We study quantum communication protocols, in which the players' storage starts out in a state where one qubit is in a pure state, and all other qubits are totally mixed (i.e. in a random state), and no other storage is available (for messages or internal computations). This restriction on the available quantum memory has been studied extensively in the model of quantum circuits, and it is known that classically simulating quantum circuits operating on such memory is hard when the additive error of the simulation is exponentially small (in the input length), under the assumption that the polynomial hierarchy does not collapse. We study this setting in communication complexity. The goal is to consider larger additive error for simulation-hardness results, and to not use unproven assumptions. We define a complexity measure for this model that takes into account that standard error reduction techniques do not work here. We define a clocked and a semi-unclocked model, and describe efficient simulations between those. We characterize a one-way communication version of the model in terms of weakly unbounded error communication complexity. Our main result is that there is a quantum protocol using one clean qubit only and using O(log n) qubits of communication, such that any classical protocol simulating the acceptance behaviour of the quantum protocol within additive error 1/poly(n) needs communication Ω(n). We also describe a candidate problem, for which an exponential gap between the one-clean-qubit communication complexity and the randomized communication complexity is likely to hold, and hence a classical simulation of the one-clean-qubit model within constant additive error might be hard in communication complexity. We describe a geometrical conjecture that implies the lower bound.

We introduce the communication problem QNDISJ, short for Quantum (Unique) Non-Disjointness, and study its complexity under different modes of communication complexity. The main motivation for the problem is that it is a candidate for the separation of the quantum communication complexity classes QMA and QCMA. The problem generalizes the Vector-in-Subspace and Non-Disjointness problems. We give tight bounds for the QMA, quantum, randomized communication complexities of the problem. We show polynomially related upper and lower bounds for the MA complexity. We also show an upper bound for QCMA protocols, and show that the bound is tight for a natural class of QCMA protocols for the problem. The latter lower bound is based on a geometric lemma, that states that every subset of the n-dimensional sphere of measure 2^-p must contain an ortho-normal set of points of size Omega(n/p). We also study a "small-spaces" version of the problem, and give upper and lower bounds for its randomized complexity that show that the QNDISJ problem is harder than Non-disjointness for randomized protocols. Interestingly, for quantum modes the complexity depends only on the dimension of the smaller space, whereas for classical modes the dimension of the larger space matters.

We study the effect that the amount of correlation in a bipartite distribution has on the communication complexity of a problem under that distribution. We introduce a new family of complexity measures that interpolates between the two previously studied extreme cases: the (standard) randomised communication complexity and the case of distributional complexity under product distributions. - We give a tight characterisation of the randomised complexity of Disjointness under distributions with mutual information k, showing that it is Theta(sqrt(n(k+1))) for all 0 <= k <= n. This smoothly interpolates between the lower bounds of Babai, Frankl and Simon for the product distribution case (k=0), and the bound of Razborov for the randomised case. The upper bounds improve and generalise what was known for product distributions, and imply that any tight bound for Disjointness needs Omega(n) bits of mutual information in the corresponding distribution. - We study the same question in the distributional quantum setting, and show a lower bound of Omega((n(k+1))^{1/4}), and an upper bound (via constructing communication protocols), matching up to a logarithmic factor. - We show that there are total Boolean functions f_d that have distributional communication complexity O(log(n)) under all distributions of information up to o(n), while the (interactive) distributional complexity maximised over all distributions is Theta(log(d)) for n <= d <= 2^{n/100}. This shows, in particular, that the correlation needed to show that a problem is hard can be much larger than the communication complexity of the problem. - We show that in the setting of one-way communication under product distributions, the dependence of communication cost on the allowed error epsilon is multiplicative in log(1/epsilon) - the previous upper bounds had the dependence of more than 1/epsilon. This result, for the first time, explains how one-way communication complexity under product distributions is stronger than PAC-learning: both tasks are characterised by the VC-dimension, but have very different error dependence (learning from examples, it costs more to reduce the error).

