## The Hidden Subgroup Problem and Quantum Computation Using Group Representations (2003)

Venue: | SIAM Journal on Computing |

Citations: | 18 - 2 self |

### BibTeX

@ARTICLE{Hallgren03thehidden,

author = {Sean Hallgren and Alexander Russell and Amnon Ta-shma},

title = {The Hidden Subgroup Problem and Quantum Computation Using Group Representations},

journal = {SIAM Journal on Computing},

year = {2003},

volume = {32},

pages = {2003}

}

### Years of Citing Articles

### OpenURL

### Abstract

The Hidden Subgroup Problem is the foundation of many quantum algorithms. An efficient solution is known for the problem over Abelian groups, employed by both Simon's algorithm and Shor's factoring and discrete log algorithms. The non-Abelian case is open; an efficient solution would give rise to an efficient quantum algorithm for Graph Isomorphism. We fully analyze a natural generalization of the Abelian case algorithm to the non-Abelian case. We show that the algorithm finds the normal core of the hidden subgroup, and that, in particular, normal subgroups can be found. We show, however, that this immediate generalization of the Abelian algorithm does not efficiently solve Graph Isomorphism. 1

### Citations

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Citation Context ...NDER RUSSELL, AND AMNON TA-SHMA Proof of Theorem 4.3. Let DH denote the probabilitydistribution over irreducible representations induced byExperiment 1.2. We now applya standard martingale bound (see =-=[20]) to pro-=-ve the theorem (based on Lemmas 4.2 and 4.1). Let σ1,... ,σk be independent random variables distributed according to DH with k = 4 log 2 |G|. Our goal is to show that Pr[Ns �= H G ] ≤ 2e − lo... |

1694 | The Probabilistic Method - Alon, Spencer - 2008 |

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Citation Context ...t stage (as the support of the first register in (1.1)). Thus, repetitions of this experiment result in the same distribution over ˆ G. We note that bythe principle of delayed measurement (see, e.g.,=-= [21]-=-), measuring the second register in the first step can in fact be delayed until the end of the experiment. It is well known that an efficient solution to the HSP for the symmetric group Sn gives, in p... |

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Citation Context ...antum algorithms, computational complexity, representation theory, finite groups AMS subject classifications. 81P68, 68Q17 DOI. 10.1137/S009753970139450X 1. Introduction. Peter Shor’s seminal articl=-=e [27]-=- presented efficient quantum algorithms for computing integer factorizations and discrete logarithms, problems thought to be intractable for classical computation models. A primaryingredient in these ... |

371 |
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Citation Context ...ory background. To define the Fourier transform (over a general group) we require the basic elements of representation theory, defined briefly below. For complete accounts, consult the books of Serre =-=[26] o-=-r Harris and Fulton [15]. Throughout, we let Id denote the d × d identitymatrix, dropping the subscript when it can be inferred from context. Linear representations. A representation ρ of a finite g... |

353 | On the power of quantum computation
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(Show Context)
Citation Context ...istributions induced on ρ in these two cases have exponentially small distance in total variation.) 1.1. Related work. The HSP plays a central role in most known quantum algorithms. Simon’s algorit=-=hm [28]-=- implicitlyinvolves distinguishing the trivial subgroup from an order 2 subgroup over the group Z n 2 . Furthermore, he has shown that a classical probabilistic oracle machine would require exponentia... |

155 |
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Citation Context ...cient algorithm for the abelian HSP using the Fourier transform is well known. Other methods have been applied to this same problem byMosca and Ekert [19]. Related problems have been studied byKitaev =-=[17], -=-who gave an algorithm using eigenvalue estimation for the abelian stabilizer problem, and Hallgren [13], who gave polynomial-time quantum algorithms for Pell’s equation and the principal ideal probl... |

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Citation Context ... ∼ = Z k 2 × B × Q, where Q = {±1, ±i, ±j, ±k} is the quaternion group and B is an abelian group with exponent b coprime with 2. For a detailed description of such groups, see Rotman’s excel=-=lent book [23]. We b-=-egin bybrieflydiscussing the case when G is abelian. If G is simplythe cyclic group Zn, the representations are the functions ρs : z ↦→ exp(2πisz/n), and the reconstruction algorithm, when it su... |

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(Show Context)
Citation Context ...plied to this same problem byMosca and Ekert [19]. Related problems have been studied byKitaev [17], who gave an algorithm using eigenvalue estimation for the abelian stabilizer problem, and Hallgren =-=[13], -=-who gave polynomial-time quantum algorithms for Pell’s equation and the principal ideal problem. As for computing the Fourier transform, Kitaev showed how to efficientlycompute the Fourier transform... |

