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17
Quantum Algorithms for Element Distinctness
 SIAM Journal of Computing
, 2001
"... We present several applications of quantum amplitude amplification to finding claws and collisions in ordered or unordered functions. Our algorithms generalize those of Brassard, Høyer, and Tapp, and imply an O(N 3/4 log N) quantum upper bound for the element distinctness problem in the comparison c ..."
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Cited by 58 (11 self)
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We present several applications of quantum amplitude amplification to finding claws and collisions in ordered or unordered functions. Our algorithms generalize those of Brassard, Høyer, and Tapp, and imply an O(N 3/4 log N) quantum upper bound for the element distinctness problem in the comparison complexity model. This contrasts with Θ(N log N) classical complexity. We also prove a lower bound of Ω ( √ N) comparisons for this problem and derive bounds for a number of related problems. 1
Lower bounds for randomized and quantum query complexity using Kolmogorov arguments
 in Proc. of the 19th IEEE Conference on Computational Complexity
, 2004
"... Abstract. We prove a very general lower bound technique for quantum and randomized query complexity that is easy to prove as well as to apply. To achieve this, we introduce the use of Kolmogorov complexity to query complexity. Our technique generalizes the weighted and unweighted methods of Ambainis ..."
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Cited by 40 (5 self)
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Abstract. We prove a very general lower bound technique for quantum and randomized query complexity that is easy to prove as well as to apply. To achieve this, we introduce the use of Kolmogorov complexity to query complexity. Our technique generalizes the weighted and unweighted methods of Ambainis and the spectral method of Barnum, Saks, and Szegedy. As an immediate consequence of our main theorem, it can be shown that adversary methods can only prove lower bounds for Boolean functions f in O(min ( √ nC0(f), √ nC1(f))), where C0,C1 is the certificate complexity and n is the size of the input.
All quantum adversary methods are equivalent
 THEORY OF COMPUTING
, 2006
"... The quantum adversary method is one of the most versatile lowerbound methods for quantum algorithms. We show that all known variants of this method are equivalent: spectral adversary (Barnum, Saks, and Szegedy, 2003), weighted adversary (Ambainis, 2003), strong weighted adversary (Zhang, 2005), an ..."
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Cited by 29 (5 self)
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The quantum adversary method is one of the most versatile lowerbound methods for quantum algorithms. We show that all known variants of this method are equivalent: spectral adversary (Barnum, Saks, and Szegedy, 2003), weighted adversary (Ambainis, 2003), strong weighted adversary (Zhang, 2005), and the Kolmogorov complexity adversary (Laplante and Magniez, 2004). We also present a few new equivalent formulations of the method. This shows that there is essentially one quantum adversary method. From our approach, all known limitations of these versions of the quantum adversary method easily follow.
Quantum and classical query complexities of local search are polynomially related
 In Proc. of 36th STOC
, 2004
"... Let f be an integer valued function on a finite set V. We call an undirected graph G(V, E) a neighborhood structure for f. The problem of finding a local minimum for f can be phrased as: for a fixed neighborhood structure G(V, E) find a vertex x ∈ V such that f(x) is not bigger than any value that f ..."
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Cited by 16 (1 self)
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Let f be an integer valued function on a finite set V. We call an undirected graph G(V, E) a neighborhood structure for f. The problem of finding a local minimum for f can be phrased as: for a fixed neighborhood structure G(V, E) find a vertex x ∈ V such that f(x) is not bigger than any value that f takes on some neighbor of x. The complexity of the algorithm is measured by the number of questions of the form “what is the value of f on x? ” We show that the deterministic, randomized and quantum query complexities of the problem are polynomially related. This generalizes earlier results of Aldous [4] and Aaronson [1] and solves the main open problem in [1]. 1
A new quantum lower bound method, with an application to strong direct product theorem for quantum search
, 2005
"... We give a new version of the adversary method for proving lower bounds on quantum query algorithms. The new method is based on analyzing the eigenspace structure of the problem at hand. We use it to prove a new and optimal strong direct product theorem for 2sided error quantum algorithms computing ..."