Motivated by the increasing need for fast distributed processing of large-scale graphs such as the Web graph and various social networks, we study a number of fundamental graph problems in the message-passing model, where we have k machines that jointly perform computation on an arbitrary n-node (typically, n ≫ k) input graph. The graph is assumed to be randomly partitioned among the k ≥ 2 machines (a common implementation in many real world systems). The communication is point-to-point, and the goal is to minimize the time complexity, i.e., the number of communication rounds, of solving various fundamental graph problems. We present lower bounds that quantify the fundamental time limitations of distributively solving graph problems. We first show a lower bound of Ω(n/k) rounds for computing a spanning tree (ST) of the input graph. This result also implies the same bound for other fundamental problems such as computing a minimum spanning tree (MST), breadth-first tree (BFS), and shortest paths tree (SPT). We also show an Ω(n/k2) lower bound for connectivity, ST verification and other related problems. Our lower bounds develop and use new bounds in random-partition communication complexity. To complement our lower bounds, we also give algorithms for various fundamental graph problems, e.g., PageRank, MST, connectivity, ST verification, shortest paths, cuts, spanners, covering problems, densest subgraph, subgraph isomorphism, finding triangles, etc. We show that problems such as PageRank, MST, connectivity, and graph covering can be solved in Õ(n/k) time (the notation Õ hides polylog(n) factors and an additive polylog(n) term); this shows that one can achieve almost linear (in k) speedup, whereas for shortest paths, we present algorithms that run in time (for (1 + ε)-factor approximation) and in time (for O(log n)-factor approximation) respectively. Our results step towards understanding the complexity of distributively solving large-scale graph problems.

We show new results about the garden-hose model. Our main results include improved lower bounds based on non-deterministic communication complexity (leading to the previously unknown $Θ(n)$ bounds for Inner Product mod 2 and Disjointness), as well as an $O(n\cdot \log^3 n)$ upper bound for the Distributed Majority function (previously conjectured to have quadratic complexity). We show an efficient simulation of formulae made of AND, OR, XOR gates in the garden-hose model, which implies that lower bounds on the garden-hose complexity $GH(f)$ of the order $Ω(n^{2+ε})$ will be hard to obtain for explicit functions. Furthermore we study a time-bounded variant of the model, in which even modest savings in time can lead to exponential lower bounds on the size of garden-hose protocols.

The focus of this paper is on quantum distributed computation, where we investigate whether quantum communication can help in speeding up distributed network algorithms. Our main result is that for certain fundamental network problems such as minimum spanning tree, minimum cut, and shortest paths, quantum communication does not help in substantially speeding up distributed algorithms for these problems compared to the classical setting.

We define a new model of data streaming algorithms that employ a prover/helper to outsource difficult computations in a verifiable way. While for the verifier the usual time (per symbol read) and space constraints of the data streaming model are in place, the prover has unbounded space. Both parties cannot look into the future (i.e., do not know data arriving later). Previous work on such models either severely restricted the total communication between the prover and the verifier, or extended the computation by a long annotation that has to be streamed from the prover to the verifier offline after the original stream has ended, delaying the computation of the result. We argue that restricting the total communication severely is unnatural and investigate a model that only bounds the communication overhead, i.e., the amount of communication sent from the prover to the verifier per symbol of the data stream. This allows for vastly more communication between prover and verifier while maintaining the online nature of the model (in particular long annotations sent after the stream has ended are not allowed). Relaxing the communication requirement allows us to find simple algorithms for problems like the Longest Increasing Subsequence Problem (LIS), finding the Median, and for deciding whether the rank of a matrix is full or not. All our algorithms have a similar structure with phases whose length shrinks geometrically, and phase i being used to verify certain properties of the stream up to phase i-1 using re-streaming of parts of the previous stream. The challenge in each case is to tie the different phases together.

