamely, those*based*on generating*and**measuring*coset*states*. ...*Quantum*Fourier Sampling (QFS) is a standard procedure*based*on*the**Quantum*Fourier Transform*to*solve*the**Hidden**Subgroup**Problem*(HSP) (see [12] for a survey). ...
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