Results 1  10
of
30
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 104 (1 self)
 Add to MetaCart
(Show Context)
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 search of spatial regions
 THEORY OF COMPUTING
, 2005
"... Can Grover’s algorithm speed up search of a physical region—for example a 2D grid of size √ n × √ n? The problem is that √ n time seems to be needed for each query, just to move amplitude across the grid. Here we show that this problem can be surmounted, refuting a claim to the contrary by Beniof ..."
Abstract

Cited by 85 (8 self)
 Add to MetaCart
Can Grover’s algorithm speed up search of a physical region—for example a 2D grid of size √ n × √ n? The problem is that √ n time seems to be needed for each query, just to move amplitude across the grid. Here we show that this problem can be surmounted, refuting a claim to the contrary by Benioff. In particular, we show how to search a ddimensional hypercube in time O ( √ n) for d ≥ 3, or O ( √ nlog 5/2 n) for d = 2. More generally, we introduce a model of quantum query complexity on graphs, motivated by fundamental physical limits on information storage, particularly the holographic principle from black hole thermodynamics. Our results in this model include almosttight upper and lower bounds for many search tasks; a generalized algorithm that works for any graph with good expansion properties, not just hypercubes; and relationships among several notions of ‘locality’ for unitary matrices acting on graphs. As an application of our results, we give an O (√ n)qubit communication protocol for the disjointness problem, which improves an upper bound of Høyer and de Wolf and matches a lower bound of Razborov.
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 66 (12 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 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 ..."
Abstract

Cited by 54 (7 self)
 Add to MetaCart
(Show Context)
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).
Quantum search on boundederror inputs
 In Proc. of 30th ICALP
, 2003
"... Abstract. Suppose we have n algorithms, quantum or classical, each computing some bitvalue with bounded error probability. We describe a quantum algorithm that uses O ( √ n) repetitions of the base algorithms and with high probability finds the index of a 1bit among these n bits (if there is such ..."
Abstract

Cited by 53 (5 self)
 Add to MetaCart
(Show Context)
Abstract. Suppose we have n algorithms, quantum or classical, each computing some bitvalue with bounded error probability. We describe a quantum algorithm that uses O ( √ n) repetitions of the base algorithms and with high probability finds the index of a 1bit among these n bits (if there is such an index). This shows that it is not necessary to first significantly reduce the error probability in the base algorithms to O(1/poly(n)) (which would require O ( √ nlog n) repetitions in total). Our technique is a recursive interleaving of amplitude amplification and errorreduction, and may be of more general interest. Essentially, it shows that quantum amplitude amplification can be made to work also with a boundederror verifier. As a corollary we obtain optimal quantum upper bounds of O ( √ N) queries for all constantdepth ANDOR trees on N variables, improving upon earlier upper bounds of O ( √ Npolylog(N)). 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 45 (9 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 39 (3 self)
 Add to MetaCart
(Show Context)
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 complexity of blockcomposed functions. Available at arXiv:0710.0095v1
, 2007
"... A major open problem in communication complexity is whether or not quantum protocols can be exponentially more efficient than classical ones for computing a total Boolean function in the twoparty interactive model. The answer appears to be “No”. In 2002, Razborov proved this conjecture for so far th ..."
Abstract

Cited by 33 (1 self)
 Add to MetaCart
A major open problem in communication complexity is whether or not quantum protocols can be exponentially more efficient than classical ones for computing a total Boolean function in the twoparty interactive model. The answer appears to be “No”. In 2002, Razborov proved this conjecture for so far the most general class of functions F (x, y) = fn(x1 · y1, x2 · y2,..., xn · yn), where fn is a symmetric Boolean function on n Boolean inputs, and xi, yi are the i’th bit of x and y, respectively. His elegant proof critically depends on the symmetry of fn. We develop a lowerbound method that does not require symmetry and prove the conjecture for a broader class of functions. Each of those functions F (x, y) is the “blockcomposition ” of a “building block ” gk: {0, 1} k × {0, 1} k → {0, 1}, and an fn: {0, 1} n → {0, 1}, such that F (x, y) = fn(gk(x1, y1), gk(x2, y2),..., gk(xn, yn)), where xi and yi are the i’th kbit block of x, y ∈ {0, 1} nk, respectively. We show that as long as gk itself is “hard ” enough, its blockcomposition with an arbitrary fn has polynomially related quantum and classical communication complexities. Our approach gives an alternative proof for Razborov’s result (albeit with a slightly weaker parameter), and establishes new quantum lower bounds. For example, when gk is the Inner Product function with k = Ω(log n), the deterministic communication complexity of its blockcomposition with any fn is asymptotically at most the quantum complexity to the power of 7.
THE PATTERN MATRIX METHOD
, 2009
"... We develop a novel and powerful technique for communication lower bounds, the pattern matrix method. Specifically, fix an arbitrary function f: {0, 1} n → {0, 1} and let Af be the matrix whose columns are each an application of f to some subset of the variables x1, x2,..., x4n. We prove that Af ha ..."
Abstract

Cited by 29 (8 self)
 Add to MetaCart
(Show Context)
We develop a novel and powerful technique for communication lower bounds, the pattern matrix method. Specifically, fix an arbitrary function f: {0, 1} n → {0, 1} and let Af be the matrix whose columns are each an application of f to some subset of the variables x1, x2,..., x4n. We prove that Af has boundederror communication complexity Ω(d), where d is the approximate degree of f. This result remains valid in the quantum model, regardless of prior entanglement. In particular, it gives a new and simple proof of Razborov’s breakthrough quantum lower bounds for disjointness and other symmetric predicates. We further characterize the discrepancy, approximate rank, and approximate trace norm of Af in terms of wellstudied analytic properties of f, broadly generalizing several recent results on smallbias communication and agnostic learning. The method of this paper has recently enabled important progress in multiparty communication complexity.
Quantum communication complexity
 Foundations of Physics
"... Can quantum communication be more efficient than its classical counterpart? Holevo’s theorem rules out the possibility of communicating more than n bits of classical information by the transmission of n quantum bits—unless the two parties are entangled, in which case twice as many classical bits can ..."
Abstract

Cited by 19 (6 self)
 Add to MetaCart
(Show Context)
Can quantum communication be more efficient than its classical counterpart? Holevo’s theorem rules out the possibility of communicating more than n bits of classical information by the transmission of n quantum bits—unless the two parties are entangled, in which case twice as many classical bits can be communicated but no more. In apparent contradiction, there are distributed computational tasks for which quantum communication cannot be simulated efficiently by classical means. In some cases, the effect of transmitting quantum bits cannot be achieved classically short of transmitting an exponentially larger number of bits. In a similar vein, can entanglement be used to save on classical communication? It is well known that entanglement on its own is useless for the transmission of information. Yet, there are distributed tasks that cannot be accomplished at all in a classical world when communication is not allowed, but that become possible if the noncommunicating parties share prior entanglement. This leads to the question of how expensive it is, in terms of classical communication, to provide an exact simulation of the spooky power of entanglement. KEY WORDS: Bell’s theorem; communication complexity; distributed computation; entanglement simulation; pseudotelepathy; spooky communication.