We use the venerable "fooling set" method to prove new lower bounds on the quantum communication complexity of various functions. Let f : X x Y -> {0,1} be a Boolean function, fool^1(f) its maximal fooling set size among 1-inputs, Q_1^*(f) its one-sided-error quantum communication complexity with prior entanglement, and NQ(f) its nondeterministic quantum communication complexity (without prior entanglement; this model is trivial with shared randomness or entanglement). Our main results are the following, where logs are to base 2: - If the maximal fooling set is "upper triangular" (which is for instance the case for the equality, disjointness, and greater-than functions), then we have Q_1^*(f) >= 1/2 log fool^1(f) - 1/2, which (by superdense coding) is essentially optimal for functions like equality, disjointness, and greater-than. No super-constant lower bound for equality seems to follow from earlier techniques. - For all f we have Q_1^*(f) >= 1/4 log fool^1(f) - 1/2. - NQ(f) >= 1/2 log fool^1(f) + 1. We do not know if the factor 1/2 is needed in this result, but it cannot be replaced by 1: we give an example where NQ(f) \approx 0.613 log fool^1(f).

We study the communication complexity of a number of graph properties where the edges of the graph G are distributed between Alice and Bob (i.e., each receives some of the edges as input). Our main results are: 1. An Omega(n) lower bound on the quantum communication complexity of deciding whether an n-vertex graph G is connected, nearly matching the trivial classical upper bound of O(n log n) bits of communication. 2. A deterministic upper bound of O(n^{3/2} log n) bits for deciding if a bipartite graph contains a perfect matching, and a quantum lower bound of Omega(n) for this problem. 3. A Theta(n^2) bound for the randomized communication complexity of deciding if a graph has an Eulerian tour, and a Theta(n^{3/2}) bound for its quantum communication complexity. 4. The first two quantum lower bounds are obtained by exhibiting a reduction from the n-bit Inner Product problem to these graph problems, which solves an open question of Babai, Frankl and Simon [Babai et al 1986]. The third quantum lower bound comes from recent results about the quantum communication complexity of composed functions. We also obtain essentially tight bounds for the quantum communication complexity of a few other problems, such as deciding if $G$ is triangle-free, or if G is bipartite, as well as computing the determinant of a distributed matrix.

We show several results related to interactive proof modes of communication complexity. First we show lower bounds for the QMA-communication complexity of the functions Inner Product and Disjointness. We describe a general method to prove lower bounds for QMA-communication complexity, and show how one can 'transfer' hardness under an analogous measure in the query complexity model to the communication model using Sherstov's pattern matrix method.Combining a result by Vereshchagin and the pattern matrix method we find a partial function with AM-communication complexity O(log n), PP-communication complexity Ω(n1/3), and QMA-communication complexity Ω(n1/6). Hence in the world of communication complexity noninteractive quantum proof systems are not able to efficiently simulate co-nondeterminism or interaction. These results imply that the related questions in Turing machine complexity theory cannot be resolved by 'algebrizing' techniques. Finally we show that in MA-protocols there is an exponential gap between one-way protocols and two-way protocols for a partial function (this refers to the interaction between Alice and Bob). This is in contrast to nondeterministic, AM-, and QMA-protocols, where one-way communication is essentially optimal.

We describe new lower bounds for randomized communication complexity and query complexity which we call the partition bounds. They are expressed as the optimum value of linear programs. For communication complexity we show that the partition bound is stronger than both the rectangle/corruption bound and the γ2/generalized discrepancy bounds. In the model of query complexity we show that the partition bound is stronger than the approximate polynomial degree and classical adversary bounds. We also exhibit an example where the partition bound is quadratically larger than the approximate polynomial degree and adversary bounds.

A strong direct product theorem states 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 of size n can only be exponentially small in k. This solves an open problem of [KSW07, LSS08]. We also show that this bound even holds for $AM$-communication protocols with limited ambiguity. The main result implies a new lower bound for Disjointness in a restricted 3-player NOF protocol, and optimal communication-space 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 so-called Intersection Sampling Lemma that generalizes a result by Razborov [Raz92].

