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Quantum measurements and the Abelian Stabilizer Problem [article]

A.Yu.Kitaev
1995 arXiv   pre-print
We present a polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm. Thus we extend famous Shor's results.  ...  The paper also contains a rather detailed introduction to the theory of quantum computation.  ...  This work was supported, in part, by the ISF grant M5R000. I am grateful to Sergei Tarasov for useful remarks.  ... 
arXiv:quant-ph/9511026v1 fatcat:5ddfh7ucgzbtrhzeyj4nxgaxwm

Normalizer Circuits and Quantum Computation [article]

Juan Bermejo-Vega
2016 arXiv   pre-print
This thesis also establishes a precise connection between Shor's quantum algorithm and the stabilizer formalism, revealing a common mathematical structure in several quantum speed-ups and error-correcting  ...  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.  ...  exponential quantum speed-ups of Shor's algorithm and the quantum algorithms for solving abelian hidden subgroup problems (HSP).  ... 
arXiv:1611.09274v1 fatcat:y6tw3ta2mfb5fkpzdyrbrmuhwu

A Hot Topic in the Quantum Hall Effect

Jason Alicea
2018 Physics  
Now David Mross and colleagues from the Weizmann Institute of Science in Israel [3] and Chong Wang and colleagues from Harvard University [4] describe a possible resolution: Because of disorder, the system  ...  Certain types of fractional quantum Hall phases, called non-Abelian, provide a potential platform for intrinsically error-resistant quantum computation.  ...  This research is published in Physical Review Letters, Physical Review B, and Nature.  ... 
doi:10.1103/physics.11.70 fatcat:vvh2kpx4h5getj47kymmbsbuha

Abelian Hypergroups and Quantum Computation [article]

Juan Bermejo-Vega, Kevin C. Zatloukal
2015 arXiv   pre-print
Motivated by a connection, described here for the first time, between the hidden normal subgroup problem (HNSP) and abelian hypergroups (algebraic objects that model collisions of physical particles),  ...  Using these tools, we develop the first provably efficient quantum algorithm for finding hidden subhypergroups of nilpotent abelian hypergroups and, via the aforementioned connection, a new, hypergroup-based  ...  Jozsa, for providing references; and the MIT Center for Theoretical Physics and the Max Planck Institute of Quantum Optics for hosting us in 2013 and 2014.  ... 
arXiv:1509.05806v2 fatcat:wyxhjpxzrbfiza34nohr4p4fxa

The Hidden Subgroup Problem and Post-quantum Group-based Cryptography [article]

Kelsey Horan, Delaram Kahrobaei
2018 arXiv   pre-print
We review the relationship between HSP and other computational problems discuss an optimal solution method, and review the known results about the quantum complexity of HSP.  ...  In this paper we discuss the Hidden Subgroup Problem (HSP) in relation to post-quantum group-based cryptography.  ...  for the abelian stabilizer problem are all special cases of HSP.  ... 
arXiv:1805.04179v2 fatcat:4pt43jdwqfhutiylezwk6y4w3q

Classical simulations of Abelian-group normalizer circuits with intermediate measurements [article]

Juan Bermejo-Vega, Maarten Van den Nest
2013 arXiv   pre-print
generalized Pauli measurements and provide a normal form of the amplitudes of generalized stabilizer states using quadratic functions and subgroup cosets.  ...  Finally we develop a generalization of the stabilizer formalism [quant-ph/9705052, quant-ph/9807006] relative to arbitrary finite Abelian groups: for example we characterize how to update stabilizers under  ...  The gate U f can now be used to measure Z(g), with a routine inspired by the coset-state preparation method used in the standard quantum algorithm to solve the Abelian hidden subgroup problem [13, 14]  ... 
arXiv:1210.3637v2 fatcat:cigwedz22vhdngncilsd7wyn3a

A non-commuting stabilizer formalism

Xiaotong Ni, Oliver Buerschaper, Maarten Van den Nest
2015 Journal of Mathematical Physics  
We propose a non-commutative extension of the Pauli stabilizer formalism. The aim is to describe a class of many-body quantum states which is richer than the standard Pauli stabilizer states.  ...  In our framework, stabilizer operators are tensor products of single-qubit operators drawn from the group 〈α I, X,S〉, where α=e^iπ/4 and S=diag(1,i).  ...  The Pauli stabilizer formalism (PSF) is a widely used tool throughout the development of quantum information (e.g. quantum error correcting codes, measurement based quantum computation, toric code).  ... 
doi:10.1063/1.4920923 fatcat:eogg5txkcjesnctkik3fbxzyxa

Engineering complex topological memories from simple Abelian models

James R. Wootton, Ville Lahtinen, Benoit Doucot, Jiannis K. Pachos
2011 Annals of Physics  
This bridges the gap between requirements for anyonic quantum computation and the potential of state-of-the-art technology.  ...  We explicitly demonstrate the control procedures for the encoding and manipulation of quantum information in specific lattice models that can be implemented in the laboratory.  ...  This work was supported by the EU grants EMALI and SCALA, the EPSRC, the Finnish Academy of Science and the Royal Society.  ... 
doi:10.1016/j.aop.2011.05.008 fatcat:2l2hdzte4zfhbfmqkzj53gguhu