74 | On Quantum Algorithms for Noncommutative Hidden Subgroups
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(Show Context)
Citation Context ...it is not possible to distinguish a 1-1 function from a 2-1 function, even with a quantum algorithm. Several specific nonabelian groups have been studied in the context of the HSP. Ettinger and Høyer=-= [7] gi-=-ve a solution for the HSP over the (nonabelian) dihedral group Dn using polynomiallymanymeasurements and exponential (classical) time. Rötteler and Beth [24] and Püschel, Rötteler, and Beth [22] ha... |

66 |
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Citation Context ...p S n gives, in particular, an efficient quantum algorithm for Graph Isomorphism. It is also known how to efficiently compute the Fourier transform over many non-Abelian groups, most notably over S n =-=[2]-=-. Nevertheless, until this work, there was no general understanding of the HSP over non-Abelian groups. In this paper we study a natural generalization of Algorithm 1.1 to non-Abelian groups. Namely, ... |

66 | Quantum mechanical algorithms for the nonabelian Hidden Subgroup Problem
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Citation Context ...sults for other specific classes of nonabelian groups. Ivanyos, Mangniez, and Santha [16] have shown how to solve certain nonabelian HSP instances using a reduction to the abelian case. Grigni et al. =-=[10]-=- independentlyshowed that measuring the representation is not enough for graph isomorphism, and theygive stronger negative results. Theyestablish the same bounds even when the row of the representatio... |

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Citation Context ...solve integer factorization and the discrete log problem. In addition to solving a special case of the HSP, he also solves specific cases when the underlying group is not even known. Boneh and Lipton =-=[3]-=- handle a case when a periodic function is not fixed on a coset. Hales and Hallgren [11, 12] generalize the results for the case when the underlying Abelian group is unknown, but an estimate is known ... |

59 |
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Citation Context ...nding a hidden subgroup over Sn cannot work using Experiment 1.2. Graph automorphism is the problem of determining whether a graph G has a nontrivial automorphism and is easier than graph isomorphism =-=[18]-=-. A natural special case occurs when the graph G consists of two disjoint connected rigid graphs G1,G2 (i.e., Aut(G1) = Aut(G2) ={e}). In this case there are two possibilities for the automorphism gro... |

59 | The hidden subgroup problem and eigenvalue estimation on a quantum computer, arXive e-print quant-ph/9903071
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Citation Context ... the underlying abelian group is unknown. The efficient algorithm for the abelian HSP using the Fourier transform is well known. Other methods have been applied to this same problem byMosca and Ekert =-=[19]-=-. Related problems have been studied byKitaev [17], who gave an algorithm using eigenvalue estimation for the abelian stabilizer problem, and Hallgren [13], who gave polynomial-time quantum algorithms... |

58 | Quantum lower bound for the collision problem
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Citation Context ...ynomial query complexity, giving an algorithm that makes an exponential number of measurements. On the other hand, if one considers arbitraryfunctions rather than those that arise from HSPs, Aaronson =-=[1]-=- shows that it is not possible to distinguish a 1-1 function from a 2-1 function, even with a quantum algorithm. Several specific nonabelian groups have been studied in the context of the HSP. Ettinge... |

49 | Fast parallel circuits for the quantum Fourier transform
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(Show Context)
Citation Context ...m for computing the Fourier transform over abelian groups was given byHales and Hallgren [12]. Shallow parallel circuits for approximating the Fourier transform have been given byCleve and Watrous in =-=[4].-=- Beals [2] showed how to efficiently compute the Fourier transform over the symmetric group Sn. For general groups, Ettinger, Høyer, and Knill [8] have shown that the HSP has polynomial query complex... |

49 |
Efficient quantum algorithms for some instances of the non-abelian hidden subgroup problem
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(Show Context)
Citation Context ...and exponential (classical) time. Rötteler and Beth [24] and Püschel, Rötteler, and Beth [22] have shown similar results for other specific classes of nonabelian groups. Ivanyos, Mangniez, and Sant=-=ha [16]-=- have shown how to solve certain nonabelian HSP instances using a reduction to the abelian case. Grigni et al. [10] independentlyshowed that measuring the representation is not enough for graph isomor... |

