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Limitations of Quantum Coset States for Graph Isomorphism [article]

Sean Hallgren, Martin Roetteler, Pranab Sen
2005 arXiv   pre-print
We show that entangled quantum measurements on at least Ω(n log n) coset states are necessary to get useful information for the case of graph isomorphism, matching an information theoretic upper bound.  ...  A common feature of the resulting algorithms is the use of quantum coset states, which encode the hidden subgroup.  ...  Acknowledgments We thank Andrew Childs, Frédéric Magniez and Umesh Vazirani for helpful discussions and comments.  ... 
arXiv:quant-ph/0511148v1 fatcat:lofr5oudnbfkxo47niwccjojje

Limitations of quantum coset states for graph isomorphism

Sean Hallgren, Cristopher Moore, Martin Rötteler, Alexander Russell, Pranab Sen
2006 Proceedings of the thirty-eighth annual ACM symposium on Theory of computing - STOC '06  
We show that entangled quantum measurements on at least Ω(n log n) coset states are necessary to get useful information for the case of graph isomorphism, matching an information theoretic upper bound.  ...  A common feature of the resulting algorithms is the use of quantum coset states, which encode the hidden subgroup.  ...  from the reduction of rigid-graph isomorphism requires superpolynomially many coset states.  ... 
doi:10.1145/1132516.1132603 dblp:conf/stoc/HallgrenMRRS06 fatcat:wwqoz6j5jzcmhgtrq7uc4ggvka

Limitations of quantum coset states for graph isomorphism

Sean Hallgren, Cristopher Moore, Martin Rötteler, Alexander Russell, Pranab Sen
2010 Journal of the ACM  
We show that entangled quantum measurements on at least Ω(n log n) coset states are necessary to get useful information for the case of graph isomorphism, matching an information theoretic upper bound.  ...  A common feature of the resulting algorithms is the use of quantum coset states, which encode the hidden subgroup.  ...  Acknowledgments We thank Andrew Childs, Frédéric Magniez and Umesh Vazirani for helpful discussions and comments.  ... 
doi:10.1145/1857914.1857918 fatcat:7hybkteqx5bkzhafo25i733sri

Quantum Fourier Sampling is Guaranteed to Fail to Compute Automorphism Groups of Easy Graphs [article]

Omar Shehab, Samuel J. Lomonaco Jr
2017 arXiv   pre-print
approach is guaranteed to always fail irrespective of the size of the graphs no matter how many copies of coset states are used.  ...  for the arbitrarily large classes of graph isomorphism problems.  ...  It was already proved in [35] that single coset state can provide only exponentially less information for the graph isomorphism problem.  ... 
arXiv:1705.00760v2 fatcat:mr5sofsp65ekvplz7ynv46nura

Degeneracy and its impact on the decoding of sparse quantum codes

Patricio Fuentes, Josu Etxezarreta Martinez, Pedro M. Crespo, Javier Garcia-Frias
2021 IEEE Access  
However, the axioms of quantum mechanics establish that direct measurement of a quantum state will force its superposition state to collapse, which would result in the loss of the information regarding  ...  If the encoded quantum state, codeword|ψ , is transmitted through an arbitrary quantum channel Ψ, a corrupted quantum state|ψ Ψ will be obtained at the channel output.  ... 
doi:10.1109/access.2021.3089829 fatcat:s3dfmpvo6rathodpgctwgnfdw4

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.  ...  As a consequence of the bounds, we show that the sample complexity for both of the decision and identification versions is Θ(||/ p) for a candidate set of hidden subgroups in the case that the candidate  ...  the graph isomorphism problem.  ... 
arXiv:quant-ph/0604174v3 fatcat:o5yz5vlv5newfpvszzvh4xrqdq

LIMDD A Decision Diagram for Simulation of Quantum Computing Including Stabilizer States [article]

Lieuwe Vinkhuijzen, Tim Coopmans, David Elkouss, Vedran Dunjko, Alfons Laarman
2022 arXiv   pre-print
Efficient methods for the representation and simulation of quantum states and quantum operations are crucial for the optimization of quantum circuits.  ...  Decision diagrams (DDs), a well-studied data structure originally used to represent Boolean functions, have proven capable of capturing relevant aspects of quantum systems, but their limits are not well  ...  The third and fourth author are funded by the Netherlands Organization for Scientific Research (NWO/OCW), as part of the Quantum Software Consortium program (Project No. 024.003.037/3368).  ... 
arXiv:2108.00931v3 fatcat:w4dv7lp7znfsba4chdra6tfgsq

