Results 1  10
of
54
Exponential lower bound for 2query locally decodable codes via a quantum argument
 Journal of Computer and System Sciences
, 2003
"... Abstract A locally decodable code encodes nbit strings x in mbit codewords C(x) in such a way that one can recover any bit xi from a corrupted codeword by querying only a few bits of that word. We use a quantum argument to prove that LDCs with 2 classical queries require exponential length: m = 2 ..."
Abstract

Cited by 123 (18 self)
 Add to MetaCart
Abstract A locally decodable code encodes nbit strings x in mbit codewords C(x) in such a way that one can recover any bit xi from a corrupted codeword by querying only a few bits of that word. We use a quantum argument to prove that LDCs with 2 classical queries require exponential length: m = 2 \Omega (n). Previously this was known only for linear codes (Goldreich et al. 02). The
Quantum Communication Complexity of Symmetric Predicates
 Izvestiya of the Russian Academy of Science, Mathematics
, 2002
"... We completely (that is, up to a logarithmic factor) characterize the boundederror quantum communication complexity of every predicate f(x; y) (x; y [n]) depending only on jx\yj. Namely, for a predicate D on f0; 1; : : : ; ng let ` 0 (D) = max f` j 1 ` n=2 ^ D(`) 6 D(` 1)g and ` 1 (D) = ..."
Abstract

Cited by 90 (2 self)
 Add to MetaCart
We completely (that is, up to a logarithmic factor) characterize the boundederror quantum communication complexity of every predicate f(x; y) (x; y [n]) depending only on jx\yj. Namely, for a predicate D on f0; 1; : : : ; ng let ` 0 (D) = max f` j 1 ` n=2 ^ D(`) 6 D(` 1)g and ` 1 (D) = max fn ` j n=2 ` < n ^ D(`) 6 D(` + 1)g. Then the boundederror quantum communication complexity of f D (x; y) = D(jx \ yj) is equal (again, up to a logarithmic factor) to ` 1 (D). In particular, the complexity of the set disjointness predicate is n). This result holds both in the model with prior entanglement and without it.
Quantum entanglement and the communication complexity of the inner product function
 IN PROCEEDINGS OF 1ST NASA QCQC CONFERENCE, VOLUME 1509 OF LECTURE NOTES IN COMPUTER SCIENCE
, 1998
"... We consider the communication complexity of the binary inner product function in a variation of the twoparty scenario where the parties have an apriori supply of particles in an entangled quantum state. We prove linear lower bounds for both exact protocols, as well as for protocols that determine ..."
Abstract

Cited by 89 (10 self)
 Add to MetaCart
We consider the communication complexity of the binary inner product function in a variation of the twoparty scenario where the parties have an apriori supply of particles in an entangled quantum state. We prove linear lower bounds for both exact protocols, as well as for protocols that determine the answer with boundederror probability. Our proofs employ a novel kind of “quantum” reduction from a quantum information theory problem to the problem of computing the inner product. The communication required for the former problem can then be bounded by an application of Holevo’s theorem. We also give a specific example of a probabilistic scenario where entanglement reduces the communication complexity of the inner product function by one bit.
Optimal lower bounds for quantum automata and random access codes
"... Consider the finite regular ¢¤£¦¥¨§�©�����©�� language ©������� �. In [3] it was shown that while this language is accepted by a deterministic finite automaton of ������ � size, any oneway quantum finite automaton (QFA) for it has ���¤ � £��� � ����£� � size. This was based on the fact that the e ..."
Abstract

