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27
The symmetric group defies strong Fourier sampling
, 2005
"... We resolve the question of whether Fourier sampling can efficiently solve the hidden subgroup problem. Specifically, we show that the hidden subgroup problem over the symmetric group cannot be efficiently solved by strong Fourier sampling, even if one may perform an arbitrary POVM on the coset state ..."
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Cited by 27 (10 self)
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We resolve the question of whether Fourier sampling can efficiently solve the hidden subgroup problem. Specifically, we show that the hidden subgroup problem over the symmetric group cannot be efficiently solved by strong Fourier sampling, even if one may perform an arbitrary POVM on the coset state. These results apply to the special case relevant to the Graph Isomorphism problem. 1 Introduction: the hidden subgroup problem Many problems of interest in quantum computing can be reduced to an instance of the Hidden Subgroup Problem (HSP). We are given a group G and a function f with the promise that, for some subgroup H ⊆ G, f is invariant precisely under translation by H: that is, f is constant on the cosets of H and takes distinct values on distinct cosets. We then wish to determine the subgroup H by querying f.
On the impossibility of a quantum sieve algorithm for Graph Isomorphism
, 2006
"... ABSTRACT. It is known that any quantum algorithm for Graph Isomorphism that works within the framework of the hidden subgroup problem (HSP) must perform highly entangled measurements across Ω(n log n) coset states. One of the only known models for how such a measurement could be carried out efficien ..."
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Cited by 16 (2 self)
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ABSTRACT. It is known that any quantum algorithm for Graph Isomorphism that works within the framework of the hidden subgroup problem (HSP) must perform highly entangled measurements across Ω(n log n) coset states. One of the only known models for how such a measurement could be carried out efficiently is Kuperberg’s algorithm for the HSP in the dihedral group, in which quantum states are adaptively combined and measured according to the decomposition of tensor products into irreducible representations. This “quantum sieve ” starts with coset states, and works its way down towards representations whose probabilities differ depending on, for example, whether the hidden subgroup is trivial or nontrivial. In this paper we show that no such approach can produce a polynomialtime quantum algorithm for Graph Isomorphism. Specifically, we consider the natural reduction of Graph Isomorphism to the HSP over the the wreath product Sn ≀ Z2. Using a recently proved bound on the irreducible characters of Sn, we show that no algorithm in this family can solve Graph Isomorphism in less than e Ω( √ n) time, no matter what adaptive rule it uses to select and combine quantum states. In particular, algorithms of this type can offer essentially no improvement over the best known classical algorithms, which run in time e O( √ nlog n) 1.
For distinguishing conjugate hidden subgroups, the pretty good measurement is as good as it gets
"... Recently Bacon, Childs and van Dam showed that the “pretty good measurement ” (PGM) is optimal for the Hidden Subgroup Problem on the dihedral group Dn in the case where the hidden subgroup is chosen uniformly from the n involutions. We show that, for any group and any subgroup H, the PGM is the opt ..."
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Cited by 11 (6 self)
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Recently Bacon, Childs and van Dam showed that the “pretty good measurement ” (PGM) is optimal for the Hidden Subgroup Problem on the dihedral group Dn in the case where the hidden subgroup is chosen uniformly from the n involutions. We show that, for any group and any subgroup H, the PGM is the optimal oneregister experiment in the case where the hidden subgroup is a uniformly random conjugate of H. We go on to show that when H forms a Gel’fand pair with its parent group, the PGM is the optimal measurement for any number of registers. In both cases we bound the probability that the optimal measurement succeeds. This generalizes the case of the dihedral group, and includes a number of other examples of interest. 1 The Hidden Conjugate Problem Consider the following special case of the Hidden Subgroup Problem, called the Hidden Conjugate Problem in [16]. Let G be a group, and H a nonnormal subgroup of G; denote conjugates of H as H g = g −1 Hg. Then we are promised that the hidden subgroup is H g for some g, and our goal is to find out which one. The usual approach is to prepare a uniform superposition over the group, entangle the group element with a second register by calculating or querying the oracle function, and then measure the oracle function. This yields a uniform superposition over a random left coset of the hidden subgroup, cH g 〉 = 1 ∑ √ ch 〉. H Rather than viewing this as a pure state where c is random, we may treat this as a classical mixture over left cosets, giving the mixed state with density matrix ρg = 1 ∑ cH G g 〉 〈cH g . (1.1) c∈G We then wish to find a positive operatorvalued measurement (POVM) to identify g. A POVM consists of a set of positive measurement operators {Ei} that obey the completeness condition Ei = 1. (1.2)
Explicit Multiregister Measurements for Hidden . . .
"... We present an explicit measurement in the Fourier basis that solves an important case of the Hidden Subgroup Problem, including the case to which Graph Isomorphism reduces. This entangled measurement uses k = log 2 G registers, and each of the 2^k subsets of the registers contributes some informat ..."
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Cited by 7 (6 self)
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We present an explicit measurement in the Fourier basis that solves an important case of the Hidden Subgroup Problem, including the case to which Graph Isomorphism reduces. This entangled measurement uses k = log 2 G registers, and each of the 2^k subsets of the registers contributes some information.
On the quantum hardness of solving isomorphism problems as nonabelian hidden shift problems
, 2005
"... ..."
Quantum Algorithm for Identifying Hidden Polynomial Function Graphs
"... We introduce the Hidden Polynomial Function Graph Problem as a natural generalization of an abelian Hidden Subgroup Problem (HSP) where the subgroups and their cosets correspond to graphs of linear functions over a finite field F with d elements. For the Hidden Polynomial Function Graph Problem the ..."
