We study the communication complexity of computing functions F: {0,1}ⁿ × {0,1}ⁿ → {0,1} in the memoryless communication model. Here, Alice is given x ∈ {0,1}ⁿ, Bob is given y ∈ {0,1}ⁿ and their goal is to compute F(x,y) subject to the following constraint: at every round, Alice receives a message from Bob and her reply to Bob solely depends on the message received and her input x (in particular, her reply is independent of the information from the previous rounds); the same applies to Bob. The cost of computing F in this model is the maximum number of bits exchanged in any round between Alice and Bob (on the worst case input x,y). In this paper, we also consider variants of our memoryless model wherein one party is allowed to have memory, the parties are allowed to communicate quantum bits, only one player is allowed to send messages. We show that some of these different variants of our memoryless communication model capture the garden-hose model of computation by Buhrman et al. (ITCS'13), space-bounded communication complexity by Brody et al. (ITCS'13) and the overlay communication complexity by Papakonstantinou et al. (CCC'14). Thus the memoryless communication complexity model provides a unified framework to study all these space-bounded communication complexity models. We establish the following main results: (1) We show that the memoryless communication complexity of F equals the logarithm of the size of the smallest bipartite branching program computing F (up to a factor 2); (2) We show that memoryless communication complexity equals garden-hose model of computation; (3) We exhibit various exponential separations between these memoryless communication models. We end with an intriguing open question: can we find an explicit function F and universal constant c > 1 for which the memoryless communication complexity is at least c log n? Note that c ≥ 2+ε would imply a Ω(n^{2+ε}) lower bound for general formula size, improving upon the best lower bound by [Nečiporuk, 1966].
We study the communication complexity of computing functions F: {0,1}ⁿ × {0,1}ⁿ → {0,1} in the memoryless communication model. Here, Alice is given x ∈ {0,1}ⁿ, Bob is given y ∈ {0,1}ⁿ and their goal is to compute F(x,y) subject to the following constraint: at every round, Alice receives a message from Bob and her reply to Bob solely depends on the message received and her input x (in particular, her reply is independent of the information from the previous rounds); the same applies to Bob. The cost of computing F in this model is the maximum number of bits exchanged in any round between Alice and Bob (on the worst case input x,y). In this paper, we also consider variants of our memoryless model wherein one party is allowed to have memory, the parties are allowed to communicate quantum bits, only one player is allowed to send messages. We show that some of these different variants of our memoryless communication model capture the garden-hose model of computation by Buhrman et al. (ITCS'13), space-bounded communication complexity by Brody et al. (ITCS'13) and the overlay communication complexity by Papakonstantinou et al. (CCC'14). Thus the memoryless communication complexity model provides a unified framework to study all these space-bounded communication complexity models. We establish the following main results: (1) We show that the memoryless communication complexity of F equals the logarithm of the size of the smallest bipartite branching program computing F (up to a factor 2); (2) We show that memoryless communication complexity equals garden-hose model of computation; (3) We exhibit various exponential separations between these memoryless communication models. We end with an intriguing open question: can we find an explicit function F and universal constant c > 1 for which the memoryless communication complexity is at least c log n? Note that c ≥ 2+ε would imply a Ω(n^{2+ε}) lower bound for general formula size, improving upon the best lower bound by [Nečiporuk, 1966].
We study the communication complexity of computing functions F:{0,1}n×{0,1}n→{0,1} in the memoryless communication model. Here, Alice is given x∈{0,1}n, Bob is given y∈{0,1}n and their goal is to compute F(x,y) subject to the following constraint: at every round, Alice receives a message from Bob and her reply to Bob solely depends on the message received and her input x; the same applies to Bob. The cost of computing F in this model is the maximum number of bits exchanged in any round between Alice and Bob (on the worst case input x,y). In this paper, we also consider variants of our memoryless model wherein one party is allowed to have memory, the parties are allowed to communicate quantum bits, only one player is allowed to send messages. We show that our memoryless communication model capture the garden-hose model of computation by Buhrman et al. (ITCS'13), space bounded communication complexity by Brody et al. (ITCS'13) and the overlay communication complexity by Papakonstantinou et al. (CCC'14). Thus the memoryless communication complexity model provides a unified framework to study space-bounded communication models. We establish the following: (1) We show that the memoryless communication complexity of F equals the logarithm of the size of the smallest bipartite branching program computing F (up to a factor 2); (2) We show that memoryless communication complexity equals garden-hose complexity; (3) We exhibit various exponential separations between these memoryless communication models. We end with an intriguing open question: can we find an explicit function F and universal constant c>1 for which the memoryless communication complexity is at least clogn? Note that c≥2+ε would imply a Ω(n2+ε) lower bound for general formula size, improving upon the best lower bound by Nečiporuk in 1966.