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Cited by 14 (2 self)
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We give a new version of the adversary method for proving lower bounds on quantum query algorithms. The new method is based on analyzing the eigenspace structure of the problem at hand. We use it to prove a new and optimal strong direct product theorem for 2sided error quantum algorithms computing k independent instances of a symmetric Boolean function: if the algorithm uses significantly less than k times the number of queries needed for one instance of the function, then its success probability is exponentially small in k. We also use the polynomial method to prove a direct product theorem for 1sided error algorithms for k threshold functions with a stronger bound on the success probability. Finally, we present a quantum algorithm for evaluating solutions to systems of linear inequalities, and use our direct product theorems to show that the timespace tradeoff of this algorithm is close to optimal. Categories and Subject Descriptors F.1.2 [Computation by Abstract Devices]: Modes of Computation; F.1.3 [Computation by Abstract Devices]: Complexity Measures and Classes—Relations among complexity
Quantum Search Algorithms
, 2005
"... We review some of quantum algorithms for search problems: Grover’s search algorithm, its generalization to amplitude amplification, the applications of amplitude amplification to various problems and the recent quantum algorithms based on quantum walks. 1 ..."
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Cited by 13 (1 self)
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We review some of quantum algorithms for search problems: Grover’s search algorithm, its generalization to amplitude amplification, the applications of amplitude amplification to various problems and the recent quantum algorithms based on quantum walks. 1
Spalek, Spanprogrambased quantum algorithm for evaluating formulas
 Proc. 40th ACM Symposium on Theory of Computing
, 2008
"... We present a timeefficient and queryoptimal quantum algorithm for evaluating adversaryboundbalanced formulas on an extended gate set. The allowed gates include arbitrary two and threebit gates, as well as bounded fanin AND, OR, PARITY and EQUAL gates. The technique behind the formula evaluatio ..."
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Cited by 10 (3 self)
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We present a timeefficient and queryoptimal quantum algorithm for evaluating adversaryboundbalanced formulas on an extended gate set. The allowed gates include arbitrary two and threebit gates, as well as bounded fanin AND, OR, PARITY and EQUAL gates. The technique behind the formula evaluation algorithm is a new framework for quantum algorithms based on span programs. For example, the classical complexity of evaluating the balanced ternary majority formula is unknown, and the natural generalization of the standard balanced ANDOR formula evaluation algorithm is known to be suboptimal. In contrast, a generalization of the optimal quantum {AND, OR, NOT} formula evaluation algorithm is optimal for evaluating the balanced ternary majority formula. 1
Quantum timespace tradeoffs for sorting
 Proceedings of 35th ACM STOC
, 2003
"... We investigate the complexity of sorting in the model of sequential quantum circuits. While it is known that a quantum algorithm based on comparisons alone cannot outperform classical sorting algorithms by more than a constant factor in time complexity, this is wrong in a space bounded setting. We o ..."
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Cited by 6 (1 self)
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We investigate the complexity of sorting in the model of sequential quantum circuits. While it is known that a quantum algorithm based on comparisons alone cannot outperform classical sorting algorithms by more than a constant factor in time complexity, this is wrong in a space bounded setting. We observe that for all storage bounds S, one can devise a quantum algorithm that sorts n numbers (using comparisons only) in time T = O(n
Roughness Theory
 In: Proceedings of the Sixth International Conference "Information Processing and Management of Uncertainty in KnowledgeBased Systems (IPMU '96)" [1
"... Let P =(X,
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Cited by 3 (1 self)
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Let P =(X, <P) be a partial order on a set of n elements X = {x1,x2, ·· ·,xn}. Define the quantum sorting problem QSORTP as: given n distinct numbers x1,x2, ·· ·,xn consistent with P, sort them by a quantum decision tree using comparisons of the form “xi: xj”. Let Qɛ(P) be the minimum number of queries used by any quantum decision tree for solving QSORTP with error less than ɛ (where 0 <ɛ<1/10 is fixed). It was proved by Høyer, Neerbek and Shi (Algorithmica 34 (2002), 429448) that, when P0 is the empty partial order, Qɛ(P0) ≥ Ω(n log n), i.e., the classical information lower bound holds for quantum decision trees when the input permutations are unrestricted. In this paper we show that the classical information lower bound holds, up to an additive linear term, for quantum decision trees for any partial order P. Precisely, we prove Qɛ(P) ≥ c log 2 e(P) − c ′ n where c, c ′> 0 are constants and e(P) is the number of linear orderings consistent with P. Our proof builds on an interesting connection between sorting and Körner’s graph entropy that was first noted and