44 | Polynomial-Time Solution to the Hidden Subgroup Problem for a Class of Non-abelian Groups, Technical report, Quantum Physics e-Print archive
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Citation Context ...e context of the HSP. Ettinger and Høyer [7] give a solution for the HSP over the (nonabelian) dihedral group Dn using polynomiallymanymeasurements and exponential (classical) time. Rötteler and Bet=-=h [24] a-=-nd Püschel, Rötteler, and Beth [22] have shown similar results for other specific classes of nonabelian groups. Ivanyos, Mangniez, and Santha [16] have shown how to solve certain nonabelian HSP inst... |

38 | Normal subgroup reconstruction and quantum computation using group representations
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(Show Context)
Citation Context ...obability of sampling ρ in Experiment 5.1 when G1 �≈ G2, and let DI(ρ) be the probability when G1 ≈ G2. Then |DN −DI|1 ≤ 2 −Ω(n) . Proof. We present the proof from [10], which simplifi=-=es the proof of [14]. When G1 �≈ G-=-2, H = {e}, soDN(ρ) =d 2 ρ/n! byTheorem 3.2. When G1 ≈ G2, and G1 and G2 are both connected and rigid, H = {e, τ}. ByTheorem 3.2, DI(ρ) = |H| |G| dρ 〈χ1 χρ〉 H . The subgroup H has onlytw... |

37 | An Improved Quantum Fourier Transform Algorithm and Applications, FOCS 2000, (also available at http://www.cs.caltech.edu/~hallgren
- Hallgren, Hales
(Show Context)
Citation Context ... of the underlying group is not fully known. Other generalizations have been studied byBoneh and Lipton [3], focusing on cases when a periodic function is not fixed on a coset, and Hales and Hallgren =-=[11, 12]-=-, who generalized the results for the case when the underlying abelian group is unknown. The efficient algorithm for the abelian HSP using the Fourier transform is well known. Other methods have been ... |

31 | The Symmetric Group,” Wadsworth and Brooks/Cole - Sagan - 1991 |

30 | Fast quantum Fourier transforms for a class of non-Abelian groups
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(Show Context)
Citation Context ...yer [7] give a solution for the HSP over the (nonabelian) dihedral group Dn using polynomiallymanymeasurements and exponential (classical) time. Rötteler and Beth [24] and Püschel, Rötteler, and Be=-=th [22]-=- have shown similar results for other specific classes of nonabelian groups. Ivanyos, Mangniez, and Santha [16] have shown how to solve certain nonabelian HSP instances using a reduction to the abelia... |

19 | Quantum Fourier sampling simplified
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Citation Context ... of the underlying group is not fully known. Other generalizations have been studied byBoneh and Lipton [3], focusing on cases when a periodic function is not fixed on a coset, and Hales and Hallgren =-=[11, 12]-=-, who generalized the results for the case when the underlying abelian group is unknown. The efficient algorithm for the abelian HSP using the Fourier transform is well known. Other methods have been ... |

18 | A quantum observable for the graph isomorphism problem
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(Show Context)
Citation Context ...h representation. Theyalso show that the problem can be solved when the intersection of the normalizers of all subgroups of G is large. Other impossibilityresults have been given byEttinger and Høyer=-= [6, 5]-=-, determining whether any measurement can distinguish certain subgroup states. 2. Representation theory background. To define the Fourier transform (over a general group) we require the basic elements... |

12 | The computational complexity of solving equations over finite groups - Goldmann, Russell |

7 |
Representation theory, Grad
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Citation Context ...the Fourier transform (over a general group) we require the basic elements of representation theory, defined briefly below. For complete accounts, consult the books of Serre [26] or Harris and Fulton =-=[15]. T-=-hroughout, we let Id denote the d × d identitymatrix, dropping the subscript when it can be inferred from context. Linear representations. A representation ρ of a finite group G is a homomorphism ρ... |

1 |
Quantum State Detection via Elimination, Los Alamos preprint, quant-ph/9905099
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Citation Context ...h representation. Theyalso show that the problem can be solved when the intersection of the normalizers of all subgroups of G is large. Other impossibilityresults have been given byEttinger and Høyer=-= [6, 5]-=-, determining whether any measurement can distinguish certain subgroup states. 2. Representation theory background. To define the Fourier transform (over a general group) we require the basic elements... |

1 |
available online from http://lanl.arxiv.org/ abs/quant-ph/9901034
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Citation Context ...a (polynomial-size) family of representation kernels. See section 6, where the above approach is applied to solve the HSP for Hamiltonian groups, where all subgroups are normal. Note that it is known =-=[8]-=- that the HSP has polynomial (in log |G|) query complexityfor anysubgroup, though the onlyknown algorithm which achieves this uses an exponential number of quantum measurements and, hence, does not gi... |