Cited by 86 (8 self)
 Add to MetaCart
Consider the finite regular ¢¤£¦¥¨§�©�����©�� language ©������� �. In [3] it was shown that while this language is accepted by a deterministic finite automaton of ������ � size, any oneway quantum finite automaton (QFA) for it has ���¤ � £��� � ����£� � size. This was based on the fact that the evolution of a QFA is required to be reversible. When arbitrary intermediate measurements are allowed, this intuition breaks down. Nonetheless, we show ���� � £�� a lower bound for such QFA ¢ £ for, thus also improving the previous bound. The improved bound is obtained from simple entropy arguments based on Holevo’s theorem [8]. This method also allows us to obtain an asymptotically op���������������� � timal bound for the dense quantum codes (random access codes) introduced in [3]. We then turn to Holevo’s theorem, and show that in typical situations, it may be replaced by a tighter and more transparent inprobability bound.
Communication complexity lower bounds by polynomials
 In Proc. of the 16th Conf. on Computational Complexity (CCC
, 2001
"... The quantum version of communication complexity allows Alice and Bob to communicate qubits and/or to make use of prior entanglement (shared EPRpairs). Some lower bound techniques are available for qubit communication [17, 11, 2], but except for the inner product function [11], no bounds are known f ..."
Abstract

Cited by 61 (12 self)
 Add to MetaCart
The quantum version of communication complexity allows Alice and Bob to communicate qubits and/or to make use of prior entanglement (shared EPRpairs). Some lower bound techniques are available for qubit communication [17, 11, 2], but except for the inner product function [11], no bounds are known for the model with unlimited prior entanglement. We show that the “log rank ” lower bound extends to the strongest model (qubit communication + prior entanglement). By relating the rank of the communication matrix to properties of polynomials, we are able to derive some strong bounds for exact protocols. In particular, we prove both the “logrank conjecture ” and the polynomial equivalence of quantum and classical communication complexity for various classes of functions. We also derive some weaker bounds for boundederror protocols. 1
The Quantum Communication Complexity of Sampling
 SIAM J. Comput
, 1998
"... Sampling is an important primitive in probabilistic and quantum algorithms. In the spirit of communication complexity, given a function f : X 1} and a probability distribution over X Y , we define the sampling complexity of (f, as the minimum number of bits Alice and Bob must communica ..."
Abstract

Cited by 54 (3 self)
 Add to MetaCart
Sampling is an important primitive in probabilistic and quantum algorithms. In the spirit of communication complexity, given a function f : X 1} and a probability distribution over X Y , we define the sampling complexity of (f, as the minimum number of bits Alice and Bob must communicate for Alice to pick x X and Bob to pick y Y as well as a value z such that the resulting distribution of (x, y, z) is close to the distribution (D, f(D)).
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 ..."
Abstract

Cited by 44 (7 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 40 (5 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 39 (12 self)
 Add to MetaCart
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
Exponential separation of quantum and classical oneway communication complexity
 SIAM J. Comput
"... Abstract. We give the first exponential separation between quantum and boundederror randomized oneway communication complexity. Specifically, we define the Hidden Matching Problem HMn: Alice gets as input a string x ∈ {0, 1} n and Bob gets a perfect matching M on the n coordinates. Bob’s goal is t ..."
Abstract

Cited by 35 (2 self)
 Add to MetaCart
Abstract. We give the first exponential separation between quantum and boundederror randomized oneway communication complexity. Specifically, we define the Hidden Matching Problem HMn: Alice gets as input a string x ∈ {0, 1} n and Bob gets a perfect matching M on the n coordinates. Bob’s goal is to output a tuple 〈i, j, b 〉 such that the edge (i, j) belongs to the matching M and b = xi ⊕ xj. We prove that the quantum oneway communication complexity of HMn is O(log n), yet any randomized oneway protocol with bounded error must use Ω ( √ n) bits of communication. No asymptotic gap for oneway communication was previously known. Our bounds also hold in the model of Simultaneous Messages (SM) and hence we provide the first exponential separation between quantum SM and randomized SM with public coins. For a Boolean decision version of HMn, we show that the quantum oneway communication complexity remains O(log n) and that the 0error randomized oneway communication complexity is Ω(n). We prove that any randomized linear oneway protocol with bounded error for this problem requires Ω ( 3 √ n log n) bits of communication. Key words. Communication complexity, quantum computation, separation, hidden matching AMS subject classifications. 68P30,68Q15,68Q17,81P68 1. Introduction. The