Location of quantum information in additive graph codes

Vlad Gheorghiu, Shiang Yong Looi, Robert B. Griffiths
2010 Physical Review A. Atomic, Molecular, and Optical Physics  
The location of quantum information in various subsets of the qudit carriers of an additive graph code is discussed using a collection of operators on the coding space which form what we call the information  ...  Partial traces of these operators down to a particular subset of carriers provide an isomorphism of a subgroup of the information group, and this gives a precise characterization of what kinds of information  ...  We use (C, |G ) to denote the corresponding graph code. For more details about graph states and graph codes for arbitrary D, see [8] .  ... 
doi:10.1103/physreva.81.032326 fatcat:gnyos3vozndtfhjp5qktbblfde

Local Hamiltonians with Approximation-Robust Entanglement [article]

Lior Eldar
2017 arXiv   pre-print
This would imply that any quantum state with energy at most, say 0.05 of the total available energy of the Hamiltonian, would be highly-entangled.  ...  In recent years however, and motivated, in part, by a quest for a quantum analog of the PCP theorem researches have tried to establish whether or not we can preserve quantum entanglement at "constant"  ...  The author thanks the Templeton foundation for they support of this work.  ... 
arXiv:1503.02269v4 fatcat:v5y3bnqcvjainclhvgskoxl3jm

Hidden Translation and Translating Coset in Quantum Computing

Katalin Friedl, Gábor Ivanyos, Frédéric Magniez, Miklos Santha, Pranab Sen
2014 SIAM journal on computing (Print)  
solvable group G is reducible to instances of Translating Coset in G/N and N, for appropriate normal subgroups N of G.  ...  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  ...  We are also grateful to Cris Moore, Csaba Schneider and Yong Zhang for helpful discussions regarding a previous version of this article, and for pointing out reference [18] .  ... 
doi:10.1137/130907203 fatcat:rwapyh6j7ffddp5phted42l4re

ON THE COMPLEXITY OF THE HIDDEN SUBGROUP PROBLEM

STEPHEN FENNER, YONG ZHANG
2013 International Journal of Foundations of Computer Science  
We show that several problems that figure prominently in quantum computing, including Hidden Coset, Hidden Shift, and Orbit Coset, are equivalent or reducible to Hidden Subgroup.  ...  For Hidden Subgroup over dihedral groups, such an equivalence can be obtained if the order of the group is smooth.  ...  Acknowledgments We thank Andrew Childs and Wim van Dam for valuable comments on a preliminary version of this paper. We also thank anonymous referees for helpful suggestions.  ... 
doi:10.1142/s0129054113500305 fatcat:jpajordpxfg4xnvyot2jxf5zra

Error Correction for Reliable Quantum Computing [article]

Patricio Fuentes
2022 arXiv   pre-print
Stabilizer codes, a particular subclass of quantum codes, have enabled fast progress in the field of quantum error correction by allowing parallels to be drawn with the widely studied field of classical  ...  This has resulted in the construction of the quantum counterparts of well-known capacity-approaching classical codes like sparse codes and quantum turbo codes.  ...  For instance, if instead of estimating the value of the depolarizing probability we fix p to p * = 0.127, which corresponds to the Hashing limit for a R Q = 1 4 quantum code, Figure 7 .1 shows that for  ... 
arXiv:2202.08599v1 fatcat:wnjowybbu5ad5m6r2nokqo43xa

The Hidden Subgroup Problem [article]

Frédéric Wang
2010 arXiv   pre-print
We recall how the problem provides a framework for efficient quantum algorithms and present the standard methods based on coset sampling.  ...  We also give a reduction of rigid graph isomorphism problem to HSP over the alternating group. For this group and other simple groups, we propose maximal subgroup reduction as a possible approach.  ...  Acknowledgments Introduction Acknowledgments Jeg vil gerne sige tak til Ivan Damgård, for at have laert mig Quantum Information Processing på først semester og have accepteret at blive min vejleder  ... 
arXiv:1008.0010v1 fatcat:mwrv3xffmzdsdg3izn6nuximym

Quantum algorithm for structured problems [article]

Hefeng Wang
2022 arXiv   pre-print
In this work, we apply a quantum algorithm of finding the ground state of a Hamiltonian via a multi-step quantum computation process (arxiv: 1912.06959) for solving some structured problems, where the  ...  Hamiltonians via quantum resonant transitions sequentially, finally obtain the ground state of the problem Hamiltonian.  ...  To find out the exact group elements for the isomorphism of two graphs, we can use the division set v 1 = (2n)!  ... 
arXiv:2204.03295v1 fatcat:wa7plrubhjbxfdqo3jrfadt5fe

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.  ...  As the best-known example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers.  ...  Acknowledgments We thank Sean Hallgren for discussions of algorithms for number fields.  ... 
doi:10.1103/revmodphys.82.1 fatcat:z6xqtmdvdffrdnyo26wsjp4vee
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