Abelian and non-Abelian states inν=2/3bilayer fractional quantum Hall systems

Michael R. Peterson, Yang-Le Wu, Meng Cheng, Maissam Barkeshli, Zhenghan Wang, Sankar Das Sarma
2015 Physical Review B  
Some of these states even possess quasiparticles with non-Abelian statistics that are powerful enough for universal topological quantum computation, and are thus of particular interest.  ...  There are several possible theoretically allowed non-Abelian fractional quantum Hall (FQH) states that could potentially be realized in one- and two- component FQH systems at total filling fraction ν =  ...  Acknowledgements -M.R.P. thanks the Office of Research and Sponsored Programs at California State University Long Beach and Microsoft Station Q. We thank M. Zaletel, P. Bonderson, and N.  ... 
doi:10.1103/physrevb.92.035103 fatcat:fs2iv4a5vfh3tmkwoqezumjnuq

Hidden translation and orbit coset in quantum computing

Katalin Friedl, Gábor Ivanyos, Frédéric Magniez, Miklos Santha, Pranab Sen
2003 Proceedings of the thirty-fifth ACM symposium on Theory of computing - STOC '03  
We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian groups including solvable groups of constant exponent and of constant length  ...  For the induction step, we introduce the problem Orbit Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a powerful self-reducibility result: Orbit Coset in a finite group G is  ...  Acknowledgements We wish to thank Mark Ettinger and Peter Høyer for sharing their knowledge and ideas about Hidden Translation with us, and Martin Rötteler for several useful discussions on Hidden Subgroup  ... 
doi:10.1145/780543.780544 fatcat:7v7sys52ybh5zcqzw2eg2zucoy

Hidden translation and orbit coset in quantum computing

Katalin Friedl, Gábor Ivanyos, Frédéric Magniez, Miklos Santha, Pranab Sen
2003 Proceedings of the thirty-fifth ACM symposium on Theory of computing - STOC '03  
We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian groups including solvable groups of constant exponent and of constant length  ...  For the induction step, we introduce the problem Orbit Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a powerful self-reducibility result: Orbit Coset in a finite group G is  ...  Acknowledgements We wish to thank Mark Ettinger and Peter Høyer for sharing their knowledge and ideas about Hidden Translation with us, and Martin Rötteler for several useful discussions on Hidden Subgroup  ... 
doi:10.1145/780542.780544 dblp:conf/stoc/FriedlIMSS03 fatcat:57cakoqgv5hs5a4qq4bjhx3z6u

Direct characterization of quantum dynamics: General theory

M. Mohseni, D. A. Lidar
2007 Physical Review A. Atomic, Molecular, and Optical Physics  
The characterization of the dynamics of quantum systems is a task of both fundamental and practical importance.  ...  Here we provide a generalization by developing a theory for direct and complete characterization of the dynamics of arbitrary quantum systems.  ...  ACKNOWLEDGMENTS The authors thank J. Emerson  ... 
doi:10.1103/physreva.75.062331 fatcat:qgmu6jxrobfgxjt3r5zx6y2zte

Hidden Translation and Translating Coset in Quantum Computing

Katalin Friedl, Gábor Ivanyos, Frédéric Magniez, Miklos Santha, Pranab Sen
2014 SIAM journal on computing (Print)  
We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian solvable groups including solvable groups of constant exponent and of constant  ...  For the induction step, we introduce the problem Translating Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a powerful self-reducibility result: Translating Coset in a finite  ...  Acknowledgements We wish to thank Mark Ettinger and Peter Høyer for sharing their knowledge and ideas about Hidden Translation with us, and Martin Rötteler for several useful discussions on Hidden Subgroup  ... 
doi:10.1137/130907203 fatcat:rwapyh6j7ffddp5phted42l4re

The computational power of normalizer circuits over black-box groups [article]

Juan Bermejo-Vega, Cedric Yen-Yu Lin, Maarten Van den Nest
2014 arXiv   pre-print
In particular, solving this problem renders black-box normalizer circuits efficiently classically simulable by exploiting the generalized stabilizer formalism in [arXiv:1201.4867v1,arXiv:1210.3637,arXiv  ...  This work presents a precise connection between Clifford circuits, Shor's factoring algorithm and several other famous quantum algorithms with exponential quantum speed-ups for solving Abelian hidden subgroup  ...  CYL acknowledges support from the ARO grant W911NF-12-0486 (Quantum Information Group), and the Natural Sciences and Engineering Research Council of Canada.  ... 
arXiv:1409.4800v1 fatcat:pjavodzw2bd2llwwgufazkqdri

An efficient quantum algorithm for finding hidden parabolic subgroups in the general linear group [article]

Thomas Decker, Gábor Ivanyos, Raghav Kulkarni, Youming Qiao, Miklos Santha
2014 arXiv   pre-print
Our main result is a quantum algorithm of time polynomial in q and n for solving the hidden subgroup problem in GL_n(F_q), when the hidden subgroup is promised to be a parabolic subgroup.  ...  Moore, and A. Russell (2010), Quantum Inf. Comput., Vol. 10, pp. 282-291), and for minimal parabolic subgroups (Borel subgroups), for the case when q is not much smaller than n (G.  ...  The research is partially funded by the Singapore Ministry of Education and the National Research Foundation, also through the Tier 3 Grant "Random numbers from quantum processes," MOE2012-T3-1-009.  ... 
arXiv:1406.6511v2 fatcat:gizb22accbf2vlcemmvoaq7ecq
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