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

Pranab Sen
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  ... 
arXiv:quant-ph/0512085v1 fatcat:tpgayxofcvcqjlx7cxmnpj6bra

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

Cristopher Moore, Daniel Rockmore, Alexander Russell, Leonard J. Schulman
2007 SIAM journal on computing (Print)  
These 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.  ... 
doi:10.1137/s0097539705447177 fatcat:47ryliu55fg33kgarhc6vxb3tq

Quantum Algorithms [article]

Michele Mosca
2008 arXiv   pre-print
amplification, 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  ... 
arXiv:0808.0369v1 fatcat:gsiyvpw7mnd2hlmki5tvgjwgvu

Quantum Measurements for Hidden Subgroup Problems with Optimal Sample Complexity [article]

Masahito Hayashi, Akinori Kawachi, Hirotada Kobayashi
2007 arXiv   pre-print
One of 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.  ... 
arXiv:quant-ph/0604174v3 fatcat:o5yz5vlv5newfpvszzvh4xrqdq

On Quantum Algorithms for Noncommutative Hidden Subgroups

Mark Ettinger, Peter Høyer
2000 Advances in Applied Mathematics  
A general framework for 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.  ... 
doi:10.1006/aama.2000.0699 fatcat:tz3gpujsknhmbnfcfwe6c36tzy

Quantum algorithms for algebraic problems

Andrew M. Childs, Wim van Dam
2010 Reviews of Modern Physics  
This article reviews 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  ... 
doi:10.1103/revmodphys.82.1 fatcat:z6xqtmdvdffrdnyo26wsjp4vee

On Quantum Algorithms for Noncommutative Hidden Subgroups [chapter]

Mark Ettinger, Peter Høyer
1999 Lecture Notes in Computer Science  
A general framework for 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.  ... 
doi:10.1007/3-540-49116-3_45 fatcat:nsuvwurevvh4bf6riimikqnwjq

Simon's Algorithm, Clebsch-Gordan Sieves, and Hidden Symmetries of Multiple Squares [article]

D. Bacon
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.  ... 
arXiv:0808.0174v1 fatcat:xeowtqof3vhwpewhfjxm3kci54

Optimal measurements for the dihedral hidden subgroup problem [article]

Dave Bacon, Andrew M. Childs, Wim van Dam
2005 arXiv   pre-print
We consider 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.  ... 
arXiv:quant-ph/0501044v1 fatcat:cgvue7uxerbmnltvowgadqxv6u

Is Grover's Algorithm a Quantum Hidden Subgroup Algorithm?

Samuel J. Lomonaco, Louis H. Kauffman
2007 Quantum Information Processing  
But we then go on 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.  ... 
doi:10.1007/s11128-007-0066-1 fatcat:qn57oocbvjaxjjhhac462up34q

Is Grover's Algorithm a Quantum Hidden Subgroup Algorithm ? [article]

Samuel J. Lomonaco, Jr., Louis H. Kauffman
2006 arXiv   pre-print
But we then go on 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.  ... 
arXiv:quant-ph/0603140v1 fatcat:djsdyy445bfqlhotv2ops377ha

Hidden symmetry detection on a quantum computer [article]

R. Schützhold, W. G. Unruh
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).  ... 
arXiv:quant-ph/0304090v4 fatcat:5jluk4ab5jcv3hozdglbdh73y4

The quantum query complexity of the abelian hidden subgroup problem

Pascal Koiran, Vincent Nesme, Natacha Portier
2007 Theoretical Computer Science  
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.  ...  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.  ... 
doi:10.1016/j.tcs.2007.02.057 fatcat:bkzyhyd4czg6tdpwyy4jmf2fwm

On the Power of Random Bases in Fourier Sampling: Hidden Subgroup Problem in the Heisenberg Group [chapter]

Jaikumar Radhakrishnan, Martin Rötteler, Pranab Sen
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.  ... 
doi:10.1007/11523468_113 fatcat:3yzkio2s2ngq3ehj6cf63wjaca

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

Hang Dinh, Cristopher Moore, Alexander Russell
2011 Lecture Notes in Computer Science  
Specifically, we show that 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).  ... 
doi:10.1007/978-3-642-22792-9_43 fatcat:5oyobilk5vb3lpqyy5v3t5gufu
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