Communication complexity has become a central completity model.In that model, we count the amount of communication bits needed between two parties in order to solve certain computational problems.We show that for certain communication complezity problems quantum communication protocols are exponentially faster than classical ores.More explicitly, we give an example for a communication complexity relation (0~ promise problem) P such that: 1) The quantum communication complexity of P is O(log m). 2) The classical probabilistic communication complexity of P is Q(m'l'/ logm).(where m is the length of the inputs).This gives an ezponential gap between quantum communication complexity and classical probabilistic communication complexity.Only a quadratic gap was previously known.Our problem P is of geometrical nature, and is a finite precision variation of the following problem: Player I gets as input a unit vector z E R" and two orthogonal subspaces MO, MI c R". Player II gets as input an orthogonal matrixT : R" + R".Their goal is to answer 0 if T(x) E MO and 1 if T(x) E M,, (and any an~luer in any other case).We give an almost tight analysis for the quantum communication complexity and for the classical-probabilistic communication complexity of this problem.
Communication complexity has become a central completity model.In that model, we count the amount of communication bits needed between two parties in order to solve certain computational problems.We show that for certain communication complezity problems quantum communication protocols are exponentially faster than classical ores.More explicitly, we give an example for a communication complexity relation (0~ promise problem) P such that: 1) The quantum communication complexity of P is O(log m). 2) The classical probabilistic communication complexity of P is Q(m'l'/ logm).(where m is the length of the inputs).This gives an ezponential gap between quantum communication complexity and classical probabilistic communication complexity.Only a quadratic gap was previously known.Our problem P is of geometrical nature, and is a finite precision variation of the following problem: Player I gets as input a unit vector z E R" and two orthogonal subspaces MO, MI c R". Player II gets as input an orthogonal matrixT : R" + R".Their goal is to answer 0 if T(x) E MO and 1 if T(x) E M,, (and any an~luer in any other case).We give an almost tight analysis for the quantum communication complexity and for the classical-probabilistic communication complexity of this problem.
We study the power of quantum proofs, or more precisely, the power of Quantum Merlin-Arthur (QMA) protocols, in two well studied models of quantum computation: the black box model and the communication complexity model. Our main results are obtained for the communication complexity model. For this model, we identify a complete promise problem for QMA protocols, the Linear Subspaces Distance problem. The problem is of geometrical nature: Each player gets a linear subspace of R m and considers the sphere of unit vectors in that subspace. Their goal is to output 1 if the distance between the two spheres is very small (say, smaller than 0.1 · √ 2) and 0 if the distance is very large (say, larger than 0.9 · √ 2). We show that: 1. The QMA communication complexity of the problem is O(log m). 2. The (classical) MA communication complexity of the problem is Ω(m ɛ) (for some ɛ> 0). 3. The (standard) quantum communication complexity of the problem is Ω ( √ m). In particular, this gives an exponential separation between QMA communication complexity and MA communication complexity.
We give the first exponential separation between quantum and bounded-error randomized one-way 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 one-way communication complexity of HMn is O(log n), yet any randomized one-way protocol with bounded error must use Ω(√n) bits of communication. No asymptotic gap for one-way 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 one-way communication complexity remains O(log n) and that the 0-error randomized one-way communication complexity is Ω(n). We prove that any randomized linear one-way protocol with bounded error for this problem requires Ω(√[3] n log n) bits of communication.
The main objective of quantum computation is to exploit the natural parallelism of quantum mechanics to solve problems using less computational resources than classical computers. Although quantum algorithms realizing an exponential time speed-up over the best known classical algorithms exist, no quantum algorithm is known performing computation using less space resources than classical algorithms. In this paper, we study, for the first time explicitly, spacebounded quantum algorithms for computational problems where the input is given not as a whole, but bit by bit. We show that there exist such problems that a quantum computer can solve using exponentiallyless work space than a classical computer. More precisely, we introduce a very natural and simple model of a space-bounded quantum online machine and prove an exponential separation of classical and quantum online space complexity, in the bounded-error setting and for a total language. The language we consider is inspired bya communication problem that Buhrman, Cleve and Wigderson used to show an almost quadratic separation of quantum and classical bounded-error communication complexity. We prove that, in the framework of online space complexity, the separation becomes exponential.
Abstract. We give the first exponential separation between quantum and bounded-error randomized one-way 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 one-way communication complexity of HMn is O(logn), yet any randomized one-way protocol with bounded error must use Ω ( √ n) bits of communication. No asymptotic gap for one-way 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 one-way communication complexity remains O(logn) and that the 0-error randomized one-way communication complexity is Ω(n). We prove that any randomized linear one-way protocol with bounded error for this problem requires Ω ( 3 √ nlog n) bits of communication.
Equality and disjointness are two of the most studied problems in communication complexity. They have been studied for both classical and also quantum communication and for various models and modes of communication. Buhrman et al. [Buh98] proved that the exact quantum communication complexity for a promise version of the equality problem is O(logn) while the classical deterministic communication complexity is n+1 for two-way communication, which was the first impressively large (exponential) gap between quantum and classical (deterministic and probabilistic) communication complexity. If an error is tolerated, both quantum and probabilistic communication complexities for equality are O(logn). However, even if an error is tolerated, the gaps between quantum (probabilistic) and deterministic complexity are not larger than quadratic for the disjointness problem. It is therefore interesting to ask whether there are some promise versions of the disjointness problem for which bigger gaps can be shown. We give a positive answer to such a question. Namely, we prove that there exists an exponential gap between quantum (even probabilistic) communication complexity and classical deterministic communication complexity of some specific versions of the disjointness problem. Klauck [Kla00] proved, for any language, that the state complexity of exact quantum/classical finite automata, which is a general model of one-way quantum finite automata, is not less than the state complexity of an equivalent one-way deterministic finite automata (1DFA). In this paper we show, using a communication complexity result, that situation may be different for some promise problems. Namely, we show for certain promise problem that the gap between the state complexity of exact one-way quantum finite automata and 1DFA can be exponential.
We give the first exponential separation between quantum and bounded-error randomized one-way 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 one-way communication complexity of HMn is O(logn), yet any randomized one-way protocol with bounded error must use Ω(n) bits of communication. No asymptotic gap for one-way 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 one-way communication complexity remains O(logn) and that the 0-error randomized one-way communication complexity is Ω(n). We prove that any randomized linear one-way protocol with bounded error for this problem requires Ω(3nlogn) bits of communication.