Consider the following simultaneous message passing (SMP) model for computing a relation f sube X times Y times Z. In this model Alice, on input x isin X and Bob, on input y isin Y, send one message each to a third party Referee who then outputs a z isin Z such that (x, y, z) isin f. We first show optimal direct sum results for all relations / in this model, both in the quantum and classical settings, in the situation where we allow shared resources (shared entanglement in quantum protocols and public coins in classical protocols) between Alice and Referee and Bob and Referee and no shared resource between Alice and Bob. This implies that, in this model, the communication required to compute k simultaneous instances of /, with constant success overall, is at least k-times the communication required to compute one instance with constant success. This in particular implies an earlier direct sum result, shown by Chakrabarti, Shi, Wirth and Yao [CSWY01] for the equality function (and a class of other so-called robust functions), in the classical SMP model with no shared resources between any parties. Furthermore we investigate the gap between the SMP model and the one-way model in communication complexity and exhibit a partial function that is exponentially more expensive in the former if quantum communication with entanglement is allowed, compared to the latter even in the deterministic case.

A basic question in complexity theory is whether the computational resources required for solving k independent instances of the same problem scale as k times the resources required for one instance. We investigate this question in various models of classical communication complexity. We introduce a new measure, the subdistribution bound , which is a relaxation of the well-studied rectangle or corruption bound in communication complexity. We nonetheless show that for the communication complexity of Boolean functions with constant error, the subdistribution bound is the same as the latter measure, up to a constant factor. We prove that the one-way version of this bound tightly captures the one-way public-coin randomized communication complexity of any relation, and the two-way version bounds the two-way public-coin randomized communication complexity from below. More importantly, we show that the bound satisfies the strong direct product property under product distributions for both one- and two-way protocols, and the weak direct product property under arbitrary distributions for two-way protocols. These results subsume and strengthen, in a unified manner, several recent results on the direct product question. The simplicity and broad applicability of our technique is perhaps an indication of its potential to solve yet more challenging questions regarding the direct product problem.

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 [Comput. Complexity, 5 (1995), pp. 205–221] 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 $\sqrt{\bar{s}(f)/\log n}$, for the average sensitivity $\bar{s}(f)$ of a function f, yields a lower bound on the bounded error quantum communication complexity of $f((x \wedge y)\oplus z)$, 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 $O(\log n)$.

In this paper the Nečiporuk method for proving lower bounds on the size of Boolean formulas is reformulated in terms of one-way communication complexity. We investigate the settings of probabilistic formulas, nondeterministic formulas, and quantum formulas. In all cases we can use results about one-way communication complexity to prove lower bounds on formula size. The main results regarding formula size are as follows: We show a polynomial size gap between probabilistic/quantum and deterministic formulas, a near-quadratic gap between the sizes of nondeterministic formulas with limited access to nondeterministic bits and nondeterministic formulas with access to slightly more such bits, and a near-quadratic lower bound on quantum formula size. Furthermore we give a polynomial separation between the sizes of quantum formulas with and without multiple read random inputs. The lower bound methods for quantum and probabilistic formulas employ a variant of the Nečiporuk bound in terms of the Vapnik–Chervonenkis dimension. To establish our lower bounds we show optimal separations between one-way and two-way protocols for limited nondeterministic and quantum communication complexity, and we show that zero-error quantum one-way communication complexity asymptotically equals deterministic one-way communication complexity for total functions.

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 imply a time‐space tradeoff $T^2S=\Om{N^3}$ for sorting N items on a quantum computer, which is optimal up to polylog factors. They also give several tight time‐space and communication‐space tradeoffs for the problems of Boolean matrix‐vector multiplication and matrix multiplication.

In some scenarios there are ways of conveying information with many fewer, even exponentially fewer, qubits than possible classically. Moreover, some of these methods have a very simple structure-they involve only few message exchanges between the communicating parties. It is therefore natural to ask whether every classical protocol may be transformed to a "simpler" quantum protocol-one that has similar efficiency, but uses fewer message exchanges. We show that for any constant k, there is a problem such that its k+1 message classical communication complexity is exponentially smaller than its k message quantum communication complexity. This, in particular, proves a round hierarchy theorem for quantum communication complexity, and implies, via a simple reduction, an Omega(N1k/) lower bound for k message quantum protocols for Set Disjointness for constant k. Enroute, we prove information-theoretic lemmas, and define a related measure of correlation, the informational distance, that we believe may be of significance in other contexts as well

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 is 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. These results imply a time-space tradeoff T/sup 2/S = /spl Omega/(N/sup 3/) for sorting N items on a quantum computer, which is optimal up to polylog factors. They also give several tight time-space and communication-space tradeoffs for the problems of Boolean matrix-vector multiplication and matrix multiplication.

