Results 1  10
of
41
Quantum walk algorithms for element distinctness
 In: 45th Annual IEEE Symposium on Foundations of Computer Science, OCT 1719, 2004. IEEE Computer Society Press, Los Alamitos, CA
, 2004
"... We use quantum walks to construct a new quantum algorithm for element distinctness and its generalization. For element distinctness (the problem of finding two equal items among N given items), we get an O(N 2/3) query quantum algorithm. This improves the previous O(N 3/4) quantum algorithm of Buhrm ..."
Abstract

Cited by 93 (9 self)
 Add to MetaCart
We use quantum walks to construct a new quantum algorithm for element distinctness and its generalization. For element distinctness (the problem of finding two equal items among N given items), we get an O(N 2/3) query quantum algorithm. This improves the previous O(N 3/4) quantum algorithm of Buhrman et al. [11] and matches the lower bound by [1]. We also give an O(N k/(k+1) ) query quantum algorithm for the generalization of element distinctness in which we have to find k equal items among N items. 1
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 ..."
Abstract

Cited by 58 (11 self)
 Add to MetaCart
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
Coins make quantum walks faster
 Proceedings of SODA’05. Also quantph/0402107
"... We show how to search N items arranged on a √ N × √ N grid in time O ( √ N log N), using a discrete time quantum walk. This result for the first time exhibits a significant difference between discrete time and continuous time walks without coin degrees of freedom, since it has been shown recently ..."
Abstract

Cited by 31 (5 self)
 Add to MetaCart
We show how to search N items arranged on a √ N × √ N grid in time O ( √ N log N), using a discrete time quantum walk. This result for the first time exhibits a significant difference between discrete time and continuous time walks without coin degrees of freedom, since it has been shown recently that such a continuous time walk needs time Ω(N) to perform the same task. Our result furthermore improves on a previous bound for quantum local search by Aaronson and Ambainis. We generalize our result to 3 and more dimensions where the walk yields the optimal performance of O ( √ N) and give several extensions of quantum walk search algorithms for general graphs. The coinflip operation needs to be chosen judiciously: we show that another “natural ” choice of coin gives a walk that takes Ω(N) steps. We also show that in 2 dimensions it is sufficient to have a twodimensional coinspace to achieve the time O ( √ N log N). 1
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 ..."
Abstract

Cited by 29 (5 self)
 Add to MetaCart
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 complexity of testing group commutativity
 Proceedings of ICALP’05
, 2005
"... Abstract. We consider the problem of testing the commutativity of a blackbox group specified by its k generators. The complexity (in terms of k) of this problem was first considered by Pak, who gave a randomized algorithm involving O(k) group operations. We construct a quite optimal quantum algorit ..."
Abstract

Cited by 26 (5 self)
 Add to MetaCart
Abstract. We consider the problem of testing the commutativity of a blackbox group specified by its k generators. The complexity (in terms of k) of this problem was first considered by Pak, who gave a randomized algorithm involving O(k) group operations. We construct a quite optimal quantum algorithm for this problem whose complexity is in Õ(k2/3). The algorithm uses and highlights the power of the quantization method of Szegedy. For the lower bound of Ω(k 2/3), we introduce a new technique of reduction for quantum query complexity. Along the way, we prove the optimality of the algorithm of Pak for the randomized model. 1
Search via quantum walk
 LOGIC PROGRAMMING, PROC. OF THE 1994 INT. SYMP
, 2007
"... We propose a new method for designing quantum search algorithms for finding a “marked ” element in the state space of a classical Markov chain. The algorithm is based on a quantum walk à la Szegedy [24] that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimat ..."
Abstract

Cited by 25 (7 self)
 Add to MetaCart
We propose a new method for designing quantum search algorithms for finding a “marked ” element in the state space of a classical Markov chain. The algorithm is based on a quantum walk à la Szegedy [24] that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator. This operator is then used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis [6] and Szegedy [24]. Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chain. In addition, it is conceptually simple, avoids several technical difficulties in the previous analyses, and leads to improvements in various aspects of several algorithms based on quantum walk.
On the power of Ambainis’s lower bounds
 Theoretical Computer Science, 339(23):241– 256, 2005. Earlier version in ICALP’04. 569 Copyright © by SIAM. Unauthorized
"... The polynomial method and Ambainis’s lower bound method are two main quantum lower bound techniques. Recently Ambainis showed that the polynomial method is not tight. The present paper aims at studying the limitation of Ambainis’s lower bounds. We first give a generalization of the three known Ambai ..."
Abstract

Cited by 24 (0 self)
 Add to MetaCart
The polynomial method and Ambainis’s lower bound method are two main quantum lower bound techniques. Recently Ambainis showed that the polynomial method is not tight. The present paper aims at studying the limitation of Ambainis’s lower bounds. We first give a generalization of the three known Ambainis’s lower bound theorems. Then it is shown that all these four Ambainis’s lower bounds have an upper bound, which is in terms of certificate complexity. This implies that for some problems such as TRIANGLE, kCLIQUE, and BIPARTITE/GRAPH MATCHING whose quantum query complexities are still open, the best known lower bounds cannot be further improved by using Ambainis’s techniques. Another consequence is that all the Ambainis’s lower bounds are not tight. Finally, we show that for total functions, this upper bound for Ambainis’s lower bounds can be further improved. This also implies limitation of Ambainis’s method on some specific problems such as ANDOR TREE, whose precise quantum complexity is still unknown. 1
Negative weights make adversaries stronger. To appear in STOC’07
 Algorithmica
, 2002
"... The quantum adversary method is one of the most successful techniques for proving lower bounds on quantum query complexity. It gives optimal lower bounds for many problems, has application to classical complexity in formula size lower bounds, and is versatile with equivalent formulations in terms of ..."
Abstract

Cited by 19 (3 self)
 Add to MetaCart
The quantum adversary method is one of the most successful techniques for proving lower bounds on quantum query complexity. It gives optimal lower bounds for many problems, has application to classical complexity in formula size lower bounds, and is versatile with equivalent formulations in terms of weight schemes, eigenvalues, and Kolmogorov complexity. All these formulations are informationtheoretic and rely on the principle that if an algorithm successfully computes a function then, in particular, it is able to distinguish between inputs which map to different values. We present a stronger version of the adversary method which goes beyond this principle to make explicit use of the existence of a measurement in a successful algorithm which gives the correct answer, with high probability. We show that this new method, which we call ADV ±, has all the advantages of the old: it is a lower bound on boundederror quantum query complexity, its square is a lower bound on formula size, and it behaves well with respect to function composition. Moreover ADV ± is always at least as large as the adversary method ADV, and we show an example of a monotone function for which ADV ± (f) = Ω(ADV(f) 1.098). We also give examples showing that ADV ± does not face limitations of ADV such as the certificate complexity barrier and the property testing barrier. 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 ..."
Abstract

Cited by 14 (2 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
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