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104
Polynomial degree vs. quantum query complexity
 Proceedings of FOCS’03
"... The degree of a polynomial representing (or approximating) a function f is a lower bound for the quantum query complexity of f. This observation has been a source of many lower bounds on quantum algorithms. It has been an open problem whether this lower bound is tight. We exhibit a function with pol ..."
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Cited by 81 (14 self)
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The degree of a polynomial representing (or approximating) a function f is a lower bound for the quantum query complexity of f. This observation has been a source of many lower bounds on quantum algorithms. It has been an open problem whether this lower bound is tight. We exhibit a function with polynomial degree M and quantum query complexity Ω(M 1.321...). This is the first superlinear separation between polynomial degree and quantum query complexity. The lower bound is shown by a new, more general version of quantum adversary method. 1
Quantum and Classical Strong Direct Product Theorems and Optimal TimeSpace Tradeoffs
 SIAM Journal on Computing
, 2004
"... 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 our overall success probability will be exponentially small in k. We establish such theorems for the classical as well as quantum ..."
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Cited by 65 (13 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 our overall success probability will be exponentially small in k. We establish such theorems for the classical as well as quantum query complexity of the OR function. This implies slightly weaker direct product results for all total functions. We prove a similar result for quantum communication protocols computing k instances of the Disjointness function. Our direct product theorems...
Lower bounds in communication complexity based on factorization norms
 In Proc. of the 39th Symposium on Theory of Computing (STOC
, 2007
"... We introduce a new method to derive lower bounds on randomized and quantum communication complexity. Our method is based on factorization norms, a notion from Banach Space theory. As we show, our bounds compare favorably with previously known bounds. Aside from the new results that we derive, our me ..."
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Cited by 60 (6 self)
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We introduce a new method to derive lower bounds on randomized and quantum communication complexity. Our method is based on factorization norms, a notion from Banach Space theory. As we show, our bounds compare favorably with previously known bounds. Aside from the new results that we derive, our method yields new and more transparent proofs of some known results as well. Among our new results we extend some known lower bounds to the realm of quantum communication complexity with entanglement. 1
Lower Bounds for Quantum Communication Complexity
 42nd IEEE Symposium on Foundations of Computer Science
"... Abstract. We prove lower bounds on the bounded error quantum communication complexity. Our methods are based on the Fourier transform of the considered functions. First we generalize a method for proving classical communication complexity lower bounds developed by Raz [35] to the quantum case. Apply ..."
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Cited by 55 (7 self)
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Abstract. We prove lower bounds on the bounded error quantum communication complexity. Our methods are based on the Fourier transform of the considered functions. First we generalize a method for proving classical communication complexity lower bounds developed by Raz [35] to the quantum case. Applying this method we give an exponential separation between bounded error quantum communication complexity and nondeterministic quantum communication complexity. We develop several other lower bound methods based on the Fourier transform, notably showing that√ s̄(f) / logn, for the average sensitivity s̄(f) of a function f, yields a lower bound on the bounded error quantum communication complexity of f((x ∧ y) ⊕ z), where x is a Boolean word held by Alice and y, z are Boolean words held by Bob. We then prove the first large lower bounds on the bounded error quantum communication complexity of functions, for which a polynomial quantum speedup is possible. For all the functions we investigate, the only previously applied general lower bound method based on discrepancy yields bounds that are O(log n).
Algebrization: A new barrier in complexity theory
 MIT Theory of Computing Colloquium
, 2007
"... Any proof of P � = NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not have linearsize circuits) that overcome both barriers simultaneously. So the question arises of whether there is a ..."
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Cited by 53 (3 self)
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Any proof of P � = NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not have linearsize circuits) that overcome both barriers simultaneously. So the question arises of whether there is a third barrier to progress on the central questions in complexity theory. In this paper we present such a barrier, which we call algebraic relativization or algebrization. The idea is that, when we relativize some complexity class inclusion, we should give the simulating machine access not only to an oracle A, but also to a lowdegree extension of A over a finite field or ring. We systematically go through basic results and open problems in complexity theory to delineate the power of the new algebrization barrier. First, we show that all known nonrelativizing results based on arithmetization—both inclusions such as IP = PSPACE and MIP = NEXP, and separations such as MAEXP � ⊂ P/poly —do indeed algebrize. Second, we show that almost all of the major open problems—including P versus NP, P versus RP, and NEXP versus P/poly—will require nonalgebrizing techniques. In some cases algebrization seems to explain exactly why progress stopped where it did: for example, why we have superlinear circuit lower bounds for PromiseMA but not for NP. Our second set of results follows from lower bounds in a new model of algebraic query complexity, which we introduce in this paper and which is interesting in its own right. Some of our lower bounds use direct combinatorial and algebraic arguments, while others stem from a surprising connection between our model and communication complexity. Using this connection, we are also able to give an MAprotocol for the Inner Product function with O ( √ n log n) communication (essentially matching a lower bound of Klauck), as well as a communication complexity conjecture whose truth would imply NL � = NP. 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.
Quantum Communication and Complexity
 Theoretical Computer Science
, 2000
"... In the setting of communication complexity, two distributed parties want to compute a function depending on both their inputs, using as little communication as possible. The required communication can sometimes be significantly lowered if we allow the parties the use of quantum communication. We sur ..."
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Cited by 37 (15 self)
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In the setting of communication complexity, two distributed parties want to compute a function depending on both their inputs, using as little communication as possible. The required communication can sometimes be significantly lowered if we allow the parties the use of quantum communication. We survey the main results of the young area of quantum communication complexity: its relation to teleportation and dense coding, the main examples of fast quantum communication protocols, lower bounds, and some applications. 1 Introduction The area of communication complexity deals with the following type of problem. There are two separated parties, called Alice and Bob. Alice receives some input x 2 X, Bob receives some y 2 Y , and together they want to compute some function f(x; y). As the value f(x; y) will generally depend on both x and y, neither Alice nor Bob will have sufficient information to do the computation by themselves, so they will have to communicate in order to achieve their go...