We investigate the power of the most important lower bound technique in randomized communication complexity, which is based on an evaluation of the maximal size of approximately monochromatic rectangles, with respect to arbitrary distributions on the inputs. While it is known that the 0-error version of this bound is polynomially tight for deterministic communication, nothing in this direction is known for constant error and randomized communication complexity. We first study a one-sided version of this bound and obtain that its value lies between the MA- and AM- complexities of the considered function. Hence the lower bound actually works for a (communication) complexity class between MA/spl cap/co - MA and AM/spl cap/co - AM, and allows to show that the MA-complexity of the disjointness problem is /spl Omega/(/spl radic/n). Following this we consider the conjecture that the lower bound method is polynomially tight for randomized communication complexity. First we disprove a distributional version of this conjecture. Then we give a combinatorial characterization of the value of the lower bound method, in which the optimization over all distributions is absent. This characterization is done by what we call a bounded error uniform threshold cover, and reduces showing tightness of the bound to the construction of an efficient protocol for a specific communication problem. We then study relaxations of bounded error uniform threshold covers, namely approximate majority covers and majority covers, and exhibit exponential separations between them. Each of these covers captures a lower bound method previously used for randomized communication complexity.

We investigate the complexity of sorting in the model of sequential quantum circuits. While it is known that in general a quantum algorithm based on comparisons alone cannot outperform classical sorting algorithms by more than a constant factor in time complexity, this is wrong in a space bounded setting. We observe that for all storage bounds n/log ≥ S ≥ log3n, one can devise a quantum algorithm that sorts n numbers (using comparisons only) in time T=O(n3/2 log3/2 n/√S). We then show the following lower bound on the time-space tradeoff for sorting n numbers from a polynomial size range in a general sorting algorithm (not necessarily based on comparisons): TS=Ω(n3/2). Hence for small values of S the upper bound is almost tight. Classically the time-space tradeoff for sorting is TS=Θ(n2).

We prove new lower bounds for 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 R. Raz (1995) 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 Fourier based lower bound methods, notably showing that /spl radic/(s~(f)/log n) n, for the average sensitivity s~(f) of a function f, yields a lower bound on the bounded error quantum communication complexity of f (x/spl and/y/spl oplus/yz), 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, only the previously applied general lower bound method based on discrepancy yields bounds that are O(log n).

One of the most intriguing facts about communication using quantum states is that these states cannot be used to transmit more classical bits than the number of qubits used, yet in some scenarios there are ways of conveying information with exponentially fewer qubits than possible classically [3, 26]. Moreover, these methods have a very simple structure---they involve only few message exchanges between the communicating parties.

We investigate the power of quantum communication protocols compared to classical probabilistic protocols.In our first result we describe a total Boolean function that has a quantum Las Vegas protocol communicating at most O(N 1°/11+~) qubits for all e > 0, while any classical probabilistic protocol (with bounded error) needs ~(N/log N) bits.Then we investigate quantum one-way communication complexity.First we show that the VC-dimension lower bound on one-way probabilistic communication of [26] holds for quantum protocols, too.Then we prove that for oneway protocols computing total functions quantum Las Vegas communication is asymptotically as efficient as exact quantum communication, which is exactly as efficient as deterministic communication.We describe applications of the lower bounds for one-way communication complexity to quantum finite automata and quantum formulae.

We investigate the effect of limiting the number of available nondeterministic bits in different computational models. First we relate formula size to one-way communication complexity and derive lower bounds of /spl Omega/R(n/sup 2-/spl epsiv///log/sup 1-/spl epsiv//n) on the size of formulae with n/sup /spl epsiv///log/sup /spl epsiv//n, nondeterministic bits for 0