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65
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 75 (9 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
Limitations of Quantum Advice and OneWay Communication
 Theory of Computing
, 2004
"... Although a quantum state requires exponentially many classical bits to describe, the laws of quantum mechanics impose severe restrictions on how that state can be accessed. This paper shows in three settings that quantum messages have only limited advantages over classical ones. ..."
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Cited by 59 (15 self)
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Although a quantum state requires exponentially many classical bits to describe, the laws of quantum mechanics impose severe restrictions on how that state can be accessed. This paper shows in three settings that quantum messages have only limited advantages over classical ones.
2006, Quantum verification of matrix products
 Proceedings of the 17th ACMSIAM Symposium on Discrete Algorithms
"... We present a quantum algorithm that verifies a product of two n×n matrices over any integral domain with bounded error in worstcase time O(n 5/3) and expected time O(n 5/3 / min(w, √ n) 1/3), where w is the number of wrong entries. This improves the previous best algorithm [ABH + 02] that runs in ..."
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Cited by 48 (0 self)
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We present a quantum algorithm that verifies a product of two n×n matrices over any integral domain with bounded error in worstcase time O(n 5/3) and expected time O(n 5/3 / min(w, √ n) 1/3), where w is the number of wrong entries. This improves the previous best algorithm [ABH + 02] that runs in time O(n 7/4). We also present a quantum matrix multiplication algorithm that is efficient when the result has few nonzero entries. 1
On computation and communication with small bias
 In Proc. of the 22nd Conf. on Computational Complexity (CCC
, 2007
"... We present two results for computational models that allow error probabilities close to 1/2. First, most computational complexity classes have an analogous class in communication complexity. The class PP in fact has two, a version with weakly restricted bias called PP cc, and a version with unrestri ..."
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Cited by 47 (3 self)
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We present two results for computational models that allow error probabilities close to 1/2. First, most computational complexity classes have an analogous class in communication complexity. The class PP in fact has two, a version with weakly restricted bias called PP cc, and a version with unrestricted bias called UPP cc. Ever since their introduction by Babai, Frankl, and Simon in 1986, it has been open whether these classes are the same. We show that PP cc � UPP cc. Our proof combines a query complexity separation due to Beigel with a technique of Razborov that translates the acceptance probability of quantum protocols to polynomials. Second, we study how small the bias of minimaldegree polynomials that signrepresent Boolean functions needs to be. We show that the worstcase bias is at worst doubleexponentially small in the signdegree (which was very recently shown to be optimal by Podolski), while the averagecase bias can be made singleexponentially small in the signdegree (which we show to be close to optimal). 1
The pattern matrix method for lower bounds on quantum communication
 In Proc. of the 40th Symposium on Theory of Computing (STOC
, 2007
"... In a breakthrough result, Razborov (2003) gave optimal lower bounds on the communication complexity of every function f of the form f (x, y) = D(x ∧ y) for some D: {0, 1,..., n} → {0, 1}, in the boundederror quantum model with and without prior entanglement. This was proved by the multidimension ..."
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Cited by 46 (9 self)
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In a breakthrough result, Razborov (2003) gave optimal lower bounds on the communication complexity of every function f of the form f (x, y) = D(x ∧ y) for some D: {0, 1,..., n} → {0, 1}, in the boundederror quantum model with and without prior entanglement. This was proved by the multidimensional discrepancy method. We give an entirely different proof of Razborov’s result, using the original, onedimensional discrepancy method. This refutes the commonly held intuition (Razborov 2003) that the original discrepancy method fails for functions such as disjointness. More importantly, our communication lower bounds hold for a much broader class of functions for which no methods were available. Namely, fix an arbitrary function f: {0, 1} n/4 → {0, 1} and let A be the Boolean matrix whose columns are each an application of f to some subset of the variables x1, x2,..., xn. We prove that the communication complexity of A in the boundederror quantum model with and without prior entanglement is Ω(d), where d is the approximate degree of f. From this result, Razborov’s lower bounds follow easily. Our result also establishes a large new class of total Boolean functions whose quantum communication complexity (regardless of prior entanglement) is at best polynomially smaller than their classical complexity. Our proof method is a novel combination of two ingredients. The first is a certain equivalence of approximation and orthogonality in Euclidean nspace, which follows by linearprogramming duality. The second is a new construction of suitably structured matrices with low spectral norm, the pattern matrices, which we realize using matrix analysis and the Fourier transform over Z n 2. The method of this paper has recently inspired important progress in multiparty communication complexity. 1
A hypercontractive inequality for matrixvalued functions with applications to quantum computing and LDCs
"... The BonamiBeckner hypercontractive inequality is a powerful tool in Fourier analysis of realvalued functions on the Boolean cube. In this paper we present a version of this inequality for matrixvalued functions on the Boolean cube. Its proof is based on a powerful inequality by Ball, Carlen, and ..."