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Cited by 6 (0 self)
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We introduce the Hidden Polynomial Function Graph Problem as a natural generalization of an abelian Hidden Subgroup Problem (HSP) where the subgroups and their cosets correspond to graphs of linear functions over a finite field F with d elements. For the Hidden Polynomial Function Graph Problem the functions are not restricted to be linear but can also be mvariate polynomial functions of total degree n ≥ 2. For fixed m and bounded n the problem is hard on a classical computer as the black box query complexity is polynomial in d. In contrast, we reduce it to a quantum state identification problem so that its query complexity is n m +n m−1 +...+n, independent of d. We derive an efficient measurement for distinguishing the resulting quantum states provided that the characteristic of F is sufficiently large. Its success probability and implementation are closely related to a classical problem involving polynomial equations.
How a ClebschGordan transform helps to solve the Heisenberg hidden subgroup problem
 Quantum Information & Computation
"... It has recently been shown that quantum computers can efficiently solve the Heisenberg hidden subgroup problem, a problem whose classical query complexity is exponential. This quantum algorithm was discovered within the framework of using pretty good measurements for obtaining optimal measurements i ..."
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Cited by 3 (0 self)
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It has recently been shown that quantum computers can efficiently solve the Heisenberg hidden subgroup problem, a problem whose classical query complexity is exponential. This quantum algorithm was discovered within the framework of using pretty good measurements for obtaining optimal measurements in the hidden subgroup problem. Here we show how to solve the Heisenberg hidden subgroup problem using arguments based instead on the symmetry of certain hidden subgroup states. The symmetry we consider leads naturally to a unitary transform known as the ClebschGordan transform over the Heisenberg group. This gives a new representation theoretic explanation for the pretty good measurement derived algorithm for efficiently solving the Heisenberg hidden subgroup problem and provides evidence that ClebschGordan transforms over finite groups are a new primitive in quantum algorithm design.
Efficient Quantum Algorithm for Hidden Quadratic and Cubic Polynomial Function Graphs, arXiv: quantph/0703195v3
, 2007
"... We introduce the Hidden Polynomial Function Graph Problem as a natural generalization of an abelian Hidden Subgroup Problem (HSP) where the subgroups and their cosets correspond to graphs of linear functions over the finite field Fp. For the Hidden Polynomial Function Graph Problem the functions are ..."
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Cited by 2 (1 self)
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We introduce the Hidden Polynomial Function Graph Problem as a natural generalization of an abelian Hidden Subgroup Problem (HSP) where the subgroups and their cosets correspond to graphs of linear functions over the finite field Fp. For the Hidden Polynomial Function Graph Problem the functions are not restricted to be linear but can also be multivariate polynomial functions of higher degree. For a fixed number of indeterminates and bounded total degree the Hidden Polynomial Function Graph Problem is hard on a classical computer as its black box query complexity is polynomial in p. In contrast, this problem can be reduced to a quantum state identification problem so that the resulting quantum query complexity does not depend on p. For univariate polynomials we construct a von Neumann measurement for distinguishing the states. We relate the success probability and the implementation of this measurement to certain classical problems involving polynomial equations. We present an efficient algorithm for hidden quadratic and cubic function graphs by establishing that the success probability of the measurement is lower bounded by a constant and that it can be implemented efficiently. 1
Random Measurement Bases, Quantum State Distinction and Applications to the Hidden Subgroup Problem
 Algorithmica
"... Abstract We show that measuring any two low rank quantum states in a random orthonormal basis gives, with high probability, two probability distributions having total variation distance at least a universal constant times the Frobenius distance between the two states. This implies that for any finit ..."
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Cited by 2 (1 self)
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Abstract We show that measuring any two low rank quantum states in a random orthonormal basis gives, with high probability, two probability distributions having total variation distance at least a universal constant times the Frobenius distance between the two states. This implies that for any finite ensemble of quantum states there is a single POVM that distinguishes between every pair of states from the ensemble by at least a constant times their Frobenius distance; in fact, with high probability a random POVM, under a suitable definition of randomness, suffices. There are examples of ensembles with constant pairwise trace distance where a single POVM cannot distinguish pairs of states by much better than their Frobenius distance, including the important ensemble of coset states of hidden subgroups of the symmetric group
Tight Results on Multiregister Fourier Sampling: Quantum Measurements for Graph Isomorphism Require
, 2008
"... We establish a general method for proving bounds on the information that can be extracted via arbitrary entangled measurements on tensor products of hidden subgroup coset states. When applied to the symmetric group, the method yields an Ω(n log n) lower bound on the number of coset states over which ..."
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Cited by 1 (1 self)
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We establish a general method for proving bounds on the information that can be extracted via arbitrary entangled measurements on tensor products of hidden subgroup coset states. When applied to the symmetric group, the method yields an Ω(n log n) lower bound on the number of coset states over which we must perform an entangled measurement in order to obtain nonnegligible information about a hidden involution. These results are tight to within a multiplicative constant and apply, in particular, to the case relevant for the Graph Isomorphism problem. Part of our proof was obtained after learning from Hallgren, Rötteler, and Sen that they had obtained similar results. 1 1 Introduction: the hidden subgroup problem Many problems of interest in quantum computing can be reduced to an instance of the Hidden Subgroup Problem (HSP). This is the problem of determining a subgroup H of a group G given oracle access to a function f: G → S with the property that f(g) = f(hg) ⇔ h ∈ H. Equivalently, f is constant on the cosets of H and takes distinct values on distinct cosets.