Communication complexity is the amount of communication needed to compute a function when the function inputs are distributed over multiple parties. In its simplest form, one-way communication complexity, Alice and Bob compute a function f(x,y) , where x is given to Alice and y is given to Bob, and only one message from Alice to Bob is allowed. A fundamental question in quantum information is the relationship between one-way quantum and classical communication complexities, i.e., how much shorter the message can be if Alice is sending a quantum state instead of bit strings? We make some progress towards this question with the following results.Let f:X×Y→Z∪{⊥} be a partial function and μ be a distribution with support contained in f−1(Z) . Denote d=|Z| . Let Rϵ1,μ(f) be the classical one-way communication complexity of f ; Qϵ1,μ(f) be the quantum one-way communication complexity of f and Qϵ1,μ,∗(f) be the entanglement-assisted quantum one-way communication complexity of f , each with distributional error (average error over μ ) at most ϵ . We show:1) If μ is a product distribution, η>0 and 0≤ϵ≤1−1/d , then, R2ϵ−dϵ2/(d−1)+η1,μ(f)≤2Qϵ1,μ,∗(f)+O(loglog(1/η)). 2)If μ is a non-product distribution and Z={0,1} , then ∀ϵ,η>0 such that ϵ/η+η<0.5 , R3η1,μ(f)=O(Qϵ1,μ(f)⋅CS(f)/η3), where CS(f)=maxyminz&am
Direct-sum questions in (two-party) communication complexity ask whether two parties, Alice and Bob, can compute the value of a function f on l inputs (x_1,y_1),...,(x_l,y_l) more efficiently than by applying the best protocol for f, independently on each input (x_i,y_i). In spite of significant efforts to understand these questions (under various communication-complexity measures), the general question is still far from being well understood. In this paper, we offer a multiparty view of these questions: The direct-sum setting is just a two-player system with Alice having inputs x_1,...,x_l, Bob having inputs y_1,...,y_l and the desired output is f(x_1,y_1),...,f(x_l,y_l). The naive solution of solving the l problems independently, is modeled by a network with l (disconnected) pairs of players Alice i and Bob i, with inputs x_i,y_i respectively, and communication only within each pair. Then, we consider an intermediate ("star") model, where there is one Alice having l inputs x_1,...,x_l and l players Bob_1,...,Bob_l holding y_1,...,y_l, respectively (in fact, we consider few variants of this intermediate model, depending on whether communication between each Bob i and Alice is point-to-point or whether we allow broadcast). Our goal is to get a better understanding of the relation between the two extreme models (i.e., of the two-party direct-sum question). If, for instance, Alice and Bob can do better (for some complexity measure) than solving the l problems independently, we wish to understand what intermediate model already allows to do so (hereby understanding the "source" of such savings). If, on the other hand, we wish to prove that there is no better solution than solving the l problems independently, then our approach gives a way of breaking the task of proving such a statement into few (hopefully, easier) steps. We present several results of both types. Namely, for certain complexity measures, communication problems f and certain pairs of models, we can show gaps between the complexity of solving f on l instances in the two models in question; while, for certain other complexity measures and pairs of models, we can show that such gaps do not exist (for any communication problem f). For example, we prove that if only point-to-point communication is allowed in the intermediate "star" model, then significant savings are impossible in the public-coin randomized setting. On the other hand, in the private-coin randomized setting, if Alice is allowed to broadcast messages to all Bobs in the "star" network, then some savings are possible. While this approach does not lead yet to new results on the original two-party direct-sum question, we believe that our work gives new insights on the already-known direct-sum results, and may potentially lead to more such results in the future.
We consider memoryless quantum communication protocols, where the two parties do not possess any memory besides their classical input and they take turns performing unitary operations on a pure quantum state that they exchange between them. Most known quantum protocols are of this type and recently a deep connection between memoryless protocols and Bell inequality violations has been explored recently by Buhrman et al. We study the information cost of memoryless quantum protocols by looking at a canonical problem: bounded-round quantum communication protocols for the one-bit AND function. We prove directly a tight lower bound of Θ(log(k)/k) for the information cost of AND for k-round memoryless quantum protocols and for the input distribution needed for the Disjointness function. It is not clear if memoryless protocols allow for a reduction between AND and Disjointness, due to the absence of private workspaces. We enhance the model by allowing the players to keep in their private classical workspace apart from their input also some classical private coins. Surprisingly, we show that every quantum protocol can be transformed into an equivalent quantum protocol with private coins that is perfectly private, i.e. the players only learn the value of the function and nothing more. Last, we consider the model where the players are allowed to use one-shot coins, i.e. private coins that can be used only once during the protocol. While in the classical case, private coins and one-shot coins are equivalent, in the quantum case, we prove that they are not. More precisely, we show that every quantum memoryless protocol with one-bit inputs that uses one-shot coins can be transformed into a memoryless quantum protocol without private coins and without increasing too much its information cost. Hence, while private coins always allow for private quantum protocols, one-shot coins do not.
In this work we introduce an intermediate setting between quantum nonlocality and communication complexity problems. More precisely, we study the value of XOR games G when Alice and Bob are allowed to use a limited amount of one-way classical communication ωo.w.−c(G) (resp. one-way quantum communication ωo.w.−c∗(G)), where c denotes the number of bits (resp. qubits). The key quantity here is the quotient ωo.w.−c∗(G)/ωo.w.−c(G). We provide a universal way to obtain Bell inequality violations of general Bell functionals from XOR games for which the quotient ωo.w.−c∗(G)/ωo.w.−2c(G) is larger than 1. This allows, in particular, to find (unbounded) Bell inequality violations from communication complexity problems in the same spirit as the recent work by Buhrman et al. (2016). We also provide an example of a XOR game for which the previous quotient is optimal (up to a logarithmic factor) in terms of the amount of information c. Interestingly, this game has only polynomially many inputs per player. For the related problem of separating the classical vs quantum communication complexity of a function, the known examples attaining exponential separation require exponentially many inputs per party.
After Bob sends Alice a bit, she responds with a lengthy reply. At the cost of a factor of two in the total communication, Alice could just as well have given Bob her two possible replies at once without listening to him at all, and have him select which one applies. Motivated by a conjecture stating that this form of “round elimination” is impossible in exact quantum communication complexity, we study the orthogonal rank and a symmetric variant thereof for a certain family of Cayley graphs. The orthogonal rank of a graph is the smallest number d for which one can label each vertex with a nonzero d-dimensional complex vector such that adjacent vertices receive orthogonal vectors. We show an exp(n) lower bound on the orthogonal rank of the graph on {0, 1} n in which two strings are adjacent if they have Hamming distance at least n/2. In combination with previous work, this implies an affirmative answer to the above conjecture.