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Cited by 39 (3 self)
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The BonamiBeckner hypercontractive inequality is a powerful tool in Fourier analysis of realvalued functions on the Boolean cube. In this paper we present a version of this inequality for matrixvalued functions on the Boolean cube. Its proof is based on a powerful inequality by Ball, Carlen, and Lieb. We also present a number of applications. First, we analyze maps that encode n classical bits into m qubits, in such a way that each set of k bits can be recovered with some probability by an appropriate measurement on the quantum encoding; we show that if m<0.7n, then the success probability is exponentially small in k. This result may be viewed as a direct product version of Nayak’s quantum random access code bound. It in turn implies strong direct product theorems for the oneway quantum communication complexity of Disjointness and other problems. Second, we prove that errorcorrecting codes that are locally decodable with 2 queries require length exponential in the length of the encoded string. This gives what is arguably the first “nonquantum” proof of a result originally derived by Kerenidis and de Wolf using quantum information theory.
A direct product theorem for discrepancy
 In Proceedings of the 23rd IEEE Conference on Computational Complexity. IEEE
, 2008
"... Discrepancy is a versatile bound in communication complexity which can be used to show lower bounds in the distributional, randomized, quantum, and even unbounded error models of communication. We show an optimal product theorem for discrepancy, namely that for any two Boolean functions f, g, disc(f ..."
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Cited by 32 (9 self)
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Discrepancy is a versatile bound in communication complexity which can be used to show lower bounds in the distributional, randomized, quantum, and even unbounded error models of communication. We show an optimal product theorem for discrepancy, namely that for any two Boolean functions f, g, disc(f ⊕ g) = Θ(disc(f)disc(g)). As a consequence we obtain a strong direct product theorem for distributional complexity, and direct sum theorems for worstcase complexity, for bounds shown by the discrepancy method. Our results resolve an open problem of Shaltiel (2003) who showed a weaker product theorem for discrepancy with respect to the uniform distribution, disc U ⊗k(f ⊗k) = O(discU(f)) k/3. The main tool for our results is semidefinite programming, in particular a recent characterization of discrepancy in terms of a semidefinite programming quantity by Linial and Shraibman (2006). 1
A strong direct product theorem for disjointness
 In 42nd ACM Symposium on Theory of Computing (STOC
, 2010
"... A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then the overall success probability will be exponentially small in k. We establish such a theorem for the randomized communication co ..."
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Cited by 28 (1 self)
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A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then the overall success probability will be exponentially small in k. We establish such a theorem for the randomized communication complexity of the Disjointness problem, i.e., with communication const · kn the success probability of solving k instances can only be exponentially small in k. We show that this bound even holds in an AM communication protocol with limited ambiguity. The main result implies a new lower bound for Disjointness in a restricted 3player NOF protocol, and optimal communicationspace tradeoffs for Boolean matrix product. Our main result follows from a solution to the dual of a linear programming problem, whose feasibility comes from a socalled Intersection Sampling Lemma that generalizes a result by Razborov [Raz92]. We also discuss a new lower bound technique for randomized communication complexity called the generalized rectangle bound that we use in our proof. 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 24 (3 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
Lower bounds on quantum query complexity
 EATCS BULLETIN
, 2005
"... Shor’s and Grover’s famous quantum algorithms for factoring and searching show that quantum computers can solve certain computational problems significantly faster than any classical computer. We discuss here what quantum computers cannot do, and specifically how to prove limits on their computation ..."
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Cited by 23 (2 self)
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Shor’s and Grover’s famous quantum algorithms for factoring and searching show that quantum computers can solve certain computational problems significantly faster than any classical computer. We discuss here what quantum computers cannot do, and specifically how to prove limits on their computational power. We cover the main known techniques for proving lower bounds, and exemplify and compare the methods.